Topological Quantum Field Theories and Homotopy Cobordisms

The motivating aim of this work is to describe certain condensed matter systems known as topological phases. This in turn has applications to quantum computation: topological quantum computing (TQC) is a proposed framework for carrying out computation by controlling the movement of emergent particles in topological phases [38, 44, 51, 52]. A topological phase may be described by assigning a space of states to each possible particle configuration, and assigning a linear operator to each allowed particle trajectory. Several categories are potentially of interest for the modelling of particle configurations and trajectories in topological phases: motion groupoids [32, 41], tangle categories ([27] for example), generalised tangle categories [4], defect cobordism categories [10], and embedded cobordism catgeories [40, 53] for example. The objective then is to find representations of these categories, in other words functors which eventually map to \({\textbf{Vect}}_{{\mathbb {C}}}\).

Here we aim to construct representations of particle trajectories which are invariant up to a notion of homotopy equivalence of the complement of the particle trajectory. Note that, in general, complements of particle trajectories in the aforementioned categories will not be compact manifolds. Moreover, even if we can compactify these spaces, if fusion is allowed, these complements will generally not be morphisms in any manifold cobordism category. Think, for example, of the complement of the ‘cap’ morphism in the tangle category ([27]). Field theories arising from homotopical properties of manifolds are common in the literature. The topological quantum field theories (TQFTs) of [28, 54] and an untwisted version of [17] can all be shown to assign to a manifold \(\Sigma \) the vector space with basis natural transformation classes of maps \(\pi (\Sigma )\rightarrow G\). In fact, in \((3+1)\)-dimensional it appears that all bosonic topological phases where all quasiparticles are bosons, are described by Dijkgraaf–Witten theories [51]. Each of these examples is generalised by [42], which gives a class of TQFTs constructed using the ‘homotopy content’. Further, the approach of looking at the homotopy of the complement is taken in certain invariants of knots [11], Artin’s representation of braids [7] and its lift to loop braids [15], for example.

Such a functor into \(\textbf{Vect}_{\mathbb {C}}\) may factor through other categories. In some cases, these categories will be more convenient to work with. Our first main result (Theorem 3.36) is the construction of a symmetric, monoidal category \({\textrm{HomCob}}\) whose objects are topological spaces and whose morphisms are equivalence classes of cofibrant cospans. Roughly, functors into the category \({\textrm{HomCob}}\), from the aforementioned categories modelling topological phases, may be constructed by taking the complement of particle trajectories. We prove that there exist functors into \({\textrm{HomCob}}\) from the motion groupoid of a manifold, as constructed in [32], as well as from \(\textbf{Cob}_{n}\), the category of cobordisms as given in [30]. The approach of constructing representations by going via categories of cospans has its origins in the work of Grandis, and Morton [20, 35]; and these can each be seen as an example of the groupoidification explained in [5].

In the second half of the paper, we construct a family of functors from \({\textrm{HomCob}}\) into \(\textbf{Vect}_{\mathbb {C}}\) based on a finite group G. Our construction follows the approach of [54]. This allows us to give an interpretation of our functor in terms of a choice of a finite set of basepoints in each of the spaces involved, and hence a method for explicit calculation. We expect it to be possible to vastly generalise the functors constructed here, to functors which take a finite n-type as input. For manifolds, the case of homotopy 2-types is considered in [55], and the case of general n-types in [42]. However, it will likely only be possible to carry out explicit calculations for the low dimensional cases.

This paper should be seen as shedding light on the study of topological phases in two key ways. The first is as a generalisation of the usual manifold TQFTs. We prove that many of the same techniques used to give functors from the cobordism categories can be used to give functors from things which are not actually cobordisms, which is potentially useful for understanding particle motion, as discussed above. Another way to see our construction is as a restriction to a specific class of TQFTs. Classifying TQFTs in general is a hard, but perhaps we can say something interesting by aiming to classify TQFTs which factor through specific target categories. From this perspective, a question one could ask is how TQFTs which factor through \({\textrm{HomCob}}\) relate to other consturctions of TQFTs. In \(2+1\)D, a TQFT can be obtained from the purely algebraic data of a modular tensor category (MTC), which in turn can be obtained from quantum groups [49]. Little is known in general about how these TQFTs might correspond to functors from \({\textrm{HomCob}}\), constructed using techniques generalising those given here. This correspondence is, however, understood in the case that the construction has input a finite group [50, App.H], which is the example we work with here (see also [12]). The appropriate functors to work with are possibly those given in [42], which take as input a general n-type.

Another interesting future direction would be to adjust the category \({\textrm{HomCob}}\) to be made up of spaces together with a subspace. On the TQFT side this can be thought of as connecting to Reshitikhin–Turaev invariants of spaces with embedded ribbons, or more generally, defect TQFT [49]. On the algebraic topology side this could be seen as extending functors from using the data of the knot group to using the knot quandle, which completely classifies a knot [25].

We now introduce our construction in more detail.

Our first objective is the construction of a new category of cospans of topological spaces. The ‘natural’ formalism for such constructions depends on one’s perspective, i.e. upon one’s aims. For example we have the categorical/‘join’ perspective following Benabou [6]. One of Benabou’s archetypes is the bicategory \(\textrm{Sp}(V)\) of spans over a category V with pullbacks and a distinguished choice of pullback for each span. And a ‘dual’, \(\textrm{Cosp}(V)\) of cospans over a category with pushouts and choices. But this comes at a cost of inducing categories with properties that are undesirable in our setting. Precisely, a manifold cobordism maps naturally to a cospan, but the image of the usual representative of the identity cobordism is of the form \(M\rightarrow M\times [0,1] \leftarrow M\). In contrast the identity in \(\textrm{Cosp}(V)\) is of the form \( 1_M:M\rightarrow M \leftarrow M:1_M\), hence the mapping does not extend to a functor. General pushouts of topological spaces also fail to preserve homotopy. If one follows this line, then a fix (the fix, essentially tautologically) is some form of ‘collaring’. The approach of [20, 35] is to explicitly construct such collars. These can be compared with [19], for example, whose decorated cospans do not include a collaring. Here we proceed by instead insisting that all maps are cofibrations.

Physically this collaring can be thought of as conditions on the way we are allowed to ‘cut’ spacetime into finite time slices; we require that sufficient information is retained at the cut to be able to reconstruct the full theory from evaluations on each slice. The existence of such a formalism is implied by the path integral formalism of quantum field theory [22]. The appropriate conditions will depend on the particular field theory.

We show that there is a category (Theorem 3.17)

whose objects are all topological spaces, elements in \(\textrm{CofCos}(X,Y)\) are equivalence classes of concrete cofibrant cospans – diagrams \(i:X \rightarrow M \leftarrow Y:j\) with the condition that \(\langle i, j\rangle :X\sqcup Y\rightarrow M\), the map obtained from the universal property of the coproduct, is a closed cofibration^{Footnote 1}. A large class of examples of cofibrant cospans comes from cospans of CW complexes and subcomplexes, which are essentially used to construct the TQFT in [42]. Composition is via pushout. We then obtain the category \({\textrm{HomCob}}\) as a subcategory of \({\textrm{HomCob}}\) with all spaces homotopically 1-finitely generated (Theorem 3.36). Pushouts of cofibrations behave well with respect to homotopy, and hence there exist maps from \({\textrm{HomCob}}\) defined using homotopical properties of the spaces involved, which preserve composition. This property is formalised by a generalisation of the van Kampen Theorem reproduced here in Theorem 2.33. The equivalence relation is a notion of homotopy equivalence which commutes with the cospans. The category \({\textrm{HomCob}}\) can be thought of as a generalisation of the usual category of cobordisms [30]; precisely we show in Proposition 3.34 that there is a functor \(\textrm{Cob}_{n}:\textbf{Cob}_{n} \rightarrow {\textrm{HomCob}}\), thus functors \({\textrm{HomCob}}\rightarrow {\textbf{Vect}}\) restrict to ordinary TQFTs. (An alternative approach is to work with cobordisms with corners, see for example [35].) We also show that \(\textrm{Cob}_{n}\) is a symmetric monoidal functor, where the monoidal product on objects in \({\textrm{HomCob}}\) is given by disjoint union.

In Sect. 4, we prove that there exist functors

from respectively the motion groupoid and the mapping class groupoid of a manifold M with fixed submanifold \(A\subset M\), each with a finiteness condition on the objects of the categories. For reference, the motion group of a subset \(X\subset M\) is the automorphism group \(\textrm{Mot}_{M}^{\varnothing }(X,X)\). In the case \(M={{\mathbb {R}}}^2\), and X a finite subset, this is a group isomorphic to the braid group on |X| strands, and in the case \(M={{\mathbb {R}}}^3\) and X is some configuration of unknotted, unlinked loops, this is a group isomorphic to the loop braid group in a number of loops equal to the number of connected components of X. Hence our results lead to representations of motion groupoids, in all dimensions and for all subsets. Existing results on representations of the loop braid group are limited, and focus on representations obtained by going via finite dimensional quotients, and extending braid representations [13, 15, 26].

In the second half of this paper we construct a family of symmetric monoidal functors \({\textsf{Z}}_G:{\textrm{HomCob}}\rightarrow {\textbf{Vect}}\), one associated to each finite group G. The construction of our functor is largely based on the approach taken in [54], although Yetter constructs the functor with triangulations of manifolds, whereas we work entirely with the fundamental groupoid, following [35]. The novel part of our construction with respect to [54] is our generalised source category. Another key aspect of our construction is that we present our functor in an explicitly calculable way, in terms of a choice of basepoints.

We construct the map on \(X\in Ob({\textrm{HomCob}})\) as a colimit over all sets of groupoid maps \(\{f:\pi (X,X_0)\rightarrow G\}\) for each choice of finite representative set \(X_0\). In general the colimit construction is a global equivalence relation on an uncountably infinite set. We prove that for each space \(X\in \chi \) we can fix a choice of finite representative subset \(X_0\), and then

where \(\cong \) denotes taking maps up to natural transformation and \(\pi (X,X_0)\) is the full subgroupoid of the fundamental groupoid with objects \(X_0\) (Theorem 5.29). This gives \({\textsf{Z}}_G\) in terms of a local equivalence relation on a finite set, making explicit calculation possible. We also prove that, for \(X\in Ob({\textrm{HomCob}})\), \({\textsf{Z}}_G(X)\) is isomorphic to the vector space with basis \(\left\{ \pi (X)\rightarrow G\right\} /\cong \) (Theorem 5.29).

The equivalence class of the cospan \(i:X \rightarrow M \leftarrow Y:j\) is mapped to the matrix where, for basis elements \([f]\in {\textsf{Z}}_G(X)\) and \([g]\in {\textsf{Z}}_G(Y)\), the element in the column corresponding to [f] and the row corresponding to [g] is

where \(M_0\subset M\) is a finite subset with some conditions, \(\pi (i)\) is the map of fundamental groupoids induced by i, and \(\sim \) indicates equivalence up to natural transformation. This is Lemma 5.26 and Remark 5.27.

1.1 Paper Overview

In Sect. 2 we have preliminaries, fixing notation and recalling results that we will make use of. First, in Sect. 2.1, we fix notation for categories, groupoids and monoidal categories, as well as introducing the terminology of ‘magmoids’. These are category-like structures without axioms that we will find a useful starting point for our constructions. Then in Sect. 2.2 we have the fundamental groupoid, and a partial product-hom adjunction in the category of topological spaces. In Sect. 2.3 we fix representative colimits in the categories we will need. Finally, in Sect. 2.4 we recall the definition of a cofibration, as well as giving some key properties. In particular, we have Corollary 2.34, which is a corollary of a version of the van Kampen theorem using cofibrations, due to [8].

In Sect. 3, we begin by constructing a magmoid whose objects are topological spaces and whose morphisms are concrete cofibrant cospans. We then quotient by a congruence in terms of homotopy equivalences to obtain, in Theorem 3.17, a category \(\textrm{CofCos}\) which has cofibrant cospans as morphisms. In Theorem 3.22 we prove there is a monoidal structure on \(\textrm{CofCos}\) with monoidal product given, on objects, by disjoint union. Imposing a finiteness condition on spaces then yields the subcategory \({\textrm{HomCob}}\) of \(\textrm{CofCos}\) (Theorem 3.33). In Theorem 3.36 we have a monoidal version of \({\textrm{HomCob}}\), with the monoidal structure inherited from \(\textrm{CofCos}\).

In Sect. 4 we prove that we have a functor \(\mathcal {MOT}_{M}^{A}:\textrm{hf}\textrm{Mot}_{M}^{A} \rightarrow {\textrm{HomCob}}\) (Theorem 4.4), and \(\mathcal {MCG}_{M}^{A}:\textrm{MCG}_{M}^{A} \rightarrow {\textrm{HomCob}}\) (Lemma 4.6) for M a manifold and \(A\subset M\) a subset. We also prove that we have \(\mathcal {MOT}_{M}^{A}={\textsf{F}}\circ \mathcal {MCG}_{M}^{A}\) where \({\textsf{F}}:\textrm{Mot}_{M}^{A}\rightarrow \textrm{MCG}_{M}^{A}\) is as constructed in [32, Sec.7], this is Theorem 4.7.

We begin Sect. 5 by constructing, in Lemma 5.5, another magmoid which has as morphisms cospans of pairs of a topological space and a subset of basepoints, we call this \(\textsf{bHomCob}\). We then construct a magmoid morphism \({\textsf{Z}}_G^{!}:\textsf{bHomCob}\rightarrow \textbf{Vect}_{\mathbb {C}}\) (Lemma 5.11), which depends on a finite group G. Under \({\textsf{Z}}_G^{!}\), pairs \((X,X_0)\) are mapped to the vector space with basis the set of maps \(\pi (X,X_0)\) to G. We then take a colimit over a diagram whose vertices are indexed by all allowed sets of basepoints. This leads to a map \({\textsf{Z}}_G:Ob({\textrm{HomCob}})\rightarrow Ob(\textbf{Vect}_{\mathbb {C}})\), given in Definition 5.20. In Lemma 5.24, this is extended to morphisms, yielding a magmoid morphism \({\textsf{Z}}_G:{\textrm{HomCob}}\rightarrow \textbf{Vect}_{\mathbb {C}}\). In Theorem 5.25 we prove the mapping is well defined and thus we have a functor \({\textsf{Z}}_G:{\textrm{HomCob}}\rightarrow \textbf{Vect}_{\mathbb {C}}\). Theorem 5.29 gives the alternative interpretation of \({\textsf{Z}}_G(X)\), as the vector space with basis \(\left\{ f:\pi (X,X_0)\rightarrow G\right\} /\cong \) for some choice of basepoints \(X_0\), where \(\cong \) denotes functors up to natural transformation. Finally, we have Theorem 5.29, which says that, for \(X\in Ob({\textrm{HomCob}})\), \({\textsf{Z}}_G(X)\) is isomorphic to the vector space with basis \(\left\{ \pi (X)\rightarrow G\right\} /\cong \).

We spend some time on this section for a number of reasons. One is that this paper draws from diverse areas of mathematics, and recalling the relevant results makes this work accessible to a wide audience. Another is to take the opportunity to fix some non-standard definitions/notation that will be helpful. In each section we give references directing the reader to more complete approaches.

We start, in Sect. 2.1, with magmoids, categories and groupoids. In Sect. 2.2 we have the fundamental groupoid and a product-hom adjunction in the category of topological spaces. In Sect. 2.3 we recall the definition of a colimit as well as fixing choices of specific colimits in the categories of sets, topological spaces and groupoids. Finally we define and recall some useful properties of cofibrations in Sect. 2.4, as well as giving examples which will demonstrate the flexibility of the subsequent construction.

2.1 Magmoids, Categories and Groupoids

In this work constructions of categories are a recurrent theme, although these are neither a convenient starting point, nor representative of the underlying physics. Such constructions will often start from something concrete with a composition. In some cases, equivalence classes of these concrete things will eventually become the morphisms of the constructed category, in other cases these structures, and maps between them, will be tools used in the construction of functors which need not become categories themselves. Since such constructions occur frequently, it is useful to have a language for referring to them. We give this here, as well as fixing various notation.

We assume familiarity with category theory such as product categories, adjunctions and natural transformations as well as with groupoids and monoidal categories; more complete introductions can be found in e.g. [1, 31, 43, 50].

We will denote a category by a quadruple \({{{\mathcal {C}}}}=(Ob({{\mathcal {C}}}),{{\mathcal {C}}}(-,-),*_{{{\mathcal {C}}}},1_{-})\), and a groupoid by a pentuple \({{{\mathcal {G}}}}= {{\mathcal {G}}}=(Ob({{\mathcal {G}}}),{{\mathcal {G}}}(-,-),*_{{\mathcal {G}}},1_-,(-)\mapsto (-)^{-1})\). A magmoid is then a triple \({{{\textsf{M}}}} = (Ob({{\textsf{M}}}),{{\textsf{M}}}(-,-),\Delta _{{{\textsf{M}}}})\) consisting objects, morphisms, and a composition which is not assumed to be associative, or to have an identity - the obvious categorification of a magma. In each case we use function order for composition, so, for composable morphisms \(f:X\rightarrow Y\) and \(g:Y\rightarrow Z\) in \({{\textsf{M}}}\), \((f,g) \mapsto g\Delta _{{\textsf{M}}}f\). Magmoid morphisms are defined in the obvious way as an appropriate collection of set maps preserving the composition.

We denote by \({\textbf{Set}}\) the usual category of sets and functions, \({{\textbf {Vect}}}_{\mathbb {k}}\) the category of \(\mathbb {k}\)-vector spaces and linear maps, and \({\textbf{Top}}\) for the category of topological spaces and continuous maps. We denote the identity map on \(X\in Ob({\textbf{Top}})\) by \(\textrm{id}_X:X\rightarrow X\). We use \({\textbf{Cat}}\) to denote the category of small categories and functors, and \({\textbf{Grpd}}\) to denote the full subcategory of groupoids.

Often the magmoids we construct are too large to be interesting objects of study themselves. Congruences and quotient magmoids are our main tool for obtaining a category from a magmoid. A congruence C on a magmoid \({{\textsf{M}}}=(Ob({{\textsf{M}}}),{{\textsf{M}}}(-,-),\Delta _{{{\textsf{M}}}})\) consists of, for each pair \(X,Y\in Ob({{\textsf{M}}})\) an equivalence relation \(R_{X,Y}\) on \({{\textsf{M}}}(X,Y)\), such that \(f'\in [f]\) and \(g'\in [g]\) implies \(g' \Delta _{{{\textsf{M}}}}f'\in [g\Delta _{{{\textsf{M}}}}f]\) where defined. The quotient magmoid of \({{\textsf{M}}}\) by C is \({{\textsf{M}}}/C=(Ob({{\textsf{M}}}),{{\textsf{M}}}(X,Y)/R_{X,Y},\Delta _{{{\textsf{M}}}/C})\), where the composition is given by composing representatives in \({{\textsf{M}}}\). Note in particular that the object set is always fixed. Allowing equivalence relations on objects in magmoids potentially leads to extra morphisms, and so is not really a quotient in the usual sense. Given a collection of relations \(R=\{R_{X,Y}\}\) on the \({{\textsf{M}}}(X,Y)\), we use \({\bar{R}}\) to denote the closure of R to a congruence - this means we take the reflexive, symmetric, transitive closure of each \(R_{X,Y}\) and insist that for any composition \(g\Delta _{{\textsf{M}}}f\sim g' \Delta _{{\textsf{M}}}f'\) if \(f\sim f'\) and \(g\sim g'\).

Any category \({{\mathcal {C}}}\) (respectively groupoid \({{\mathcal {G}}}\)) clearly has an underlying magmoid given by forgetting all but the first three elements of the tuple, by abuse of notation we may refer to the underlying magmoid also as \({{\mathcal {C}}}\) (respectively \({{\mathcal {G}}}\)). Notice that the identities and inverses of a groupoid are uniquely determined from the data of the underlying magmoid. Similarly, a magmoid morphism between the underlying magmoids of a pair of groupoids always lifts to functor preserving the groupoid structure. The construction of the quotient magmoid extends to categories giving the quotient category construction of [31, Ch.2].

A category \({{\mathcal {C}}}\) is called finitely generated if there exists a finite set X of morphisms (including identities) in \({{\mathcal {C}}}\) such that every morphism in \({{\mathcal {C}}}\) can be obtained by composing morphisms in X.

2.1.1 Monoidal categories

Here we fix notation for monoidal categories, and of examples that we will make use of later. A good reference for this section is [50].

We will denote a monoidal category by a pentuple \(({{\mathcal {C}}},\otimes , \mathbb {1},\alpha _{-,-,-},\lambda _{-},\rho _{-})\) consisting of, following the terminology of [50], a category, a monoidal product, a monoidal unit, an associativity constraint, and left and right unitality constraints. Recall that we have a monoidal structure on \({\textbf{Top}}\):

where the associators and unitors are the obvious isomorphisms. Similarly there exists a monoidal structure on \({\textbf{Vect}}_{\mathbb {k}}\)

where \(\otimes _{\mathbb {k}}\) is the usual tensor product of vector spaces and the linear maps are given by \(\alpha _{V,W,X}^\mathbb {k}((v\otimes _{\mathbb {k}}w)\otimes _{\mathbb {k}}x)= (v\otimes _{\mathbb {k}}(w\otimes _{\mathbb {k}}x))\), \(\lambda _{V}^\mathbb {k}(v\otimes k)=kv\) and \(\rho _V^\mathbb {k}(k\otimes v)=kv\).

We use strong monoidal functor to refer to a monoidal functor where all coherence maps are isomoprhisms - all monoidal functors we consider here will be strong.

We denote a braided monoidal category by a six-tuple \( ({{\mathcal {C}}},\otimes , \mathbb {1},\alpha _{-,-,-},\lambda _{-},\rho _{-},\beta _{-,-}). \) where the first five elements give a monoidal category, and the last element is a family of natural isomorphisms \(\beta _{X,Y}:X\otimes Y\rightarrow Y\otimes X\). We say the category is symmetric monoidal if all \(\beta _{Y,X}*\beta _{X,Y}=1_{X\otimes Y}\).

It is straightforward to check that the monoidal structures on \({\textbf{Top}}\) and \({\textbf{Vect}}_{\mathbb {k}}\) given above extend to braided monoidal categories with the braidings given by the morphisms \(\beta _{X,Y}:X\sqcup Y\rightarrow Y\sqcup X\), \((x,y)\mapsto (y,x)\) in \({\textbf{Top}}\) and \(\beta _{V,W}:V\otimes _\mathbb {k} W\rightarrow W\otimes _\mathbb {k} V\), \(v\otimes _\mathbb {k} w\mapsto w\otimes _\mathbb {k} v\) in \({\textbf{Vect}}_{\mathbb {k}}\).

When speaking about (braided) monoidal categories we may drop entries of the tuple corresponding to the natural isomorphisms, or even refer to a braided monoidal category as just \({{\mathcal {C}}}\) where \({{\mathcal {C}}}\) is the notation of the underlying category. We note however, that there will often be many (braided) monoidal categories with the same underlying category, monoidal product and monoidal unit.

2.2 A Product-Hom Adjunction in \({\textbf{Top}}\) and the Fundamental Groupoid

We begin by recalling a partial lift of the classical product-hom adjunction in \({\textbf{Set}}\) to \({\textbf{Top}}\). We will make heavy use of this result in Sect. 4.

We then fix notation for the fundamental groupoid, and discuss the relationship between fundamental groupoids obtained by varying a finite number of basepoints. This will be necessary for the construction in Sect. 5 (Lemmas 2.5 and 2.6). Throughout the rest of this paper we will encounter several equivalence relations so we also introduce some careful notation for path equivalence here.

For spaces X, Y, we use \({\textbf{TOP}}(X,Y)\) to denote the set \({\textbf{Top}}(X,Y)\) together with the compact open topology. For the precise definition see [36, p.285]; a useful characterisation is given by the fact that if Y is a metric space, the compact-open topology coincides with the topology obtained from the sup-norm metric on \({\textbf{Top}}(X,Y)\) – the distance between two maps f, g is given by the least upper bound of the distance between all pairs f(x), g(x).

There exists a functor \(-\times Y:{\textbf{Top}}\rightarrow {\textbf{Top}}\) called the product functor which sends a space X to the product space \(X\times Y\), and a continuous map \(f:X\rightarrow X'\) to the map \(f\times \textrm{id}_Y :X\times Y\rightarrow X'\times Y\), \((x,y)\mapsto (f(x),y)\). There also exists a functor \({\textbf{TOP}}(Y,-):{\textbf{Top}}\rightarrow {\textbf{Top}}\) called the hom functor which sends a space Z to the space \( {\textbf{TOP}}(Y,Z)\), and which sends a continuous map \(f:Z\rightarrow Z'\) to \(f\circ -:{\textbf{TOP}}(Y,Z)\rightarrow {\textbf{TOP}}(Y,Z')\), \(g\mapsto f\circ g\).

We will make extensive use of the following classical result in Sect. 4. For full details see for example Section 2.2 of [32].

Lemma 2.1

Let Y be a locally compact Hausdorff topological space. The product functor \(-\times Y \) is left adjoint to the hom functor \({\textbf{TOP}}(Y,-)\). In particular, for objects \(X,Y,Z\in {\textbf{Top}}\), this gives a set map

that is a bijection, natural in the variables X and Z.

Spanier [45] and Brown [8] were among the first to consider fundamental groupoids, and careful constructions can be found in [16] and [8] for example.

We use \({{\mathbb {I}}}\) to denote \([0,1]\subset {{\mathbb {R}}}\) with the subset topology. For a space X, a path is the an element of \({\textbf{Top}}({{\mathbb {I}}},X)\). We use \(e_x\) to denote the constant path at a point \(x\in X\). There is then a magmoid \({{\mathfrak {P}}}X\) whose objects are the elements of X, morphisms are paths and composition is the usual path composition. Path homotopy gives a congruence on \({{\mathfrak {P}}}X\), and the quotient is the fundamental groupoid, which we denote by \(\pi (X)\).

If \(\gamma \) and \(\gamma '\) are path homotopic, we write \(\gamma {\mathop {\sim }\limits ^{p}}\gamma '\), and we use \([\gamma ]_{\!\tiny \hbox {p}}\) for the path-equivalence class of \(\gamma \).

It is straightforward to check that the map sending a space to its fundamental groupoid extends to a functor.

Lemma 2.2

There is a functor \(\pi :\textbf{Top} \rightarrow \textbf{Grpd}\) which sends a space X to the fundamental groupoid \(\pi (X)\) and is defined on morphisms as follows. Let \(f:X \rightarrow Y\) be a continuous map, \(\pi (f):\pi (X) \rightarrow \pi (Y) \) defined by \(x\mapsto f(x)\) for a point \(x\in X=Ob(\pi (X))\) and by \([\gamma ]_{\!\tiny \hbox {p}}\mapsto [ f \circ \gamma ]_{\!\tiny \hbox {p}}\) for a path \(\gamma \) in X. \(\square \)

For a topological space X, and a subset \(A\subseteq X\), we call the full subgroupoid of \(\pi (X)\) with object set A the fundamental groupoid of X with respect to A, and denote it by \(\pi (X,A)\). We refer to A as the set of basepoints. We thus have \(\pi (X,X)=\pi (X)\), and, for X a path-connected topological space \( \pi (X,\{x\}) \) is the fundamental group based at \(x \in X\). Notice that for any \(A'\subseteq A\), there is an inclusion \(\iota :\pi (X,A')\rightarrow \pi (X,A)\).

Definition 2.3

For topological spaces X and \(A\subseteq X\), then A is called representative in X if A contains a point in every path-component of X. (The nomenclature (X, A) is a 0-connected pair is also used.)

Lemma 2.4

Suppose \(f:X\rightarrow Y\) is a surjection and A is a representative subset of X, then f(A) is representative in Y.

Proof

Let \(y\in Y\) be any point. We must construct a path from y to an element of f(A). Let \(y'\in f^{-1}(y)\) be any preimage, then there exists a path \(\gamma \) from \(y'\) to a point in A and \(f\circ \gamma \) is a path from y to an element of f(A). \(\square \)

We will need the following results about the fundamental groupoid with finite sets of basepoints in Sect. 5. In what follows we use the same labels for paths, and their equivalence classes to keep the notation readable. The meaning will be clear from context.

Lemma 2.5

Let \({{\mathcal {G}}}\) be a groupoid, X a topological space, \(X_0\subseteq X\) a finite subset and \(y \in X \setminus X_0\) any point. Given a groupoid map \(f:\pi (X,X_0) \rightarrow {{\mathcal {G}}}\), a path \(\gamma :x \rightarrow y\) where \(x\in X_0\) and a morphism \(g:f(x)\rightarrow {\textsf{g}}\) in \({{\mathcal {G}}}\) there exists a unique \(F:\pi (X,X_0 \cup \{y\}) \rightarrow {{\mathcal {G}}}\) extending f such that

• the diagram

  

  commutes, where \(\iota \) is the inclusion map, and

• \(F(\gamma ) = g \).

Proof

First we construct such an F. On objects we have,

For a path \(\phi :a\rightarrow y\) with \(a \in X_0\) we must have

Arguing similarly for all cases we have that for a morphism \(\phi :a\rightarrow b\),

Notice that in each case F is inferred from the conditions set out in the theorem and by functoriality. This gives uniqueness. Now it remains to check that functoriality is always preserved, i.e. for any two composable paths \(\phi ,\phi ' \in \pi (X,X_0 \cup \{y\})\) we have \(F(\phi ') F(\phi )=F(\phi ' \phi )\). This can be checked case by case, we give two examples, the other cases are checked similarly.

1. (I)

   If we have \(\phi :y \rightarrow y\), \( \phi ' :y \rightarrow y\), then

   $$\begin{aligned} F(\phi ')F(\phi )= & {} gf(\gamma ^{-1}\phi '\gamma )g^{-1}gf(\gamma ^{-1}\phi \gamma )g^{-1}=gf(\gamma ^{-1}\phi '\gamma \gamma ^{-1}\phi \gamma ) g^{-1} \\= & {} gf(\gamma ^{-1}\phi '\phi \gamma )g^{-1} =F(\phi ' \phi ). \end{aligned}$$
2. (II)

   If we have \(\phi : a \rightarrow y, a \in X_0\), \( \phi ': y \rightarrow b\), \(b\in X_0\), then

   $$\begin{aligned} F(\phi ')F(\phi )&= f(\phi ' \gamma )g^{-1}g f(\gamma ^{-1}\phi ) = f(\phi ' \gamma \gamma ^{-1}\phi ) = f(\phi '\phi )=F(\phi '\phi ). \end{aligned}$$

\(\square \)

Given a group G, there is a groupoid \({{\mathcal {G}}}_G=(\{*\},{{\mathcal {G}}}_G(*,*),\circ _G,e_G,g\mapsto g^{-1})\), where \({{\mathcal {G}}}_G(*,*)\) is the underlying set of G.

Lemma 2.6

Let X be a topological space, G a group, \(X_0\subseteq X\) a finite representative subset and \(y\in X\) a point with with \(y\notin X_0\). There is a non-canonical bijection of sets

where \(\gamma \) is a choice of a path from some \(x\in X_0\) to y and F is the extension along \(\gamma \) and g as described in Lemma 2.5.

Proof

First notice that for any \(g\in G\), \(g:f(x)\rightarrow f(x)\) is a morphism in \({{\mathcal {G}}}_G\), so the map is well-defined.

The map \(\Theta _{\gamma }\) has inverse which sends a map \(f'\in {\textbf{Grpd}}(\pi (X,X_0\cup \{y\}),G)\) to the pair \((f'\circ \iota ,f'(\gamma ))\) where \(\iota :\pi (X,X_0)\rightarrow \pi (X,X_0\cup \{y\})\) is the inclusion. \(\square \)

2.3 Colimits

Colimits will play an integral role in the construction in Sect. 5 so we use this section to recall the definition and review some key properties that we will use. We also fix representative colimits in the categories we will work with throughout the paper.

The topics covered here can be found, for example, in [39, Ch.3].

Definition 2.7

Let \({{\mathcal {C}}}\) be a category and \({\textbf{I}}\) a small category. A functor \(D:{\textbf{I}} \rightarrow {{\mathcal {C}}}\) is called a diagram in \({{\mathcal {C}}}\) of shape \({\textbf{I}}\)

Let \(D:{\textbf{I}} \rightarrow {{\mathcal {C}}}\) be a diagram in \({{\mathcal {C}}}\). A cocone is an object \(C \in Ob({{\mathcal {C}}})\) together with a family of morphisms

indexed by the objects in \({\textbf{I}}\), such that for all morphisms \(f:i \rightarrow j\) in \({\textbf{I}}\) the following triangle commutes.

A colimit of D is a cocone \((C,\phi )\) with the universal property that for any other cocone \((C',\psi )\) there exists a unique morphism \(C\rightarrow C'\) making the following diagram commute for all morphisms \(f:i \rightarrow j\) in \({\textbf{I}}\).

We will refer to the object C as \({\textrm{colim}}(D)\).

Definition 2.8

Let \({\textbf{T}}\) be the category with two objects and no non identity morphisms, then a colimit of a diagram of shape \({\textbf{T}}\) is a coproduct. Let \({\textbf{P}} = \bullet \leftarrow \bullet \rightarrow \bullet \) be a category with three objects and two non identity morphisms as shown. A colimit of a diagram of shape \({\textbf{P}}\) is a pushout. Let be the category with two objects and two non identity morphisms as shown. A colimit of a diagram of shape E is called a coequaliser.

In general colimits do not exist. However it is straightforward to show that, where colimits do exist, they are unique up to isomorphism. Thus, where colimits do exist, we are free to choose a representative element of each isomorphism class to work with. Indeed we will fix representative elements for various colimits so a coproduct is uniquely defined by giving the appropriate diagram. For example, in \({\textbf{Set}}\) we will fix the representative coproduct of a pair \(X,Y\in Ob({\textbf{Set}})\), to be the disjoint union

with the natural inclusions (i.e. those given by \(\iota _i(x) \;{:}{=}(x,i)\)).

We explain our convention for pushouts in \({\textbf{Set}}\). Consider morphisms \(f:Z \rightarrow X\) and \(g:Z\rightarrow Y\) in \({\textbf{Set}}\). Then the diagram below:

is a pushout in \({\textbf{Set}}\). Here

where \(\sim \) is the reflexive, symmetric, transitive closure of the relation

on \(X\sqcup Y\), \(p_X(x)\) is the equivalence class of \(\iota _1(x)\) in \(X\sqcup Y/\sim \) and \(p_Y(y)\) is the equivalence class of \(\iota _2(y)\) in \(X\sqcup Y/\sim \).

We will fix the following representative general colimit in \({\textbf{Set}}\). Let \(D:{\textbf{I}}\rightarrow {\textbf{Set}}\) be a diagram. Denote by \(\sqcup _{i\in Ob({\textbf{I}})}D(i)\) the set of all pairs (x, i), \(i\in Ob({\textbf{I}})\), \(x\in D(i)\). This is the disjoint union of the D(i), and is a colimit of D if D has no non-identity morphisms. For \(i\in Ob({\textbf{I}})\), let \(\iota _i:D(i) \rightarrow \sqcup _{i\in Ob({\textbf{I}})}D(i)\) denote the map \(x\mapsto (x,i)\). Consider the relation

on \(\sqcup _{i\in {\textbf{I}}}D(i)\). The colimit of D is given by

with maps \(\phi _i:D(i)\rightarrow \nicefrac {\sqcup _{i\in {\textbf{I}}}D(i)}{{\bar{R}}}\) which send x to the equivalence class of \(\iota _i(x)\).

The following theorem says that adjunctions interact nicely with colimits, allowing us to fix colimits in other categories in terms of those in \({\textbf{Set}}\).

Theorem 2.9

([43, Thm. 4.5.3]) Left adjoints preserve colimits. This means for any left adjoint \(F:{{\mathcal {C}}}\rightarrow {{\mathcal {D}}}\), any diagram \(D:{\textbf{I}}\rightarrow {{\mathcal {C}}}\) then

2.3.1 Colimits in \({\textbf{Top}}\)

We denote by \(\mathrm {U_T}:{\textbf{Top}}\rightarrow {\textbf{Set}}\) the forgetful functor which sends a space to its underlying set and a continuous map to its underlying function. It is straightforward to prove the following:

Lemma 2.10

(I) There is a functor \(\mathrm {G_T}:{\textbf{Set}}\rightarrow {\textbf{Top}}\) which sends a set X to the space with underlying set X and the indiscrete topology, and which sends a function to the map which has the same action on the underlying sets.

(II) The functor \(\mathrm {G_T}\) is right adjoint to \(U_T\). \(\square \)

Since \(\mathrm {U_T}:{\textbf{Top}}\rightarrow {\textbf{Set}}\) is a left adjoint, by Theorem 2.9, \(\mathrm {U_T}\) preserves colimits. This means coproducts and pushouts of diagrams in \({\textbf{Top}}\) have the same underlying set as the coproducts and pushouts of their images in \({\textbf{Set}}\).

Let X and Y be spaces. Then

is a topology on \(X\sqcup Y\). It is straightforward to prove that \((X\sqcup Y,\tau _{X\sqcup Y})\) is a coproduct in \({\textbf{Top}}\) (see for example (3.1.2) of [8]). We will use the notation indicated by the following diagram to refer to the map given by the universal property of the coproduct in \({\textbf{Top}}\).

(If we have maps of spaces \(i:X\rightarrow M\) and \(j:Y\rightarrow N\), we will use \(i\sqcup j\) for the obvious map \(X\sqcup Y \rightarrow M\sqcup N\).)

Let X, Y, Z be topological spaces. Consider continuous maps \(f:Z \rightarrow X\) and \(g:Z\rightarrow Y\). The topology on \(X { \sqcup _{Z} } Y\) which makes it into a pushout in \({\textbf{Top}}\) is the following:

This topology can be equivalently defined as the finest topology on \(X { \sqcup _{Z} } Y\) making \(p_X\) and \(p_Y\) continuous.

2.3.2 Colimits in \({\textbf{Grpd}}\)

The construction of the category \({\textrm{HomCob}}\) will rely on the fact that the pushout of two finitely generated groupoids is a finitely generated groupoid.

Let \({{\mathcal {G}}}_1,{{\mathcal {G}}}_2\in Ob({\textbf{Grpd}})\) be groupoids. Then we fix a representative coproduct, denoted \({{\mathcal {G}}}_1\sqcup {{\mathcal {G}}}_2\), as follows. The object set \(Ob({{\mathcal {G}}}_1\sqcup {{\mathcal {G}}}_2)=Ob({{\mathcal {G}}}_1)\sqcup Ob({{\mathcal {G}}}_2)\), the coproduct in \({\textbf{Set}}\), and the morphism set in \({{\mathcal {G}}}_1\sqcup {{\mathcal {G}}}_2 \) is the coproduct in \({\textbf{Set}}\) of the morphisms in \({{\mathcal {G}}}_1\) and the morphisms in \({{\mathcal {G}}}_2\), where for \(f:w\rightarrow x\) in \({{\mathcal {G}}}_i\), (f, i) is a morphism from (w, i) to (x, i).

The difficulty in constructing more general colimits stems from the fact that the image of a functor of groupoids is often not a groupoid, as it is not closed. More precisely, suppose \(F :{{\mathcal {G}}}\rightarrow {\mathcal {H}}\) is a functor of groupoids, and \(g_1:w\rightarrow x\) and \(g_2:y\rightarrow z\) are morphisms in \({{\mathcal {G}}}\). Then \(g_2*g_1\) is defined if and only if \(x=y\) and then we must have \(F(g_2*g_1)=F(g_2)*F(g_1)\), meaning \(F(g_2)*F(g_1)\) must be the image of a morphism in \({{\mathcal {G}}}\). Suppose, however, that \(x\ne y\) but \(F(x)=F(y)\), then \(F(g_2)*F(g_1)\) is defined in \({\mathcal {H}}\) but will not be the image of any single element in \({{\mathcal {G}}}\). A consequence is that it is possible for the coequaliser of finite groupoids to be infinite. This is illustrated by the following example.

Let \(({\mathbb {Z}},+)\) denote the category with one object and morphisms labelled by elements of \({\mathbb {Z}}\), with composition given by addition in \({\mathbb {Z}}\).

Example 2.11

Let \(\{*\}\) be the groupoid with one object and only the identity morphism, and let \({\textbf{I}}\) be the groupoid with two objects \(\{a,b\}\) and one non-identity morphism from a to b. Let \(\iota _{a}\) be the functor uniquely defined by \(\iota _a(*)=a\), and \(\iota _{b}\) the functor uniquely defined by \(\iota _b(*)=b\). The following diagram is a coequaliser

where p is a functor which maps the only non-identity morphism to \(1\in {\mathbb {Z}}\). (Note p must send \(\{a\}\) and \(\{b\}\) to the only object in \(({\mathbb {Z}},+)\).)

A general pushout in \({\textbf{Grpd}}\) is constructed in [23, Ch. 8,9], and it is immediate after understanding the construction that the following theorem is true.

Theorem 2.12

Let \({{\mathcal {G}}}_0\), \({{\mathcal {G}}}_1\) and \({{\mathcal {G}}}_2\) be finitely generated groupoids and let \(f:{{\mathcal {G}}}_0 \rightarrow {{\mathcal {G}}}_1\) and \(g:{{\mathcal {G}}}_0 \rightarrow {{\mathcal {G}}}_2\) be functors. The pushout of f and g is a finitely generated groupoid. \(\square \)

Although it is by no means required to understand the paper, we found it helpful to reproduce the results present in [23] in our notation, and thus retain this in the appendix of the arXiv version of this paper.

2.4 Cofibrations in \({\textbf{Top}}\)

In Sect. 3 we define a magmoid whose morphisms are cofibrant cospans, and quotient by a congruence defined in terms of cofibre homotopy equivalence. Then, in Sect. 5, our TQFT construction will rely on a version of the van Kampen Theorem for cofibrations. Here we recall the results we will need. More detail on cofibrations can be found in [16, Ch. 5] or [8, Ch. 7], definitions and results on cofibre homotopy equivalence follow [33, Ch. 6], and the version of the van Kampen theorem reproduced here can be found in [8, Thm. 9.1.2].

2.4.1 Cofibrations

A cofibration can be thought of as a homotopically well behaved embedding. Specifically, an embedding \(i:A \rightarrow X\) is a cofibration if we have both that there is an open neighbourhood of the image which strongly deformation retracts onto i(A), and that this retraction can be extended to a homotopy on the whole of X. In other words there is an open neighbourhood of the image which, up to a homotopy of X, is equivalent to the image. This characterisation of a cofibration is not immediately obvious from the below definition but we will see it is equivalent in Theorem 2.21.

One can think of the aforementioned characterisation of a cofibration as a version of the Collar Neighbourhood Theorem of a boundary of a manifold ( [9]). To construct cobordism categories (see [34]) the collar neighbourhood is required to prove the identity axiom. The cofibration condition we impose will play a similar role in our category construction, we will see this in Theorem 3.17.

The following definition is from [16, Sec. 5.1].

Definition 2.13

Let A and X be spaces. A map \(i:A \rightarrow X\) has the homotopy extension property, with respect to the space Y, if for any pair of a homotopy \(h:A \times {{\mathbb {I}}}\rightarrow Y\) and a map \(f:X \rightarrow Y\) satisfying \((f\circ i)(a)=h(a,0)\), there exists a homotopy \(H:X \times {{\mathbb {I}}}\rightarrow Y\), extending h, with \(H(x,0)=f(x)\) and \(H(i(a),t)=h(a,t)\). This is illustrated by the following diagram.

(Where for any space X, \(\iota ^X_0:X \rightarrow X \times {{\mathbb {I}}}\) is the map \(x\mapsto (x,0)\).)

Definition 2.14

Let A and X be spaces. We say that \(i:A \rightarrow X\) is a cofibration if i satisfies the homotopy extension property for all spaces Y. A closed cofibration is a cofibration with image a closed set. If \(A\subseteq X\) is a subspace and the inclusion \(\iota :A\rightarrow X\) is a cofibration, we say (X, A) is a cofibred pair.

The following useful well known results are completely straightforward to prove.

Lemma 2.15

The composition of two cofibrations is a cofibration. \(\square \)

Lemma 2.16

Every homeomorphism is a cofibration. \(\square \)

Lemma 2.17

Let X and Y be topological spaces. The map \(\iota _1 :X \rightarrow X\sqcup Y\), \(x\mapsto (x,1)\) is a cofibration. \(\square \)

Proposition 2.18

Let A and X be spaces and \(i:A \rightarrow X\) a map which is a homeomorphism onto its image. Then i is a cofibration if and only if (X, i(A)) is a cofibred pair.

Proof

Suppose \(i:A \rightarrow X\) is a cofibration and K is any space. Consider a map \(f:X\rightarrow K\) and a homotopy \(h:i(A)\times {{\mathbb {I}}}\rightarrow K\) such that for all \(x\in i(A)\), \(h(x,0)=f(x)\). Applying the homotopy extension property to f and \(h\circ (i\times \textrm{id})\) gives a homotopy \(H:X\times {{\mathbb {I}}}\rightarrow K\). This same H is also a homotopy extending h, and hence (X, i(A)) is a cofibred pair.

Suppose \(i:A\rightarrow X\) a map which is a homeomorphism onto its image and (X, i(A)) is a cofibred pair. From Lemma 2.16 we have that the homeomorphism \( A\rightarrow i(A)\), \(a\mapsto i(a)\) is a cofibration and the inclusion \(\iota :i(A)\rightarrow X\) is a cofibration by assumption. Hence \(i:A\rightarrow X\) is a composition of cofibrations, and so a cofibration by Lemma 2.15. \(\square \)

Theorem 2.19

(See for example [46, Th. 1]) Let A and X be spaces. If \(i:A \rightarrow X\) is a cofibration then i is a homeomorphism onto i(A) with the subspace topology (i.e. i is an embedding). \(\square \)

To prove specific pairs are cofibred we have the following two classical results.

Proposition 2.20

(See for example [16, Prop. 5.1.2]) Let A be a closed subspace of X. The pair (X, A) is cofibred if and only if \(X\times \{0\}\, \cup \, A \times {{\mathbb {I}}}\) is a retract of \(X \times {{\mathbb {I}}}\).

(Recall \(N\subset M\) is a retract of M if there is a continuous map \(r:M\rightarrow N\) such that \(r(n)=(n)\) for all \(n\in N\).) \(\square \)

The following theorem characterises cofibrations as inclusions such that there is a neighbourhood of the image that deformation retracts onto, and hence is homotopy equivalent to the image. This is reminiscent of the Collar Neighbourhood Theorem for manifolds.

Theorem 2.21

[46, Th. 2] Let A be a closed subspace of space X. Then (X, A) is a cofibred pair if and only if there exists

1. i)

   a neighbourhood \(U\subseteq X\) of A (not necessarily open) and a homotopy \(H :U\times {{\mathbb {I}}}\rightarrow X\) such that for all \(t\in {{\mathbb {I}}}\), \(x\in U\) and \(a\in A\), we have \(H(x,0)=x\), \(H(a,t)=a\) and \(H(x,1)\in A\), and

2. ii)

   a map \(\phi :X \rightarrow {{\mathbb {I}}}\) such that \(A=\phi ^{-1}(0)\) and \(\phi (x)=1\) for all \(x\in X-U\). \(\square \)

There is also more general version of the previous theorem (see [47, Lem. 4]) which characterises cofibred pairs in a similar way without restricting to closed subspaces. However we will only need the case of closed subspaces.

Examples of Cofibrations

These key examples will be useful to demonstrate the flexibility of our construction in Sect. 3.

Example 2.22

For any space X, the pairs (X, X) and \((X,\varnothing )\) are cofibred.

Example 2.23

The pair \(({{\mathbb {I}}},\{0,1\})\) is cofibred.

Consider \({{\mathbb {I}}}\times {{\mathbb {I}}}\) as a subset of \({\mathbb {R}}^2\) and let \(z=(\frac{1}{2},\frac{3}{2})\in {\mathbb {R}}^2\). For any \(x\in {{\mathbb {I}}}\times {{\mathbb {I}}}\), let \(x'\) be the unique point of \({{\mathbb {I}}}\times \{0\} \, \cup \, \{0,1\} \times {{\mathbb {I}}}\) such that \(z,x,x'\) are colinear. Then \(\rho :x \mapsto x'\) is a retraction \({{\mathbb {I}}}\times {{\mathbb {I}}}\) to \({{\mathbb {I}}}\times \{0\} \,\cup \, \{0,1\} \times {{\mathbb {I}}}\).

Define topological spaces \(S^n=\{x\in {{\mathbb {R}}}^{n+1} \;\vert \; |x|=1\}\) and \(D^n=\{x\in {{\mathbb {R}}}^n \;\vert \; |x|\le 1\}\), each topologised with the subset topology.

Example 2.24

The pair \((D^{n},S^{n-1})\) is cofibred.

A retraction \(r:D^{n} \rightarrow S^{n-1} \times {{\mathbb {I}}}\; \cup \; D^{n} \times \{0\}\) can be constructed in a similar way to the previous example, see [16, Ex. 2.3.5].

The previous two examples are special cases of the following proposition for manifolds.

Proposition 2.25

Let M be a smooth compact manifold with boundary. The inclusion \(i:\partial M \rightarrow M\) is a cofibration.

Proof

The Collar Neighbourhood Theorem for smooth manifolds [34, Cor. 3.5] says that there exists an open neighbourhood \(N'\) of \(\partial M\) such that there exists a diffeomorphism \(f:N' \rightarrow \partial M \times [0,1)\) satisfying \(f(\partial M)=\partial M \times \{0\}\). It follows that we can choose a closed neighbourhood N of \(\partial M\) such that the we can identify N with \(\partial M \times {{\mathbb {I}}}\), where \(\partial M \cong \partial M \times \{0\}\). Then the function

is a homotopy satisfying condition (i) of Theorem 2.21. Define a map \(\phi :M \rightarrow {{\mathbb {I}}}\) as follows.

Notice that these definitions agree on the overlap so \(\phi \) is continuous. Hence by Theorem 2.21 the inclusion \(i:\partial M\rightarrow M\) is a cofibration. \(\square \)

Recall that a smooth submanifold of M is a subset \(N\subseteq M\) such that the identity inclusion \(\iota :N \rightarrow M\) is a diffeomorphism onto its image and a topological embedding, as in [29, Ch. 5]. A submanifold \(N\subseteq M\) is neatly embedded if \(\partial N\subseteq \partial M\).

We have the following stronger proposition, which slightly generalises [37, Ex.2.3.11].

Proposition 2.26

Let N be a closed smooth submanifold of a smooth manifold M which is neatly embedded. Then the inclusion \(\iota :N \rightarrow M\) is a cofibration.

Proof

By the tubular neighbourhood theorem [24, Ch.4,Th.6.3], there exists an open neighbourhood \(U\subset M\) of N which can be identified with the normal bundle of \(\textrm{v}(N,M)\) of N in M, such that N is identified with the zero section of the normal bundle. This allows us to identify points in \(u\in U\) with pairs (n, v) where \(n\in N\) and v is a vector in the tangent space. Further, a choice of Riemannian metric on M allows us to define \(V=\textrm{D}_1(\textrm{v}(N,M))\), which has as underlying set pairs (n, v) such that \(||v||\le 1\). Label the corresponding subset in M by \({\bar{V}}\). Similarly let V be the open set obtained from taking the set with pairs (n, v) where \(||v||<1\).

Now, as in the previous proposition, we can define a homotopy

and a map \(\phi :M \rightarrow {{\mathbb {I}}}\) with

\(\square \)

In many cases it will be easier to find a CW complex structure than to prove we have a smooth manifold submanifold pair. In this case we have the following proposition.

Proposition 2.27

Let X be a CW complex and let A be a subcomplex of X. Then the inclusion \(i:A \rightarrow X\) is a closed cofibration.

Proof

By [21, Prop.0.16], the assumption implies \(X\times \{0\}\cup A\times {{\mathbb {I}}}\) is a deformation retract of \(X\times {{\mathbb {I}}}\), so we may apply Prop.2.20. \(\square \)

Cofibre homotopy equivalence

To construct a category of cofibrant cospans, we will will require a notion of homotopy equivalence of spaces relative to maps from a shared space.

Definition 2.28

Let A, X, Y be spaces. A space under A is a map \(i :A \rightarrow X\). A map of spaces under A from \(i :A \rightarrow X\) to \(j:A\rightarrow Y\) is a map \(f:X\rightarrow Y\) such that we have a commuting diagram

Suppose \(f,f':X\rightarrow Y\) are maps under A from \(i :A \rightarrow X\) to \(j:A\rightarrow Y\). A homotopy under A from f to \(f'\) is a continuous map \(H:X\times {{\mathbb {I}}}\rightarrow Y\) such that

• for all \(a \in A\) and \(t \in {{\mathbb {I}}}\), \(H(i(a),t)=j(a)\),

• for all \(x\in X\), \(H(x,0)=f(x)\),

• for all \(x\in X\), \(H(x,1)=f'(x)\).

The proof of the following Proposition proceeds exactly as for the usual notion of homotopy equivalence.

Proposition 2.29

Define a relation on spaces under A as follows. We have \((i:A\rightarrow X)\sim (j:A\rightarrow Y)\) if there exists maps of spaces under A, \(f:X\rightarrow Y\) from \(i:A\rightarrow X\) to \(j:A\rightarrow Y\), and \(f':Y\rightarrow X\) from \(j:A\rightarrow Y\) to \(i:A\rightarrow X\), such that there exists a homotopy under A from \(f\circ f'\) to \(\textrm{id}_Y\) and from \(f'\circ f\) to \(\textrm{id}_X\). Note that this is well defined since \(f\circ f'\circ j(a)=f\circ i(a)=j(a)\), \(f'\circ f\circ i(a)=f'\circ j(a) = i(a)\)

This is an equivalence relation. \(\square \)

Definition 2.30

Given a space A, the equivalence relation described in Proposition 2.29 is called cofibre homotopy equivalence.

The following technical result, which justifies the choice of name, will be crucial for our construction. Proofs are present in [8, 33, 48].

Theorem 2.31

([8, Thm. 7.2.8] for example.) Let A, X and Y be spaces. Let \(i:A \rightarrow X\) and \(j:A \rightarrow Y\) be cofibrations and let \(f:X \rightarrow Y\) be a map such that \(f\circ i=j\). Suppose that f is a homotopy equivalence from X to Y, then f is a cofibre homotopy equivalence \(i:A \rightarrow X\) to \(j:A \rightarrow Y\). \(\square \)

2.4.2 A van Kampen Theorem for Cofibrations

Here we give a generalisation of the van Kampen Theorem, due to Brown [8], which says that pushouts are preserved by the functor \(\pi \) if at least one of the maps we take the pushout over is a cofibration. Hence, in this case, we can obtain the fundamental groupoid of a pushout in \({\textbf{Top}}\) as a pushout of fundamental groupoids.

Suppose we have spaces \(X_0\), \(X_1\) and \(X_2\) and maps \(f:X_0 \rightarrow X_1\) and \(g:X_0 \rightarrow X_2\). Consider the pushout square:

Now let A, B and C be representative subsets of \(X_0\), \(X_1\) and \(X_2\) respectively, with \(f(A)= B\cap f(X_0)\) and \(g(A) \subseteq C\). Let \(D=B \sqcup _A C\), the pushout of \(f|_A:A\rightarrow B\) and \(g|_A:A\rightarrow C\).

Lemma 2.32

Under the above conditions conditions D is representative (Definition 2.3) in \( X_1\sqcup _{X_0} X_2\).

Proof

It is clear that \(B\sqcup C\) is representative in \(X_1\sqcup X_2\). By Lemma 2.4 surjections send representative subsets to representative subsets, hence we have the result by considering the surjection \(\langle p_1,p_2\rangle :X_1\sqcup X_2\rightarrow X_1\sqcup _{X_0}X_2\). \(\square \)

Theorem 2.33

(See [8, Thm. 9.1.2].) Now suppose in addition to the above conditions we take \(X_0\subseteq X_1\) and \(f=\iota :X_0 \rightarrow X_1\) the inclusion map in (3). Then the following diagram is a pushout if \((X_1,X_0)\) is a cofibred pair.

\(\square \)

Corollary 2.34

Now suppose in addition to the above conditions, we instead consider \(f=i:X_0 \rightarrow X_1\) any cofibration in (3). Then the following square is a pushout.

Proof

Using Proposition 2.18 and Theorem 2.19 we can separate the cofibration i into two maps, a homeomorphism \({\tilde{i}}:X_0 \rightarrow i(X_0)\) and a cofibred inclusion \(\iota :i(X_0)\rightarrow X_1\). Consider the following commuting pushouts.

Observe that the pushout of g and i is also the pushout of \(g\circ {\tilde{i}}^{-1}\) and \(\iota \), since \({\tilde{i}}\) is a homeomorphism. Choosing the subset i(A) in \(i(X_0)\) and keeping all other subsets as in the statement of the corollary, the outer pushout is preserved by the fundamental groupoid functor by Theorem 2.33. Hence, by functoriality, so is the inner pushout. \(\square \)

In this section the main result is Theorem 3.36 which says that we have a symmetric monoidal category, \({\textrm{HomCob}}\), of homotopy cobordisms.

We proceed by first constructing a magmoid whose morphisms are concrete cofibrant cospans, which compose via pushouts. We then quotient by a congruence, defined in terms of a homotopy equivalence, to obtain the category \(\textrm{CofCos}\) (Theorem 3.17) and show that there exists a symmetric monoidal structure on \(\textrm{CofCos}\) (Theorem 3.22). We add a finiteness condition to the topological spaces in the cospans to arrive at the category \({\textrm{HomCob}}\) as a symmetric monoidal subcategory of \(\textrm{CofCos}\) (Theorems 3.33 and 3.36).

We note that the choices we make in this section are made with the type of functor into \(\textbf{Vect}_{\mathbb {C}}\) that we will construct already in mind. We have already said that we are interested in functors which depend on the homotopy of the spaces, and Corollary 2.34 tells us that cofibrations are the maps that behave well with respect to the fundamental groupoid. Further, the congruence is defined in terms of a suitable version of homotopy equivalence. As a consequence, the category \({\textrm{HomCob}}\) contains interesting subgroupoids which are finitely generated, such as those coming from the images of the functors constructed in Sect. 4. Thus these subgroupoids are manageable structures to work with. Moreover, equivalent morphisms are mapped to the same linear map under the mapping we construct in Sect. 5, meaning that we are able to extend this mapping to a well defined functor.

We note that our cospan categories deviate from those of e.g. [19], in our choice of identity. For Fong, the category identity at an object X is the equivalence class of the cospan \(\textrm{id}_X:X \rightarrow X \leftarrow X:\textrm{id}_X\). In a topological quantum field theory we require that any arbitrary time evolution of a state X is evaluated as the identity if the state X does not change. Hence we insist on identities being the equivalence classes of cospans of the form \(\iota _0^X:X \rightarrow X\times {{\mathbb {I}}} \leftarrow X:\iota _1^X\) (where \(\iota ^X_a:X \rightarrow X \times {{\mathbb {I}}}\) is the map \(x\mapsto (x,a)\)). As a result more work is required to prove that this is, in fact, an identity; see Theorem 3.17.

3.1 Magmoid of Concrete Cofibrant Cospans \(\textsf{CofCos}\)

Here we define concrete cofibrant cospans, construct a composition and organise them into a magmoid.

Definition 3.1

Let X, Y and M be spaces. A concrete cofibrant cospan from X to Y is a diagram \(i:X \rightarrow M \leftarrow Y:j\) such that \(\langle i, j\rangle :X \sqcup Y \rightarrow M\) is a closed cofibration. (The map \(\langle i,j\rangle \) is obtained via the universal property of the coproduct, see Diagram (2).)

For spaces \(X,Y\in {\textbf{Top}}\), we define the collection of all concrete cofibrant cospans from X to Y

Remark 3.2

The previous definition forces the images of i and j to be disjoint since a cofibration is a homeomorphism onto its image (Theorem 2.19).

Here i is a diffeomorphism from \(S^1\) to the boundary of \(D^2\), and j is a smooth embedding of \(S^1\) into the interior of the disk \(D^2\). We have that \(i:S^1 \rightarrow D^2 \leftarrow S^1:j\) is a concrete cofibrant cospan (Proposition 3.5)

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Example 3.3

Let X be a space. The cospan \(\textrm{id}_X:X \rightarrow X \leftarrow X:\textrm{id}_X\) is not a concrete cofibrant cospan, unless \(X=\varnothing \). This is clear from the previous remark.

Physically we expect the identity cospan to be (the equivalence class of) \(\iota ^X_0:X \rightarrow X\times {{\mathbb {I}}} \leftarrow X:\iota ^X_1\).

Proposition 3.4

For X a topological space, the cospan \(\iota ^X_0:X \rightarrow X\times {{\mathbb {I}}} \leftarrow X:\iota ^X_1\) is a concrete cofibrant cospan (where \(\iota ^X_a:X \rightarrow X \times {{\mathbb {I}}}\) is the map \(x\mapsto (x,a)\)).

Proof

The complement of the image of \(\langle \iota ^X_0, \iota ^X_1 \rangle :X\sqcup X \rightarrow X\times {{\mathbb {I}}}\) is \(X\times (0,1)\) which is open, so the image is a closed set.

We now show \(\langle \iota ^X_0, \iota ^X_1 \rangle :X\sqcup X \rightarrow X\times {{\mathbb {I}}}\) is a cofibration. Let K be any space and suppose we have a homotopy \(h:(X\sqcup X)\times {{\mathbb {I}}}\rightarrow K\). By Lemma 2.1 and Theorem 2.9, the product with \({{\mathbb {I}}}\) preserves colimits. Using this together with the universal property of the coproduct, the map \(h:(X\sqcup X)\times {{\mathbb {I}}}\rightarrow K\) is uniquely defined by a pair of maps \(h_0:X\times {{\mathbb {I}}}\rightarrow K\) and \(h_1:X\times {{\mathbb {I}}}\rightarrow K\).

Now suppose we have a map \(f:X \times {{\mathbb {I}}}\rightarrow K\) such that for all \(x\in X\) we have \(h_0(x,0)=f(x,0)\) and \(h_1(x,0)=f(x,1)\). (Notice this implies \(h({\tilde{x}},0)=f(\langle \iota _0^X,\iota _0^Y\rangle ({\tilde{x}}))\) for \({\tilde{x}}\in X\sqcup X\).) We can construct a homotopy \(H:(X \times {{\mathbb {I}}}) \times {{\mathbb {I}}}\rightarrow K\) which commutes with h and f as follows. Let \(L = \left\{ 0,1 \right\} \times {{\mathbb {I}}}\cup {{\mathbb {I}}}\times \left\{ 0\right\} \) be the subset of the unit square consisting of the two vertical edges and the bottom horizontal edge. Let \(\Gamma :{{\mathbb {I}}}\times {{\mathbb {I}}}\rightarrow L\) be a retraction sending the unit square to the subset L, see Example . We denote elements of \(X\times L\subset (X\times {{\mathbb {I}}})\times {{\mathbb {I}}}\) as triples (x, s, t) and define \(g:X\times L\rightarrow K\) as

By assumption these agree on the overlap and so g is continuous. Now define \(H:(X\times {{\mathbb {I}}}) \times {{\mathbb {I}}}\rightarrow K\) by \(g(x,\Gamma (s,t))\). \(\square \)

Proposition 3.5

(See Fig. 1.) Consider \(S^1\) and \(D^2\) as smooth manifolds. There is a concrete cofibrant cospan \(i:S^1 \rightarrow D^2 \leftarrow S^1:j\) where i is any diffeomorphism sending \(S^1\) to the boundary of \(D^2\), and j is any smooth embedding of \(S^1\) into the interior of \(D^2\).

Proof

The map \(\langle i,j\rangle :S^1\sqcup S^1\rightarrow D^2\) is the composition of a homeomorphism from \(S^1\sqcup S^1\) to \(i(S^1)\sqcup j(S^1)\), and an inclusion \(\iota :i(S^1)\sqcup j(S^1) \rightarrow D^2\). Proposition 2.26 gives that \(\iota \) is a cofibration, by Proposition 2.18 the homeomorphism is a cofibration, and by Proposition 2.15 the composition is a cofibration. \(\square \)

The following definition can be found in e.g. [30], where it is referred to as a bordism.

Definition 3.6

An n-dimensional concrete cobordism from an \((n-1)\)-dimensional smooth oriented manifold X to an \((n-1)\)-dimensional smooth oriented manifold Y, is an n-dimensional smooth oriented manifold M equipped with an orientation preserving diffeomorphism \(\phi :{\bar{X}}\sqcup Y \rightarrow \partial M\) (where the bar denotes the opposite orientation).

Proposition 3.7

There is a canonical way to map a concrete cofibration to a concrete cofibrant cospan. Precisely, let X, Y and M be smooth oriented manifolds, and let M be a concrete cobordism from X to Y. Hence there exists a diffeomorphism \(\phi :{\bar{X}}\sqcup Y\rightarrow \partial M\). Define maps \(i(x)=\phi (x,0)\) and \(j(y)=\phi (y,1)\). Then, using X, Y and M to denote the underlying topological spaces, \(i:X \rightarrow M \leftarrow Y:j\) is a concrete cofibrant cospan.

Proof

The pair \((M,\partial M)\) is cofibred by Proposition 2.25. The map \(\langle i,j \rangle \) is a homeomorphism onto its image \(\partial M\) as \(\phi \) is a diffeomorphism, hence, using Proposition 2.18, \(\langle i,j \rangle \) is a cofibration. The boundary \(\partial M\) is closed so \(\langle i,j \rangle \) a closed cofibration. \(\square \)

Here the grey lines represent the corners of the manifold \({{\mathbb {I}}}^3\), and the black lines represent an embedded submanifold \(M'\subset {{\mathbb {I}}}^3\). Let X, be the complement of \(M'\) in the bottom boundary, \({{\mathbb {I}}}^2\times \{0\}\), Y the complement in the top boundary, \({{\mathbb {I}}}^2\times \{1\}\), and M the complement in \({{\mathbb {I}}}^3\). Then there is a concrete cofibrant cospan \(i:X \rightarrow M \leftarrow Y:j\) where i and j are subspace inclusions

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Example 3.8

Consider the manifold (with corners) \({{\mathbb {I}}}^3\), and let \(M'\) be an embedded submanifold as illustrated by the black part of Fig. 2. Let \(M={{\mathbb {I}}}^3\setminus M'\), \(X=({{\mathbb {I}}}^2\times \{0\})\setminus (M\cap ({{\mathbb {I}}}^2\times \{0\}))\) and \(Y=({{\mathbb {I}}}^2\times \{1\})\setminus (M\cap ({{\mathbb {I}}}^2\times \{1\}))\), i.e. X is the complement of \(M'\) in bottom boundary in the figure and Y the top boundary. There is a concrete cofibrant cospan \(i:X \rightarrow M \leftarrow Y:j\) where i and j are subspace inclusions. We can see this by noticing that there are non-intersecting neighbourhoods of the top and bottom boundary of M that are homeomorphic to \(X\times [0,\epsilon ]\) and \(Y\times [0,\epsilon ']\) with \(\epsilon ,\epsilon '\in {{\mathbb {R}}}^{+}\). Thus an H and \(\phi \) satisfying the conditions of Theorem 2.21 can be constructed as in the proof of Proposition 2.25.

Example 3.9

Figure 3 represents a concrete cofibrant cospan. Proposition 2.25 gives that \(\langle i,j\rangle \) is a cofibration. Notice also that the boundary is a closed subset of M.

Let M be a manifold represented by the figure, with boundary homeomorphic to \(S^1\cup S^1\) at the bottom, and \(S^1\) at the top. Let X be the bottom boundary and Y the top boundary. Then there is a concrete cofibrant cospan \(i:X \rightarrow M \leftarrow Y:j\) where i and j are subspace inclusions

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Lemma 3.10

For any pair \(X,Y\in Ob({\textbf{Top}})\) there is a bijection

Proof

We first check \(\textrm{rev}\) is well defined. The image of \(\langle j,i\rangle \) is the same as the image of \(\langle i,j \rangle \) so it is a closed. We show it is a cofibration. Suppose we have a space K, and maps \(h:(Y\sqcup X)\times {{\mathbb {I}}}\rightarrow K\) and \(f:M\rightarrow K\) satisfying the commutativity conditions of Definition 2.13. The map h canonically determines a map \(h':(X\sqcup Y)\times {{\mathbb {I}}}\rightarrow K\). The map \(\langle i,j\rangle \) is a cofibration so we can apply the homotopy extension property to give a map \(H:M\times {{\mathbb {I}}}\rightarrow K\) which extends f and \(h'\). This H also commutes with f and h.

It is clear that \(\textrm{rev}\) is its own inverse, thus it is a bijection. \(\square \)

Lemma 3.11

If \(i:X \rightarrow M \leftarrow Y:j\) is a concrete cofibrant cospan, then \(i:X\rightarrow M\) and \(j:{Y}\rightarrow {M}\) are closed cofibrations.

Proof

The map \(i:X\rightarrow M\) is equal to the composition \(X\xrightarrow {\iota _1} X\sqcup Y \xrightarrow {\langle i,j\rangle } M\). The map \(\iota _1\) is a cofibration by Lemma 2.17 and the composition of cofibrations is a cofibration by Lemma 2.15, hence i is a cofibration.

We now prove that the image of X under the composition is closed in M. Here we use primes to denote images of \(\langle i,j\rangle \). The map \(\langle i,j\rangle \) is an embedding by Theorem 2.19, hence a homeomorphism onto its image, and it is straightforward to see that \(\iota _1(X)\) is closed in \(X\sqcup Y\). Hence there exists an open \(U\subseteq M\) with \(U\cap (X\sqcup Y)'= (X\sqcup Y)'\setminus \iota _1(X)'\). The image of \(X\sqcup Y\) is closed since \(\langle i,j\rangle \) is a closed cofibration, so \(M\setminus (X\sqcup Y)'\) is an open set. Thus there is an open set \(M{\setminus } (X\sqcup Y)'\cup U=M{\setminus } \iota _1(X)'\), hence the image of X under \(\langle i,j \rangle \circ \iota _1\) is closed.

The same argument gives that j is a closed cofibration. \(\square \)

Lemma 3.12

(I) For any spaces X, Y and Z in \(Ob({\textbf{Top}})\) there is a composition of concrete cofibrant cospans

where \({\tilde{i}}=p_M\circ i\) and \({\tilde{l}}=p_N\circ l\) are obtained via the following diagram

the middle square of which is the pushout of \(j:M \leftarrow Y \rightarrow N :k\) in Top.

(II) Hence there is a magmoid

We will also use notation for composition.

Proof

We need to prove that \(\langle {\tilde{i}}, {\tilde{l}} \rangle \; :X\sqcup Z \rightarrow M{ \sqcup _{Y} }N\) is a closed cofibration. We first check the map is closed. The image of \(\langle {\tilde{i}}, {\tilde{l}} \rangle \) is equal to \(p_M(i(X))\cup p_N(l(Y))\). Sets in \(M\sqcup _{Y} N\) are closed if the preimage under \(p_{M}\) and \(p_N\) is closed in M and N respectively. By Proposition 2.18, \(\langle i,j\rangle \) is a homeomorphism onto its image, hence we have \(i(X)\cap j(Y)=\varnothing \). This implies \(p_N^{-1}(p_M(i(X)))=\varnothing \), which is closed, and \(p_M^{-1}(p_M(i(X)))=i(X)\) is closed by Lemma 3.11. Hence \(p_M(i(X))\) is closed in \(M\sqcup _Y N\). Similarly \(p_N(l(Y))\) is closed.

We now check \(\langle {\tilde{i}}, {\tilde{l}} \rangle \) is cofibration. Define J to be the map obtained by taking either route around the pushout square:

We will prove that there exists a cofibration \(\big \langle \langle {\tilde{i}},J\rangle ,{\tilde{l}} \big \rangle :(X\sqcup Y) \sqcup Z \rightarrow M \sqcup _Y N \). By Lemma 2.15 and a straightforward extension of Lemma 2.17, this implies that the composition

which is equal to \(\langle {\tilde{i}}, {\tilde{l}} \rangle \), is a cofibration. Let K be a space and suppose we have maps \(f :M\sqcup _Y N \rightarrow K\) and \(h :((X\sqcup Y) \sqcup Z)\times {{\mathbb {I}}}\rightarrow K\) satisfying the commutation conditions of Definition 2.13. We construct a map \(H :(M\sqcup _Y N) \times {{\mathbb {I}}}\rightarrow K\) extending f and h as follows. First note that by Lemma 2.1 and Theorem 2.9, the product with \({{\mathbb {I}}}\) preserves coproducts and thus we have canonical isomorphisms, \(((X\sqcup Y) \sqcup Z)\times {{\mathbb {I}}}\cong ((X \times {{\mathbb {I}}}) \sqcup (Y \times {{\mathbb {I}}})) \sqcup (Z \times {{\mathbb {I}}})\) and \((M\sqcup _Y N) \times {{\mathbb {I}}}\cong M \times {{\mathbb {I}}}\sqcup _{Y\times {{\mathbb {I}}}} N\times {{\mathbb {I}}}\). Thus, by the universal property of the coproduct we have that the map h is in one to one correspondence with a triple of maps \(h_X :X \times {{\mathbb {I}}}\rightarrow K\), \(h_Y :Y \times {{\mathbb {I}}}\rightarrow K\) and \(h_Z :Z \times {{\mathbb {I}}}\rightarrow K\). Now using the homotopy extension property of \(\langle i,j\rangle \) on the maps \(\langle h_X,h_Y \rangle \) and the restriction of f to M, we obtain a map \({\mathcal {H}}_L :M \times {{\mathbb {I}}}\rightarrow K\). Similarly we obtain a map \({\mathcal {H}}_R :N \times {{\mathbb {I}}}\rightarrow K\). These two homotopies agree on the images of \(Y \times {{\mathbb {I}}}\) by construction so we can use the universal property of the pushout to obtain a map \(\langle {\mathcal {H}}_L,{\mathcal {H}}_R \rangle :M \times {{\mathbb {I}}}\sqcup _{Y\times {{\mathbb {I}}}} N\times {{\mathbb {I}}}\rightarrow K\) which, precomposed with the canonical isomorphism \((M\sqcup _Y N) \times {{\mathbb {I}}}\cong M \times {{\mathbb {I}}}\sqcup _{Y\times {{\mathbb {I}}}} N\times {{\mathbb {I}}}\), is a homotopy extending h. \(\square \)

3.2 Category of Cofibrant Cospans \(\textrm{CofCos}\)

Notice that the composition in \(\textsf{CofCos}\) is not strictly associative. Here we impose a congruence on concrete cofibrant cospans such that we obtain a category.

One option would be cospan isomorphism, by which we mean \(i:X \rightarrow M \leftarrow Y:j\) is equivalent to \(i':X \rightarrow N \leftarrow Y:j'\) if there exists a homeomorphism \(M\rightarrow N\) which commutes with the cospans. This is a direct analogue of the equivalence usually used for smooth manifold cobordisms in e.g. [30]. This equivalence would be sufficient to give an associative composition. However it will not be sufficient to ensure the cospan \(\iota ^X_0:X \rightarrow X\times {{\mathbb {I}}} \leftarrow X:\iota ^X_1\) behaves as an identity. (This is the image of a representative of the smooth manifold cobordism identity under the map described in Proposition 3.7.) One way to see this is by thinking about the cospan in Example 3.5: taking a pushout over the map from \(S^1\) into the interior of the disk, and the map into \(S^1 \times {{\mathbb {I}}}\) will not give a space homeomorphic to the disk. Hence we use a stronger equivalence relation.

Definition 3.13

For each pair \(X,Y\in Ob(\textsf{CofCos})\), we define a relation on \(\textsf{CofCos}(X,Y)\) by

if there exists a commuting diagram

where \(\psi \) is a homotopy equivalence.

Lemma 3.14

The relation \({\mathop {\sim }\limits ^{ch}}\) an equivalence relation.

Proof

Using the universal property of the coproduct we can rewrite the relation in terms of a homotopy equivalence \(\psi :M\rightarrow M'\) which is a map of spaces under \(X\sqcup Y\) from \(\langle i,j\rangle :X\sqcup Y\rightarrow M\) to \(\langle i',j'\rangle :X\sqcup Y\rightarrow M'\). Then since the maps \(X\sqcup Y\rightarrow M\) are defined to be cofibrations, Theorem 2.31 gives that this relation is precisely a cofibre homotopy equivalence of spaces under \(X\sqcup Y\), thus is an equivalence relation by Proposition 2.29. \(\square \)

Remark 3.15

The fact that, by Theorem 2.31, cospan homotopy equivalence is equivalent to cofibre homotopy equivalence of spaces under the disjoint union of the objects, will be used to prove that cospan homotopy equivalence yields a congruence. We could instead have defined cospan homotopy equivalence to be cofibre homotopy equivalence of spaces under the disjoint union of the objects. Then we would use Theorem 2.31 in the proof of the identity axiom instead.

We call the map \(\psi \) a cospan homotopy equivalence, and refer to an equivalence class of concrete cofibrant cospans as just a cofibrant cospan, denoted \(\left[ i:X \rightarrow M \leftarrow Y:j\right] _{\!\tiny \hbox {ch}}\). We have

Lemma 3.16

For each pair \(X,Y\in {\textbf{Top}}\) the relations \((\textsf{CofCos}(X,Y),{\mathop {\sim }\limits ^{ch}})\) are a congruence on \(\textsf{CofCos}\) and hence we have a magmoid

Proof

We have from Lemma 3.14 that the \({\mathop {\sim }\limits ^{ch}}\) are equivalence relations for each pair \(X,Y\in {\textbf{Top}}\), thus we only need to check that the relations respect composition.

Let \(i:X \rightarrow M \leftarrow Y:j\) and \(i':X \rightarrow M' \leftarrow Y:j'\) be two representatives of the same cofibrant cospan from X to Y and similarly let \(k:Y \rightarrow N \leftarrow Z:l\) and \(k':Y \rightarrow N' \leftarrow Z:l'\) be representatives of the same cofibrant cospan from Y to Z.

Using Theorem 2.31, and the universal property of the coproduct, we have the following commuting diagram where \(\phi ,\phi ',\psi \) and \(\psi '\) are cofibre homotopy equivalences between spaces under of X, Y and Z as shown.

This means there exists a homotopy under \(X\sqcup Y\), say \(H_{\phi }:M\times {{\mathbb {I}}}\rightarrow M\), from \(\phi '\circ \phi \) to the identity and a homotopy under \(Y\sqcup Z\), say \(H_{\psi }:N\times {{\mathbb {I}}}\rightarrow N\), from \(\psi '\circ \psi \) to the identity. In particular, for all \(y\in Y\), we have \(H_\phi (j(y),t)=j(y)\) and \(H_\psi (k(y),t)=k(y)\).

By the universal property of the pushout, the commuting pair \(p_{M'}\circ \phi \) and \(p_{N'}\circ \psi \) uniquely determine a map \(F:M\sqcup _{Y}N \rightarrow M' \sqcup _{Y} N'\) making the diagram commute. We will show F is a homotopy equivalence.

We can similarly construct a map \(F':M' \sqcup _{Y} N' \rightarrow M\sqcup _{Y}N\) using the pair \( p_{M}\circ \psi '\) and \( p_{N}\circ \phi '\). Notice the maps \( p_M\circ H_\phi \circ (j\times \textrm{id}_{{\mathbb {I}}}):Y\times {{\mathbb {I}}}\rightarrow M\sqcup _{Y}N\) and \(p_N\circ H_\psi \circ (k\times \textrm{id}_{{\mathbb {I}}}):Y\times {{\mathbb {I}}}\rightarrow M\sqcup _{Y}N\) commute using that for all \(y\in Y\) we have \(H_\psi (k(y),t)=k(y)\) and \(H_\phi (j(y),t)=j(y)\), and the commutativity of the diagram. Taking the product with \({{\mathbb {I}}}\) of the pushout of j and k is still a pushout, by Lemma 2.1. Using the universal property of this pushout on the maps \(p_M\circ H_\phi \) and \(p_N\circ H_\psi \) gives a map \((M\sqcup _Y N)\times {{\mathbb {I}}}\rightarrow M\sqcup _{Y} N\) which is a homotopy from \(F'\circ F\) to the identity functor.

In the same way we can construct a homotopy \(F\circ F'\) to the identity. \(\square \)

Theorem 3.17

The quadruple

is a category.

Proof

Note that is a magmoid by Lemma 3.16.

We first check that the given class is indeed an identity. Note that \(\iota _{0}^X:X \rightarrow X\times {{\mathbb {I}}} \leftarrow X:\iota _1^X\) is a concrete cofibrant cospan by Proposition 3.4. Suppose we have a cofibrant cospan represented by \(i:X \rightarrow M \leftarrow Y:j\). We will show there is a cospan homotopy equivalence from to \(i:X \rightarrow M \leftarrow Y:j\). Consider the following diagram.

The map \(\phi \) is constructed using the universal property of the pushout. By construction \(\phi \) commutes with the cospans and \(i:X \rightarrow M \leftarrow Y:j\). We claim \(\phi \) is a homotopy equivalence with homotopy inverse \(p_M\). It is by construction that \(\phi \circ p_M=\textrm{id}_M\).

We construct a homotopy \( p_M \circ \phi \rightarrow \textrm{id}_{M\sqcup _Y(Y\times {{\mathbb {I}}})}\) as follows. Since \(M\sqcup _Y(Y\times {{\mathbb {I}}})\) is a pushout, the map \(p_M \circ \phi \) is uniquely determined by the pair of maps \(M\rightarrow M\sqcup _Y(Y\times {{\mathbb {I}}})\), \(m\mapsto p_M(m)\) and \(Y\times {{\mathbb {I}}}\rightarrow M\sqcup _Y(Y\times {{\mathbb {I}}})\), \((y,t)\mapsto p_M(j(y))\), or equivalently \((y,t)\mapsto p_{Y\times {{\mathbb {I}}}}(\iota _{0}^Y(y))\). Similarly the identity is determined by the pair \(M\rightarrow M\sqcup _Y(Y\times {{\mathbb {I}}})\), \(m\mapsto p_M(m)\) and \(Y\times {{\mathbb {I}}}\rightarrow M\sqcup _Y(Y\times {{\mathbb {I}}})\), \((y,t)\mapsto p_{Y\times {{\mathbb {I}}}}(y,t)\). The map \(H_{Y\times {{\mathbb {I}}}} :(Y\times {{\mathbb {I}}}) \times {{\mathbb {I}}}\rightarrow M\sqcup _Y (Y \times {{\mathbb {I}}})\), \(((y,t),s) \mapsto p_{Y\times {{\mathbb {I}}}}(y,ts)\) is a homotopy between the two maps from \(Y\times {{\mathbb {I}}}\). And for M we can use the homotopy \(H_M:M\times {{\mathbb {I}}}\rightarrow M\sqcup _Y (Y \times {{\mathbb {I}}})\), \((m, t) \mapsto p_M(m)\).

By Lemma 2.1 the product with \({{\mathbb {I}}}\) preserves pushouts. Notice that \(H_M\circ (j\times \textrm{id}):Y\times {{\mathbb {I}}}\rightarrow M\sqcup _Y (Y \times {{\mathbb {I}}})\) is \((y,t)\mapsto p_M(j(y))\) and \(H_{Y\times {{\mathbb {I}}}}\circ (\iota _0^Y\times \textrm{id}):Y\times {{\mathbb {I}}}\rightarrow M\sqcup _Y (Y \times {{\mathbb {I}}})\) is \((y,s)\mapsto p_{Y\times {{\mathbb {I}}}} (\iota _0^Y(y))\), so we can use the universal property of the pushout of \(j\times \textrm{id}\) and \(\iota _0^Y\times \textrm{id}\) to obtain a homotopy \({\mathcal {H}} :(M\sqcup _Y (Y \times {{\mathbb {I}}})) \times {{\mathbb {I}}}\rightarrow M\sqcup _Y (Y \times {{\mathbb {I}}})\) from \(p_M \circ \phi \) to \(\textrm{id}_{M\sqcup _Y(Y\times {{\mathbb {I}}})}\).

We can similarly construct a cospan homotopy equivalence to \(i:X \rightarrow M \leftarrow Y:j\).

We now check that the composition is associative. Let \(i:W \rightarrow M \leftarrow X:j\), \(k:X \rightarrow N \leftarrow Y:l\) and \(r:Y \rightarrow O \leftarrow Z:s\) be concrete cofibrant cospans. The two ways to compose these three cospans corresponds to taking a pushout first over X or first over Y as shown in the following diagram

We can use the universal property of the pushout on the pair of maps \(M\rightarrow M\sqcup _X \left( N\sqcup _Y O \right) \) and \(N\rightarrow N\sqcup _YO\rightarrow M\sqcup _X \left( N\sqcup _Y O \right) \) to obtain a map \( M\sqcup _X N\rightarrow M\sqcup _X \left( N\sqcup _Y O \right) \). We can then apply the universal property again to this map \( M\sqcup _X N\rightarrow M\sqcup _X \left( N\sqcup _Y O \right) \) and the map \(O\rightarrow N\sqcup _YO\rightarrow M\sqcup _X \left( N\sqcup _Y O \right) \) to obtain a map \(\left( M\sqcup _X N\right) \sqcup _Y O \rightarrow M\sqcup _X \left( N\sqcup _Y O \right) \) which commutes with the diagram. In a similar way we can obtain an inverse \(M\sqcup _X \left( N\sqcup _Y O \right) \rightarrow \left( M\sqcup _X N\right) \sqcup _Y O\). \(\square \)

Let \(i:X \rightarrow M \leftarrow Y:j\) and \(k:Y \rightarrow N \leftarrow Z:l\) be concrete cofibrant cospans. In an attempt to avoid excessive notation, from here we may use i and l to refer also to the maps \({\tilde{i}}=p_M\circ i\) and \({\tilde{l}}=p_N\circ l\) obtained in the composition.

Proposition 3.18

The family of bijections \(\textrm{rev}_{X,Y}:\textsf{CofCos}(X,Y)\rightarrow \textsf{CofCos}(Y,X)\) from Lemma 3.10 extend to an involutive functor

Proof

Lemma 3.10 gives that \(\textrm{rev}\) is well defined, and that it is its own inverse. To show composition is preserved, let \(i:X \rightarrow M \leftarrow Y:j\) and \(k:Y \rightarrow N \leftarrow Z:l\) be concrete cofibrant cospans. Then the universal property of the pushout gives an isomorphism between \(M\sqcup _Y N\) and \(N\sqcup _Y M\), which gives a cospan homotopy equivalence from to . \(\square \)

3.2.1 Monoidal Structure on \(\textrm{CofCos}\)

We now construct a functor from \(\textrm{CofCos}\times \textrm{CofCos}\) to \(\textrm{CofCos}\), and show that this leads to a symmetric monoidal category with underlying category \(\textrm{CofCos}\).

Lemma 3.19

There is a functor

where \(i\sqcup j\) is the image of a pair of maps under the monoidal product on \({\textbf{Top}}\) (where \({\textbf{Top}}\) is as in Sect. 2.1.1).

Proof

We first check that \(i\sqcup k:W\sqcup Y \rightarrow M\sqcup N \leftarrow X\sqcup Z:j\sqcup l\) is a concrete cofibrant cospan. In particular we show that the map \(\langle i\sqcup k,j\sqcup l\rangle :(W\sqcup Y)\sqcup (X\sqcup Z)\rightarrow M\sqcup N\) is a closed cofibration. Let K be a space and suppose we have maps \(h:((W\sqcup Y)\sqcup (X\sqcup Z))\times {{\mathbb {I}}}\rightarrow K\) and \(f:M\sqcup N\rightarrow K\) satisfying the commutation conditions of Definition 2.13. By Lemma 2.1, the product with \({{\mathbb {I}}}\) preserves colimits so the map h uniquely determines a pair \(h':(W\sqcup Y)\times {{\mathbb {I}}}\rightarrow K\) and \(h'':(X\sqcup Z)\times {{\mathbb {I}}}\rightarrow K\). Similarly the map f determines maps \(f':M\rightarrow K\) and \(f'':N\rightarrow K\). We can use the homotopy extension property of \(\langle i,j\rangle \) on the pair \(h'\) and \(f'\) to obtain a map \(H':M\times {{\mathbb {I}}}\rightarrow K\) and similarly of \(\langle k,l\rangle \) on the pair \(h''\) and \(f''\) to obtain \(H'':N\times {{\mathbb {I}}}\rightarrow K\). Now using Lemma 2.1 again, \(H'\) and \(H''\) determine uniquely a map \(H:(M\sqcup N)\times {{\mathbb {I}}}\rightarrow K\) extending f and h. The image of \(\langle i\sqcup k,j\sqcup l\rangle \) is the union of the images of \(\langle i,j\rangle \) and \(\langle k,l\rangle \), thus is closed.

We now check that the monoidal product respects the equivalence relation. Suppose we have a concrete cofibrant cospan \(i':W \rightarrow M' \leftarrow X:j'\) which is cospan homotopy equivalent to \(i:W \rightarrow M \leftarrow X:j\) via some cospan homotopy equivalence \(\phi :M\rightarrow M'\), and a concrete cofibrant cospan \(k':Y \rightarrow N' \leftarrow Z:l'\) equivalent to \(k:Y \rightarrow N \leftarrow Z:l\) via cospan homotopy equivalence \(\psi :N\rightarrow N'\). It follows that there exist homotopy inverses \(\phi '\) of \(\phi \) and \(\psi '\) of \(\psi \), and that the following diagram commutes.

Using the universal property of the coproduct on the appropriate homotopies, it is straightforward to check that \(\phi \sqcup \psi \) is a homotopy equivalence with homotopy inverse \(\phi '\sqcup \psi '\).

We now check that \(\otimes \) is a functor, starting with checking that \(\otimes \) preserves identities. Let X and Y be any spaces. The canonical isomorphism \((X\sqcup Y)\times {{\mathbb {I}}}\rightarrow (X\times {{\mathbb {I}}})\sqcup (Y\times {{\mathbb {I}}})\), which in particular is a homotopy equivalence, is sufficient to show that \(\iota _0^X\sqcup \iota ^Y_0:X\sqcup Y \rightarrow (X\times {{\mathbb {I}}})\sqcup (Y\times {{\mathbb {I}}}) \leftarrow X\sqcup Y:\iota _0^X\sqcup \iota _1^Y\) is cospan homotopy equivalent to \(\iota _0^{X\sqcup Y}:X\sqcup Y \rightarrow (X\sqcup Y)\times {{\mathbb {I}}} \leftarrow X\sqcup Y:\iota _1^{X\sqcup Y}\).

Finally we check that \(\otimes \) preserves composition. Given two pairs of composable concrete cofibrant cospans, there are distinct cospans obtained from first appplying \(\otimes \) and then composing and from composing and then applying \(\otimes \). A commuting isomorphism is constructed between these cospans using the universal properties of the coproduct and the pushout. \(\square \)

To construct a monoidal structure on \(\textrm{CofCos}\), we will give all associators and unitors in the form of the cospan in the following lemma.

Lemma 3.20

Let X and \(X'\) be spaces and \(f:X\rightarrow X'\) a homeomorphism. Then the cospan

is a concrete cofibrant cospan and its cospan homotopy equivalence class is an isomorphism in \(\textrm{CofCos}\).

(Recall that \(\iota ^X_a:X \rightarrow X \times {{\mathbb {I}}}\) is the map \(x\mapsto (x,a)\).)

Proof

We first prove that the cospan is a concrete cofibrant cospan. Note that the map \(\langle \iota _0^{X'}\circ f, \iota _1^{X'}\rangle \) is equal to the composition

The first map is a homeomorphism; hence it is a cofibration by Lemma 2.16. We proved that the second map is a cofibration in Lemma 3.4. Hence, by Lemma 2.15, the composition is a cofibration. Since the first map is a homeomorphism, the image of the composition is equal to the image of the second map, so is closed by Lemma 3.4.

To see that the cospan homotopy equivalence class is an isomorphism notice that the composition

is equivalent to \(\iota _0^{X'}\circ f:X \rightarrow X'\times {{\mathbb {I}}} \leftarrow X:\iota _1^{X'}\circ f\) via the obvious isomorphism \(X'\times {{\mathbb {I}}}\cong (X'\times {{\mathbb {I}}})\sqcup _{X'}(X'\times {{\mathbb {I}}})\), which is equivalent to \(\iota _0^X:X \rightarrow X\times {{\mathbb {I}}} \leftarrow X:\iota _1^X\) via the homeomorphism \(f\times \textrm{id}_{{{\mathbb {I}}}}:X\times {{\mathbb {I}}}\rightarrow X'\times {{\mathbb {I}}}\). \(\square \)

Lemma 3.21

Recall the monoidal category \(({\textbf{Top}},\sqcup ,\varnothing ,\alpha _{X,Y,Z}^T,\lambda _X^T,\rho _X^T)\) from Sect. 2.1.1. There is a monoidal category

where \(\otimes \) is as in Lemma 3.19,

• for any spaces \(X,Y,Z\in Ob(\textrm{CofCos})\), \(\alpha _{X,Y,Z}:(X\sqcup Y)\sqcup Z \rightarrow X \sqcup (Y\sqcup Z)\) is the cospan homotopy equivalence of the cospan

  

• for any space \(X\in Ob(\textrm{CofCos})\), \(\lambda _X:\varnothing \sqcup X \rightarrow X\) is the cospan homotopy equivalence class of the cospan

  

• for any space \(X\in Ob(\textrm{CofCos})\), \(\rho _X:X\sqcup \varnothing \rightarrow X\) is the cospan homotopy equivalence class of the cospan

  

Proof

First note that Lemma 3.20 gives that all associators and unitors are isomorphisms.

The proofs of all required identities are similar, so we only give the proof that \((\rho _X\otimes 1_Y)=(1_X\otimes \lambda _Y)\alpha _{X,\varnothing , Y}\) here.

We must construct a cospan homotopy equivalence from the composition

to the cospan

By the universal property of the coproduct and Lemma 2.1, a map \(f:(X\times {{\mathbb {I}}})\sqcup ((\varnothing \times {{\mathbb {I}}})\sqcup (Y\times {{\mathbb {I}}}))\rightarrow (X\times {{\mathbb {I}}})\sqcup (Y\times {{\mathbb {I}}})\) is uniquely determined by

and

Similarly a map \(g:(X\times {{\mathbb {I}}})\sqcup (Y\times {{\mathbb {I}}})\rightarrow (X\times {{\mathbb {I}}})\sqcup (Y\times {{\mathbb {I}}})\) is determined by the pair

and

We have that \(f\circ \iota _1^{X\sqcup (\varnothing \sqcup Y)}= g\circ \iota _0^X\sqcup (\iota _0^Y\circ \lambda ^T_Y)\) commute, so by the universal property of the pushout, these maps determine a map

which is a homeomorphism, and it is straightforward to check this commutes with the cospans, hence is a cospan homotopy equivalence. \(\square \)

Theorem 3.22

There is a symmetric monoidal category

where \((\textrm{CofCos},\otimes ,\varnothing ,\alpha _{X,Y,Z},\lambda _{X},\rho _{X})\) is as in Lemma 3.21, and for any spaces \(X,Y\in Ob(\textrm{CofCos})\), \(\beta _{X,Y} :X\otimes Y \rightarrow Y\otimes X \) is the cospan homotopy equivalence class of the cospan

where \(\beta ^T_{X,Y}\) is the braiding in \({\textbf{Top}}\) as in Sect. 2.1.1

By abuse of notation we will refer to this symmetric monoidal category as \(\textrm{CofCos}\).

Proof

As with the previous theorem, the proofs of all necessary identities are similar. Here we give the proof that \(\beta \) is symmetric.

We must construct a cospan homotopy equivalence from the composition

to the cospan

Define maps

and

Note that \(f_1\circ \iota _1^{Y\sqcup X}=f_2\circ (\iota _0^{X\sqcup Y}\circ \beta ^T_{Y,X})\), hence applying the universal property of the pushout determines a map

Notice that f is a homeomorphism, and it is straightforward to check that it commutes with the cospans, and so is a cospan homotopy equivalence. \(\square \)

Remark 3.23

Denote by \({\textbf{Top}}^h\) is the wide subcategory of \({\textbf{Top}}\) where all maps are homeomorphisms. There is a functor \(\kappa :{\textbf{Top}}^h \rightarrow \textrm{CofCos}\), which sends a homeomorphism \(f:X\rightarrow Y\) to the cospan homotopy equivalence class of the cospan \(\iota _0^Y\circ f:X \rightarrow Y\times {{\mathbb {I}}} \leftarrow Y:\iota ^Y_1\). Aside from naturality, the identities required to to prove \(\textrm{CofCos}\) becomes a monoidal category, as well as the identities required to prove braiding and symmetry, then commute in \(\textrm{CofCos}\) as they are precisely the images of the corresponding identities in \({\textbf{Top}}\). The same construction leads to a map from the classical mapping class group of a manifold into \(\textrm{CofCos}\). This is why TQFTs give representations of mapping class groups. A related mapping leads to a functor from the mapping class groupoid of a space X into \(\textrm{CofCos}\). See Sect. 4 for more.

3.3 Category of Homotopy Cobordisms \({\textrm{HomCob}}\)

Here we construct the symmetric monoidal category \({\textrm{HomCob}}\) (Theorem 3.36), which we will use as the source category of the TQFT we construct in Sect. 5. We obtain \({\textrm{HomCob}}\) as a subcategory of \(\textrm{CofCos}\) with a finiteness condition on spaces.

Definition 3.24

A space X is called homotopically 1-finitely generated if \(\pi (X,A)\) is finitely generated for all finite sets of basepoints A.

Let \(\chi \) denote the class of all homotopically 1-finitely generated spaces.

Lemma 3.25

There exists a submagmoid

of \(\textsf{CofCos}\) where

Morphisms in \(\textsf{HomCob}\) are called concrete homotopy cobordisms.

Proof

We check \(\textsf{HomCob}\) is closed under composition. Suppose \(i:X \rightarrow M \leftarrow Y:j\) and \(k:Y \rightarrow N \leftarrow Z:l\) are concrete homotopy cobordisms. Consider the pushout

We may choose finite representative subsets \(Y_0\subseteq Y\), \(M_0\subseteq M\) and \(N_0\subseteq N\) such that \(j(Y_0)= M_0\cap j(Y)\) and \(k(Y_0)= N_0\cap k(Y)\). Applying Corollary 2.34 the following square is also a pushout.

We have, from Theorem 2.12, that the pushout of finitely generated groupoids is finitely generated, so that \(\pi (M{ \sqcup _{Y} } N,M_0{ \sqcup _{Y_0} } N_0)\) is finitely generated follows from the fact that \(\pi (M,M_0)\) and \(\pi (N,N_0)\) are. Hence the composition is a concrete homotopy cobordism. \(\square \)

To give specific examples of concrete homotopy cobordism we will use the following result which says that, to check a space X is homotopically 1-finitely generated, it will be sufficient to find a single representative subset \(A\subseteq X\) such that \(\pi (X,A)\) is finitely generated.

Lemma 3.26

If \(\pi (X,A)\) is finitely generated for some finite representative set A, then \(\pi (X,A')\) is finitely generated for all finite representative sets \(A'\).

Proof

Let \(A=\{a_1,...,a_N\}\subset X\) be a finite representative subset, and suppose \(\pi (X,A)\) is finitely generated. Then there exists a finite set of generating morphisms. Let \(S=\{s_1,...,s_K\}\) be a finite set of representative paths, such that taking path-equivalence classes of each path gives a set of generating morphisms for \(\pi (X,A)\).

Let \(B=\{b_1,...,b_M\}\subset X\) be another finite representative subset. For each pair \(\{n,m\}\) such that \(a_n\) and \(b_m\) are in the same path connected component, choose a path \(\gamma _{n,m} :a_n \rightarrow b_m\). We denote the set of all such paths by \(\Gamma \). Note that this is finite since A and B are finite. We will show that \(\pi (X,B)\) is generated by the set of path-equivalence classes of all paths of the form \(\gamma _{n',m'} s\gamma _{n,m}^{-1}\), where \(s\in S\), \(s_0=a_n\) and \(s_1=a_{n'}\). Note this is again finite.

Let \(t:b_m\rightarrow b_{m'}\) be any path. Choose n such that \(a_n\) is in the same path connected component as \(b_m\), then \({\tilde{t}}=\gamma ^{-1}_{n,m'}t \gamma ^{}_{n,m}:a_n\rightarrow a_n\) is a path and \(\gamma ^{}_{n,m'}{\tilde{t}} \gamma ^{-1}_{n,m}{\mathop {\sim }\limits ^{p}}t\). Now we have \({\tilde{t}}\sim p_L\dots p_2p_1\), a finite sequence of \(p_l \in S\), since the equivalence classes of the \(s_k\) generate \(\pi (X,A)\). Hence \(t{\mathop {\sim }\limits ^{p}}\gamma _{n,m'}^{} p_L...p_2p_1\gamma _{n,m}^{-1}\). For each \(p_l\), choose a path \(\gamma _{p_l}\in \Gamma \) such that \((\gamma _{p_l})_0=(p_l)_1\), so \(\gamma ^{-1}_{p_l}\gamma _{p_l}^{}{\mathop {\sim }\limits ^{p}}e_{(p_l)_1}\), the identity path. Now \(t{\mathop {\sim }\limits ^{p}}\gamma _{n,m'}^{}p_L\gamma _{p_{L-1}}^{-1}...\gamma _{p_2}^{}p_2\gamma _{p_1}^{-1}\gamma _{p_1}^{}p_1\gamma _{n,m}^{-1}\), which is of the desired form. \(\square \)

Example 3.27

The concrete cofibrant cospan from Proposition 3.5 is a concrete homotopy cobordism, as the fundamental groups of \(D^2\) and of \(S^1\) are finitely generated.

Example 3.28

The concrete cofibrant cospan, \(i:X \rightarrow M \leftarrow Y:j\), in Example 3.9 is a concrete homotopy cobordism. We have \(X\cong S^1\sqcup S^1\), hence, letting \(X_0\) be a subset with a single point in each path connected component, \(\pi (X,X_0)\cong {\mathbb {Z}}\sqcup {\mathbb {Z}}\). Similarly \(Y\cong S^1\), so \(\pi (Y,\{y\})\cong {\mathbb {Z}}\) for any \(y\in Y\). The manifold M is a homotopy equivalent to the twice punctured disk, hence has fundamental group \(\pi (M,\{m\})\cong {\mathbb {Z}}*{\mathbb {Z}}\). Hence X, Y and M are homotopically 1-finitely generated.

Example 3.29

The concrete cofibrant cospan, \(i:X \rightarrow M \leftarrow Y:j\), in Example 3.8 is a concrete homotopy cobordism. The space X is homotopy equivalent to the disjoint union of two copies of the open disk and a twice punctured disk. Thus, choosing \(X_0\subset X\) with a point in each connected component, it is clear that \(\pi (X,X_0)\) is finitely generated. The space Y is homotopy equivalent to the disjoint union of the circle and the open disk. Choosing \(Y_0\) in the same way implies that \(\pi (Y,Y_0)\) finitely generated. The space M is the disjoint union of a contractible space, and a space which is homotopy equivalent to a 3 punctured sphere and thus via a stereographic projection, homotopy equivalent to the twice punctured disk. Hence \(\pi (M,M_0)\) is finitely generated for any choice of \(M_0\subset M\) finite.

To see the homotopy equivalence to the punctured sphere, notice that a ball with the centre removed is equivalent to the sphere, and this can be seen by choosing a homotopy equivalence which sends each point in the ball to the closest point on the sphere. Now consider the ball with three line segments removed, each with one end point on the boundary and the other at the centre of the ball. The same homotopy equivalence sends this space to the 3 punctured sphere. Notice that this extends to a homotopy gradually pushing points to the boundary, and at some point during this homotopy, the image will be a space homeomorphic to the space represented in Fig. 2.

Example 3.30

Let \(\Gamma \) be a finite graph. Choose disjoint sets \(V_1,V_2\subseteq V(\Gamma )\) of vertices. Then \(i:V_1 \rightarrow \Gamma \leftarrow V_2:j\) is a concrete homotopy cobordism where i and j are inclusions. To see that \(\langle i,j\rangle \) is a closed cofibration, we can think of \(\Gamma \) as a CW complex, and \(V_1\cup V_2\) a subcomplex, and then apply Proposition 2.27. That the spaces are homotopically 1-finitely generated can be seen by taking the set of basepoints to be all vertices, and generating paths to be edges.

Example 3.31

Let M be a CW complex, and X and Y disjoint subcomplexes. Then \(i:X \rightarrow M \leftarrow Y:j\), where i and j are inclusions, is a concrete homotopy cobordism. Proposition 2.27 gives that \(\langle i, j\rangle \) is a closed cofibration. That finitely generated CW complexes have finite fundamental group follows from Proposition 1.26 of [21].

Definition 3.32

A cofibrant cospan is called a homotopy cobordism if there exists a representative which is a concrete homotopy cobordism.

For homotopically 1-finitely generated spaces \(X,Y\in {\textbf{Top}}\) define

Notice that if \(i:X \rightarrow M \leftarrow Y:j\) is a concrete cofibrant cospan with all spaces homotopically 1-finitely generated, then it is clear from the definition of cospan homotopy equivalence that every cospan in the equivalence class also has all spaces homotopically 1-finitely generated.

Theorem 3.33

There is a subcategory of \(\textrm{CofCos}\) (Thm 3.17)

with

• all homotopically 1-finitely generated spaces as objects;

• for spaces \(X,Y\in Ob({\textrm{HomCob}})\), morphisms in \({\textrm{HomCob}}(X,Y)\) are homotopy cobordisms i.e. cospan homotopy equivalence classes (Lem 3.14) of cospans

  

  with all spaces homotopically 1-finitely generated and \(\langle i, j \rangle \) a closed cofibration;

• composition is as follows

  

  where \({\tilde{i}}=p_Mi\) and \({\tilde{l}}=p_Nl\) are obtained via the pushout diagram

  

• for a space \(X\in Ob({\textrm{HomCob}})\) the identity morphism is the equivalence class of the cospan

  

Proof

We have from Lemma 3.25 that is a magmoid, in particular the composition is closed. It remains only to prove that all identities are in \({\textrm{HomCob}}\).

Let X be a homotopically 1-finitely generated space. Then \(X\times {{\mathbb {I}}}\) is homotopy equivalent to X, and so \([\iota _0^X:X \rightarrow X\times {{\mathbb {I}}} \leftarrow X:\iota _1^X]\) is a homotopy cobordism. \(\square \)

Proposition 3.34

Let \(\textbf{Cob}(n)\) be the category where objects are \((n-1)\)-dimensional closed oriented smooth manifolds and morphisms are equivalence classes of concrete cobordisms (Definition 3.6) as in [30, Ch. 1]. For all \(n\in {\mathbb {N}}\) there is a functor

which maps objects to their underlying space and maps a morphism to the equivalence class of the concrete cofibrant cospan which is the image of a representative cobordism under the mapping described in Proposition 3.7.

Proof

We first check that \(\textrm{Cob}_{n}\) is well defined. Chapter 6 of [24] proves that compact smooth manifolds have the homotopy type of finite CW complexes (see in particular the start of Sect. 3 and Theorems 1.2 and 4.1). If we choose the set of basepoints to be the 0-cells of the corresponding CW complex, then the generators of the fundamental groupoid are the 1-cells and so the fundamental groupoids of smooth manifolds with a finite set of basepoints are finitely generated. If two concrete cobordisms are equivalent up to boundary preserving diffeomorphism then they are certainly equivalent up to cofibre homotopy equivalence using the same map. So we have that the functor is well defined.

Let X, Y, Z be a triple of objects in \(\textbf{Cob}(n)\) and \(M:X\rightarrow Y\), \(M':Y\rightarrow Z\) a pair of cobordisms. Then we have maps \(\phi :X\sqcup Y \rightarrow M\) and \(\phi ':Y\sqcup Z\rightarrow M'\) between the underlying topological spaces. The image of the composition in \(\textbf{Cob}(n)\) is the cospan \(i:X \rightarrow M\sqcup N/((y,0)\sim (y,1)) \leftarrow Z:j\) where \(i(x)=\phi (x,0)\) and \(j(y)=\phi '(z,1)\). This is precisely the composition of the images of \(M:X\rightarrow Y\) and \(M':Y\rightarrow Z\) in \({\textrm{HomCob}}\).

The identity for a manifold X in \(\textbf{Cob}(n)\) is represented by the cylinder \(X\times {{\mathbb {I}}}\) with \(\langle \iota ^X_0, \iota _1^X \rangle :{\bar{X}}\sqcup X\rightarrow X\times {{\mathbb {I}}}\), this clearly maps to a representative of the identity cospan of X. \(\square \)

3.3.1 Monoidal Structure on \({\textrm{HomCob}}\)

The category \({\textrm{HomCob}}\) becomes a symmetric monoidal category, just like \(\textrm{CofCos}\).

We will need the following result about homotopically 1-finitely generated spaces to ensure \(\otimes \) restricts to a closed composition in \({\textrm{HomCob}}\).

Lemma 3.35

If X and Y are homotopically 1-finitely generated spaces, then \(X\sqcup Y\) is homotopically 1-finitely generated.

Proof

Suppose \(X_0\) and \(Y_0\) are finite representative subsets of X and Y respectively. The images of X and Y in \(X\sqcup Y\), under the maps into the coproduct, are disjoint, hence there is an isomorphism \(\pi (X\sqcup Y,X_0 \sqcup Y_0)\cong \pi (X,X_0)\amalg \pi (Y,Y_0)\) of groupoids given by sending a path equivalence class \([\gamma ]\) to \(([\gamma ],1)\) if \(\gamma \) is a path in X and to \(([\gamma ],2)\) if \(\gamma \) is a path in Y. By Theorem 2.12 we have that \(\pi (X,X_0)\amalg \pi (Y,Y_0)\) is finitely generated if and only if \(\pi (X,X_0)\) and \(\pi (Y,Y_0)\) are. Notice all finite representative subsets of \(X\sqcup Y\) are of the form \(X_0\sqcup Y_0\), so this is sufficient. \(\square \)

Theorem 3.36

There is a symmetric monoidal subcategory

of \(\textrm{CofCos}\). Here \(\otimes \) is as in Lemma 3.19, the \(\alpha _{X,Y,Z}\), \(\lambda _{X}\), \(\rho _{X}\) are as in Lemma 3.21 and the \(\beta _{X,Y}\) are as in Lemma 3.22.

Proof

The empty set is homotopically 1-finitely generated. For each pair of homotopically 1-finitely generated spaces, the disjoint union is homotopically 1-finitely generated by Lemma 3.35 so \(\otimes \) sends a pair of homotopy cobordisms to a homotopy cobordism.

Using again Lemma 3.35 along with the fact that for any space X and finite representative \(A\subseteq X\) we have \(\pi (X,A)\cong \pi (X\times {{\mathbb {I}}}, A\times \{0\})\), and \(A\times \{0\}\) is representative in \(X\times {{\mathbb {I}}}\), thus the \(\alpha _{X,Y,Z}\), \(\lambda _{X}\), \(\rho _{X}, \beta _{X,Y}\) are all in \({\textrm{HomCob}}\). \(\square \)

Proposition 3.37

The functor \(\textrm{Cob}_{n}:\textbf{Cob}_n\rightarrow {\textrm{HomCob}}\) as in Proposition 3.34 is a symmetric strong monoidal functor with \((\textrm{Cob}_{n})_0=\left[ \varnothing :\varnothing \rightarrow \varnothing \leftarrow \varnothing :\varnothing \right] _{\!\tiny \hbox {ch}}\) and \((\textrm{Cob}_{n})_2(X,Y)=\left[ \iota _0^{X\sqcup Y}:X\sqcup Y \rightarrow (X\sqcup Y)\times {{\mathbb {I}}} \leftarrow X\sqcup Y:\iota _1^{X\sqcup Y}\right] _{\!\tiny \hbox {ch}}\).

Proof

The monoidal product \(\otimes '\) in \(\textbf{Cob}(n)\) is given by disjoint union, thus we have \(\otimes \circ (\textrm{Cob}_{n} \times \textrm{Cob}_{n})=\textrm{Cob}_{n} \circ \otimes '\) and thus the appropriate identity is the required natural transformation. It is straightforward to check all necessary identities as \((\textrm{Cob}_{n})_0\) and \((\textrm{Cob}_{n})_2\) are identity morphisms, and the functor \(\textrm{Cob}_{n}\) maps all associators, unitors and braidings to exactly the corresponding associators, unitors and braidings in \({\textrm{HomCob}}\). \(\square \)

In this section we construct functors into \({\textrm{HomCob}}\) from (subgroupoids of) the motion groupoid \(\textrm{Mot}_{M}^{A}\) and from the mapping class groupoid \(\textrm{MCG}_{M}^{A}\) of a manifold M and a subset \(A\subset M\), as constructed in [32]. Thus functors from \({\textrm{HomCob}}\) into \({\textbf{Vect}}_{{\mathbb {C}}}\) give representations of motion groupoids and mapping class groupoids.

We briefly recall each of these constructions below. For a point of reference note that, for \(K\subset D^2\) a finite subset in the 2-disk, both \(\textrm{Mot}_{D^2}^{\partial D^2}(K,K)\) and \(\textrm{MCG}_{D^2}^{\partial D^2}(K,K)\) are isomorphic to the Artin braid group on |K| strands [3]. Similarly, considering \(L\subset B^3\) a configuration of unknotted, unlinked circles in the 3-ball, we have \(\textrm{Mot}_{B^3}^{\partial B^3}(L,L)\) and \(\textrm{MCG}_{B^3}^{\partial B^3}(L,L)\) are isomorphic to the corresponding loop braid group as in [14].

Let \({\textbf{Top}}^h\subset {\textbf{Top}}\) denote the subcategory of homeomorphisms. Then let \({\textbf{TOP}}^h(X,Y)\) denote the space with underlying set \({\textbf{Top}}^h(X,Y)\) and the compact open topology (see e.g. [16, Sec.2.4]). Let X be a space and \(A\subset X\) a subset, denote by \({\textbf{TOP}}^h_A(X,X)\subset {\textbf{TOP}}^h(X,X)\) the subspace containing maps \(f:X\rightarrow X\) such that \(f(a)=a\) for all \(a\in A\). For X a set, \({{\mathcal {P}}}X\) denotes the power set of X.

4.1 Functor from the Motion Groupoid, \(\mathcal {MOT}_{M}^{A}:\textrm{hf}\textrm{Mot}_{M}^{A} \rightarrow {\textrm{HomCob}}\)

We briefly recall the construction of the motion groupoid from [32]. Let \({{\mathbb {I}}}=[0,1]\subset {{\mathbb {R}}}\) with the usual topology. Fixing a manifold M, and a subset \(A\subset M\)

- a set of gradual deformations of M fixing A, over some standard unit of time. For \((f,g)\in \textrm{Flow}_{M}^{A}\times \textrm{Flow}_{M}^{A}\) note that \(g*f\) given by

is in \(\textrm{Flow}_{M}^{A}\), as is \({\bar{f}}\) given by \( {\bar{f}}_t \;=\; f_{(1-t)}\circ f_1^{-1}. \)   Thus \((\textrm{Flow}_{M}^{A},*)\) is a magma.

Let \(\text {Mt} _M^A \; =\; ( {{\mathcal {P}}}M, \text {Mt} _M^A(-,-), * )\) be the magmoid constructed as follows. The object set is the power set \({{\mathcal {P}}}M\), and morphisms are triples \((f,N,N')\) consisting of an \(f\in \textrm{Flow}_{M}^{A}\) together with a pair of spaces \(N,N'\subseteq M\) and such that \(f_1(N)=N'\). We denote triples \((f,N,N')\in \text {Mt} _M^A\) by . A morphism in \(\text {Mt} _M^A\) is a gradual deformation of M that takes the initial object subset to the final object subset, hence called a motion. The worldline of a motion in a manifold M is

There is a partial composition of motions given by . For \(f,g\in \textrm{Flow}_{M}^{A}\) and \(N,N'\) subsets, let

Theorem 4.1

(Thms. 3.49 and 4.6 of [32]) Let M be a manifold. The relation \(f{\mathop {\sim }\limits ^{\scriptscriptstyle {rp}}}g\) if \({\textbf{Top}}({{\mathbb {I}}}^2,{\textbf{TOP}}_A^h(M,M))(f^{N'}_{N}\!g) \ne \varnothing \) gives a congruence on the magmoid \(\text {Mt} _M^A\). The quotient is a groupoid — the motion groupoid of M:

Definition 4.2

Let \(\textrm{hf}\textrm{Mot}_{M}^{A}\subset \textrm{Mot}_{M}^{A}\) be the full subgroupoid with \(N\in Ob(\textrm{hf}\textrm{Mot}_{M}^{A})\) if \(M{\setminus } N\) is homotopically 1-finitely generated.

Definition 4.3

Let

Notice that , thus there is a well-defined, continuous map , \(m\mapsto (m,t)\).

Theorem 4.4

Let M be a manifold. There is a well-defined functor

which sends an object \(N \in Ob(\textrm{hf}\textrm{Mot}_{M}^{A})\) to \(M{\setminus } N\), and which sends a morphism to the cospan homotopy equivalence class of

where the maps \(\iota _{{f}_t}\) are as explained before the lemma.

Proof

We first check that \(\mathcal {MOT}_{M}^{A}\) is well defined.

From [32, Thm.3.72], for any motion , there exists a homeomorphism \(\Theta (f):M\times {{\mathbb {I}}}\rightarrow M\times {{\mathbb {I}}}\) given by \(\Theta (f)(m,t)=(f_t(m),t)\), and notice that , and hence the restriction of \(\Theta (f)\) gives a homeomorphism .

Now suppose is a motion representing a morphism in \(\textrm{hf}\textrm{Mot}_{M}^{A}\). By construction, \(M{\setminus } N\) and \(M{\setminus } N'\) are homotopically 1-finitely generated. Let \(X\subset M{\setminus } N\) be a finite representative subset. This implies \(X\times \{0\}\) is representative in \((M\setminus N)\times {{\mathbb {I}}}\). Then we have \(\pi ((M{\setminus } N)\times {{\mathbb {I}}},X\times \{0\})\cong \pi (M{\setminus } N, X)\times \pi ({{\mathbb {I}}}, \{0\})\cong \pi ((M{\setminus } N),X)\), where the first isomorphism follows from the fact that \(\pi \) preserves products ( [8, 6.4.4]). Hence \((M\setminus N)\times {{\mathbb {I}}}\) is homotopically 1-finitely generated. Using that , it follows that is homotopically 1-finitely generated.

We can see the map is a cofibration as follows. The map \(\langle \iota _{{f}_0}, \iota _{{f}_1} \rangle \) is equal to the composition

Now the first and last maps are homeomorphisms, hence cofibrations by Lemma 2.16. That the middle map is a cofibration is proved in Proposition 3.4. Finally, by Lemma 2.15, the composition of cofibrations is a cofibration.

Suppose are representatives of the same morphism in \(\textrm{hf}\textrm{Mot}_{M}^{A}(N,N')\). By Theorem 4.18 of [32] this implies there is a level preserving ambient isotopy of \(H:(M\times {{\mathbb {I}}})\times {{\mathbb {I}}}\rightarrow M\times {{\mathbb {I}}}\) taking to which fixes \((M\times \{0,1\})\cup (A\times {{\mathbb {I}}})\) pointwise. The full definition of level preserving ambient isotopy can be found in Definition 4.12 of [32]. The property we will need is that \((m,t)\mapsto H(m,t,1)\) defines a homeomorphism of \(M\times {{\mathbb {I}}}\), and, in particular, it is a homotopy equivalence. It follows that the restriction to , defined by \((m,t)\mapsto H(m,t,1)\), is a well defined homeomorphism, and hence a homotopy equivalence. Moreover, J fixes and . Hence .

We now check that \(\mathcal {MOT}_{M}^{A}\) preserves composition. Let and be motions in M representing composable morphisms in \(\textrm{hf}\textrm{Mot}_{M}^{A}\). Then

By Lemma 3.38 of [32]

and hence

We have that

The pushout is given by the set of elements of the form ((m, t), 1) and ((m, t), 2) quotiented by the relation \(((m,1),1)\sim ((m,0),2)\). There is a map given by sending (m, t) to the [((m, 2t), 1)] if \(t\in [0,1/2]\) and (m, t) to \([((m,2t-1),2)]\) if \(t\in [1/2,1]\). It is straightforward to check that this map commutes appropriately with the relevant maps, and it is clearly a homotopy equivalence. Thus we have proved that composition is preserved. \(\square \)

4.2 Functor from the Mapping Class Groupoid, \(\mathcal {MCG}_{M}^{A} :\textrm{hf}\textrm{MCG}_{M}^{A} \rightarrow {\textrm{HomCob}}\)

We briefly recall the construction of the mapping class groupoid \(\textrm{MCG}_{M}^{A}\), of a manifold M together with a fixed subset \(A\subset M\), from [32, Thm.6.10]. The object set is the power set \({{\mathcal {P}}}M\). Morphisms in \(\textrm{MCG}_{M}^{A}(N,N')\) are equivalence classes of triples, denoted , consisting of a pair of subsets \(N,N'\subseteq M\) and a self-homeomorphism \(\mathfrak {f}\in {\textbf{TOP}}_A^h(M,M)\) such that \(\mathfrak {f}(N)=N'\). Triples and are related if

is non-empty. We denote the equivalence class as The partial composition is where \(\circ \) denotes function composition. The inverse of a class is given by .

Definition 4.5

Let \(\textrm{hf}\textrm{MCG}_{M}^{A}\subset \textrm{MCG}_{M}^{A}\) be the full subgroupoid with \(N\in Ob(\textrm{hf}\textrm{MCG}_{M}^{A})\) if \(M{\setminus } N\) is homotopically 1-finitely generated.

For a triple representing a morphism in \(\textrm{MCG}_{M}^{A}\), define \(\mathfrak {f}_{{\tiny \textrm{mc}}}:M{\setminus } N \rightarrow (M{\setminus } N') \times {{\mathbb {I}}}\) to be the composition \(\iota _0^{M{\setminus } N'}\circ \mathfrak {f}|_{M{\setminus } N}:M{\setminus } N\rightarrow (M{\setminus } N')\times {{\mathbb {I}}}\), \(m\mapsto (\mathfrak {f}(m),0)\). The map \(\mathfrak {f}_{{\tiny \textrm{mc}}}\) is well defined since \(\mathfrak {f}(N)=N'\) and \(\mathfrak {f}\) is a homeomorphism, thus \(\mathfrak {f}(M\setminus N)=M\setminus N'\).

Lemma 4.6

Let M be a manifold. There is a functor

which sends an object \(N\in Ob(\textrm{hf}\textrm{MCG}_{M}^{A})\) to \(M{\setminus } N\), and which sends a morphism to the cospan homotopy equivalence class of

Proof

We first check that \(\mathcal {MCG}_{M}^{A}\) is well defined.

Suppose a represents a morphism in \(\textrm{hf}\textrm{MCG}_{M}^{A}\), then, by construction, \(M\setminus N\) and \(M\setminus N'\) are homotopically 1-finitely generated.

We now check that the map \(\langle \mathfrak {f}_{{\tiny \textrm{mc}}}, \iota _1^{M{\setminus } N'} \rangle :(M{\setminus } N)\sqcup (M{\setminus } N') \rightarrow (M{\setminus } N')\times {{\mathbb {I}}}\) is a cofibration. The map \(\langle \mathfrak {f}_{{\tiny \textrm{mc}}}, \iota _1^{M{\setminus } N'} \rangle \) is equal to the composition

Now the first map is a homeomorphism, thus a cofibration by Lemma 2.16. The second is a cofibration by Proposition 3.4. Using Lemmas 2.16 and 2.15 it follows that the composition is a cofibration.

Suppose that and are representatives of the same morphism in \(\textrm{hf}\textrm{MCG}_{M}^{A}\). We show by constructing a homeomorphism from \((M\setminus N')\times {{\mathbb {I}}}\) to itself, which commutes with the appropriate cospans. It follows from the assumption that there exists a continuous map \(H:{{\mathbb {I}}}\rightarrow {\textbf{TOP}}^h_A(M,M)\) from \(\mathfrak {f}\) to \(\mathfrak {f}'\), satisfying \(H(t)(N)=N'\) for all \(t\in {{\mathbb {I}}}\). We denote H(t) by \(H_t\). Using the product-hom adjunction (Lemma 2.1), H corresponds to a continuous map \(H':M \times {{\mathbb {I}}}\rightarrow {{\mathbb {I}}}\), \((m,t)\mapsto H_t(m)\). Define a map \({\hat{H}}:M\times {{\mathbb {I}}}\rightarrow M\times {{\mathbb {I}}}\) by \((m,t)\mapsto (H_{1-t}\circ \mathfrak {f}^{-1}(m),t)\). It is straightforward to see that this map is continuous. We show that \({\hat{H}}\) is a homeomorphism by giving a continuous inverse. Since \({\textbf{TOP}}^h(M,M)\) is a topological group (original result [2, Th.4], see also [32, Th.2.11]) the map from \({\textbf{TOP}}^h(M,M)\) to itself sending each homeomorphism to its inverse is continuous. Thus \(J:{{\mathbb {I}}}\rightarrow {\textbf{TOP}}^h(M,M)\) given by \(t\mapsto H_t^{-1}\) is a continuous map. Applying the product-hom adjunction again, there is a continuous map \(J':M\times {{\mathbb {I}}}\rightarrow M\) given by \((m,t)\mapsto H_t^{-1}(m)\). Hence there exists a continuous map \({\hat{J}}:M\times {{\mathbb {I}}}\rightarrow M\times {{\mathbb {I}}}\) given by \((m,t)\mapsto (f\circ J'(m,1-t),t)=(f\circ H_{1-t}^{-1}(m),t)\). It is straightforward to see \({\hat{J}}\circ {\hat{H}}=\textrm{id}_{M\times {{\mathbb {I}}}}={\hat{H}}\circ {\hat{J}}\). Next notice that \({\hat{H}}(N'\times {{\mathbb {I}}})=\cup _{t\in {{\mathbb {I}}}}H_{1-t}(f^{-1}(N'))\times \{t\}=\cup _{t\in {{\mathbb {I}}}}H_{1-t}(N)\times \{t\}=\cup _{t\in {{\mathbb {I}}}}N'\times \{t\}=N'\times {{\mathbb {I}}}\), thus \({\hat{H}}\) restricts to a homeomorphism \((M\setminus N')\times {{\mathbb {I}}}\). Finally we check the commutation relations. We have \({\hat{H}}( \mathfrak {f}_{{\tiny \textrm{mc}}}(m))={\hat{H}}(\mathfrak {f},0)=(H_{1}\circ \mathfrak {f}^{-1}\circ \mathfrak {f} (m),0)=(\mathfrak {f}'(m),0)=\mathfrak {f'}_{{\tiny \textrm{mc}}}(m)\) and \({\hat{H}}( \iota _1^{M{\setminus } N'}(m))={\hat{H}}(m,1)=(H_0\circ \mathfrak {f}^{-1}(m)),1)=(m,1)=\iota _1(m)\). Thus \({\hat{H}}\) is a cospan homotopy equivalence from to .

We now check that \(\mathcal {MCG}_{M}^{A}\) preserves composition. Let and be representatives of composable morphisms in \(\textrm{hf}\textrm{MCG}_{M}^{A}\).

and we have that

The pushout \(((M{\setminus } N')\times {{\mathbb {I}}})\sqcup _{M{\setminus } N'}((M{\setminus } N'')\times {{\mathbb {I}}})\) is given by the set of elements of the form ((m,��t), 1) and ((m, t), 2) quotiented by the relation \(((m,1),1)\sim ((\mathfrak {g}(m),0),2)\). To show that the two compositions are equivalent, we must construct a cospan homotopy equivalence \(K:(M{\setminus } N'')\times {{\mathbb {I}}}\rightarrow ((M{\setminus } N')\times {{\mathbb {I}}})\sqcup _{M{\setminus } N'}((M{\setminus } N'')\times {{\mathbb {I}}})\). A suitable choice is given by

Notice that \(\mathfrak {g}^{-1}(N'')=N'\), thus \(\mathfrak {g}^{-1}(M{\setminus } N'')=M{\setminus } N'\), and furthermore, the images are the same equivalence class at \(t=1/2\), thus K is well defined. It can be seen that K is continuous by considering it as a composition of maps into the disjoint union and then projecting to the pushout.

Each piece is clearly continuous, and defined on closed sets, thus K is continuous. It is straightforward to see that K commutes with the relevant maps in the cospans. We can see that K is a homotopy equivalence by noting that it is, in particular a homeomorphism. A continuous inverse is given by the map \(L:((M{\setminus } N')\times {{\mathbb {I}}})\sqcup _{M{\setminus } N'}(M{\setminus } N'')\times {{\mathbb {I}}}\rightarrow (M{\setminus } N')\times {{\mathbb {I}}}\),

It is straightforward to check that L is well-defined and continuous. \(\square \)

4.3 Composition of Functor \({\textsf{F}}:\textrm{Mot}_{M}^{A}\rightarrow \textrm{MCG}_{M}^{A}\) with \(\mathcal {MCG}_{M}^{A}\)

From [32] we have that, for any manifold M and subset \(A\subset M\), there is a functor \({\textsf{F}}:\textrm{Mot}_{M}^{A}\rightarrow \textrm{MCG}_{M}^{A}\), which is the identity on objects and which sends the \({\mathop {\sim }\limits ^{\scriptscriptstyle {rp}}}\) equivalence class of a motion to the \({\mathop {\sim }\limits ^{i}}\) equivalence class of .

Theorem 4.7

Let M be a manifold, with \(A\subset M\) a subset, we have that

Proof

Suppose we have a morphism , then

and

We must construct a homotopy equivalence commuting with the cospans. From [32, Thm.3.72], for any motion , there is homeomorphism of \(\Theta (f):M\times {{\mathbb {I}}}\rightarrow M\times {{\mathbb {I}}}\) given by \(f(m,t)=(f_t(m),t)\).

Let \(K':M\times {{\mathbb {I}}}\rightarrow M\times {{\mathbb {I}}}\) be the map \((m,t) \mapsto \Theta (f)(f_1^{-1}(m),t)\), and notice that this is a homeomorphism. Notice also that \(K'\) restricts to a well defined homeomorphism , since .

We now check that it commutes with the cospans. Consider \(m\in M{\setminus } N\). Then \(K({f_1}_{{\tiny \textrm{mc}}} (m))=K(f_1(m),0)=(\Theta (f)(f_1^{-1}(f_1(m)),0)=(f_0(m),0)=(m,0)=\iota _{{f}_0}(m)\), and \(K(\iota _1^{M{\setminus } N'}(m))=K(m,1)=\Theta (f)(f_1^{-1}(m),1)=(f_1(f_1^{-1}(m)),1)=(m,1)\). \(\square \)

Remark 4.8

The previous theorem says that any representation of \(\textrm{hf}\textrm{Mot}_{M}^{A}\) obtained using the functor \(\mathcal {MOT}_{M}^{A}\), i.e. factoring through \({\textrm{HomCob}}\), will map any pair of motion classes to the same linear map if their images under \({\textsf{F}}\) are equivalent in \(\textrm{MCG}_{M}^{A}\). For many cases we are interested in, the functor \({\textsf{F}}\) is an isomorphism, but when information is lost by the functor \({\textsf{F}}\), we would need to look for functors from \(\textrm{hf}\textrm{Mot}_{M}^{A}\) which do not factor through \({\textrm{HomCob}}\) to see this information.

In this section we explicitly construct, for each finite group G, a functor

We begin by defining a magmoid morphism from a modification of \(\textsf{HomCob}\) whose objects are pairs of a space and a set of basepoints, into \({\textsf{Vect}_\mathbb {C}}\). We arrive at the functor \({\textsf{Z}}_G\), in Theorem 5.25, by constructing a colimit over all representative finite subsets \(A\subseteq X\) of basepoints and maps \(g:\pi (X,A)\rightarrow G\). We subsequently show, in Sect. 5.4, that \({\textsf{Z}}_G\) can be calculated on objects by choosing a convenient fixed choice of finite representative subset \(A\subseteq X\) (Theorem 5.29). Thus our construction is explicitly calculable, see Example 5.36. We will ultimately prove, in Theorem 5.38, that \({\textsf{Z}}_G\) maps an object space X to the vector space with basis the set of equivalence classes of maps \(f:\pi (X)\rightarrow G\), up to natural transformation. In Sect. 5.5 we prove that \({\textsf{Z}}_G\) is a symmetric monoidal functor.

5.1 Magmoid of Based Cospans, \(\textsf{bHomCob}\)

Let \({\pmb \chi }\) denote the class of pairs of the form \((X,X_0)\) where X is a homotopically 1-finitely generated space (Def. 3.24) and \(X_0\) is a representative finite subset of X (representative means one point in each path component). We will refer to the set \(X_0\) as a set of basepoints.

Definition 5.1

Let X, Y be spaces and \(A \subseteq X\) and \( B \subseteq Y\) be subsets. A map of pairs \(f:(X,A) \rightarrow (Y,B)\) is a map \(f:X \rightarrow Y\) such that \(f(A)\subseteq B\).

Definition 5.2

Let \((X,X_0)\), \((Y,Y_0)\) and \((M,M_0)\) be pairs in \({\pmb \chi }\). A concrete based homotopy cobordism from \((X,X_0)\) to \((Y,Y_0)\) is a diagram

such that:

1. (i)

   \(i :X \rightarrow M \rightarrow Y :j\) is a concrete homotopy cobordism.

2. (ii)

   i and j are maps of pairs.

3. (iii)

   \(M_0 \cap i(X) =i(X_0)\) and \(M_0 \cap j(Y) = j(Y_0)\).

For any pairs \((X,X_0),(Y,Y_0)\) in \({\pmb \chi }\),

Example 5.3

Consider the concrete cofibrant cospan described in Proposition 3.5, which is a homotopy cobordism (see Example 3.27). We can add basepoints to obtain a based homotopy cobordism as shown in Fig. 4.

Here the dots represent basepoints. The three blue points in the leftmost copy of \(S^1\) are \(S^1_0\) and the two red basepoints in the rightmost copy of \(S^1\) are \({S^1}_0'\). The set of basepoints \(D^2_0\) then contains \(i(S^1_0)\cup j({S^1}'_0)\) as well as two extra basepoints marked in green which do not intersect \(i(S^1)\cup j(S^1)\). Then \(i:(S^1,S^1_0) \rightarrow (D^2,D^2_0) \leftarrow {S^1}'_0 :j\) is a concrete based homotopy cobordism. (Color figure online)

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Proposition 5.4

Let \(i:X \rightarrow M \leftarrow Y:j\) be a concrete homotopy cobordism, then there exists a based homotopy cobordism \(i:(X,X_0) \rightarrow (M,M_0) \leftarrow (Y,Y_0):j\) for some representative finite subsets \(X_0,Y_0\) and \(M_0\) of X, Y and M respectively.

Proof

A suitable choice of \(X_0\), \(Y_0\) and \(M_0\) is constructed as follows. Choose a point in each path-connected component of X, and let \(X_0\) be the union of these points. Choose \(Y_0\) in the same way. Let \(M_0\) be the union of \(i(X_0)\) and \(j(Y_0)\), together with a choice of point in each path-connected component not containing a point \(i(X_0)\cup j(Y_0)\).

Notice that \(X_0,Y_0\) and \(M_0\) are finite, as X, Y and M are homotopically 1-finitely generated, and thus contain a finite number of path-connected components. \(\square \)

Of course for any manifold X consisting of a single path connected component, there is an uncountably infinite number of choices of \(\{x\}\) such that \((X,\{x\})\in {\pmb \chi }\). Hence there will usually be many ways to obtain a based homotopy cobordism from a homotopy cobordism.

The following Lemma says that the composition extends to a composition of based homotopy cobordisms.

Lemma 5.5

(I) For any spaces X, Y and Z in \(Ob(\textsf{HomCob})\) there is a composition

where \({\tilde{i}}:X \rightarrow M \sqcup _Y N \leftarrow Z:{\tilde{l}}\) is the composition with as in Lemma 3.12, and \(M_0\sqcup _{Y_0}N_0\) the set pushout of \(M_0\xleftarrow {j}Y_0\xrightarrow {k}Z_0\) (where we use j and k also for the obvious restrictions).

(II) Hence there is a magmoid

Proof

We check that the composition is well defined. We first show that \(M_0\sqcup _{Y_0} N_0\) is representative in \(M\sqcup _Y N\). Recall that our chosen representatives of pushouts in \({\textbf{Top}}\) have the same underlying set and set maps as the representative of the corresponding pushout of the underlying set maps in \({\textbf{Set}}\). Also j and k are homeomorphisms, by Lemma 2.16. Thus \(M_0\sqcup _{Y_0} N_0\subseteq M\sqcup _{Y} N\) and \(M_0\sqcup _{Y_0} N_0=p_M(M_0)\cup p_N(N_0)\) where \(p_M\) and \(p_N\) are as in Proposition 3.12. Let \(m\in M\sqcup _Y N\) be any point, then it has a preimage \(p^{-1}(m)\) in M or N, and thus there is a path in M or N connecting \(p^{-1}(m)\) to a point in \(M_0\) or \(N_0\). The image of this path under \(p_M\) or \(p_N\) connects m to a point in \(M_0\sqcup _{Y_0} N_0\).

We have from Proposition 3.33 that \(i:X \rightarrow M\sqcup _{Y}N \leftarrow Z:l\) is a concrete homotopy cobordism.

Since \(M_0\sqcup _{Y_0} N_0=p_M(M_0)\cup p_N(N_0)\), and \(i(X_0)\subseteq M_0\) and \(l(Z_0)\subseteq N_0\), we have \({\tilde{i}}(X_0)\subseteq M_0\sqcup _{Y_0}N_0\) and \({\tilde{l}}(Z_0)\subseteq M_0\sqcup _{Y_0}N_0\), thus \({\tilde{i}}\) and \({\tilde{k}}\) are maps of pairs.

The map \(\langle i,j \rangle :X\sqcup Y \rightarrow M\) is a cofibration, hence by Theorem 2.19 it is a homeomorphism onto its image. This means \(i(X)\cap j(Y)=\varnothing \) in M, and similarly \(k(Y)\cap l(Z)=\varnothing \). Hence there is no equivalence on points in X or Z in the pushout. Thus \(\left( M_0\sqcup _{Y_0} N_0\right) \cap {\tilde{i}}(X)={\tilde{i}}(X_0)\) follows directly from the fact that \(\left( M_0\right) \cap i(X)=i(X_0)\) and similarly \(\left( M_0 \sqcup _{Y_0}N_0\right) \cap {\tilde{l}}(Z) = {\tilde{l}}(Z_0)\). \(\square \)

5.2 Magmoid Morphism from \(\textsf{bHomCob}\) to \({\textsf{Vect}_\mathbb {C}}\)

Here we construct a magmoid morphism \( {\textsf{Z}}_G^{!}:\textsf{bHomCob}\rightarrow {\textsf{Vect}_\mathbb {C}}\).

Recall that there is a groupoid \({{\mathcal {G}}}_G=(\{*\},{{\mathcal {G}}}_G(*,*),\circ _G,e_G,g\mapsto g^{-1})\) obtained from any group G with morphisms the elements of G, and composition and inverses given by group composition and inverse. Throughout this section, by abuse of notation we will use G for \({{\mathcal {G}}}_G\).

Definition 5.6

Let G be a group. For a pair \((X,X_0)\in {\pmb \chi }\), define

That is, \({\textsf{Z}}_G^{!}(X,X_0)\) is the \({\mathbb {C}}\) vector space whose basis is the set of functors from the fundamental groupoid of X with respect to \(X_0\) into G.

Example 5.7

Let \(X=S^1\sqcup S^1\) be the topological space, and let \(X_0\subset X\) contain two points, one in each copy of \(S^1\). Then \(\pi (X,X_0)\cong {\mathbb {Z}}\sqcup {\mathbb {Z}}\), where \({\mathbb {Z}}\) denotes the groupoid with one object and endomorphism group isomorphic to the group \({\mathbb {Z}}\), and \(\sqcup \) is the groupoid coproduct described in Section. 2.3.2. Hence maps from \(\pi (X,X_0)\) to G are determined by pairs in the underlying set \(G\times G\), where the elements of G denote the image of the generating elements of each copy of \({\mathbb {Z}}\). So we have \({\textsf{Z}}_G^{!}(X,X_0)\cong {\mathbb {C}}(G\times G)\).

In the following example note that \(X_0\) must be representative by the definition of \({\pmb \chi }\), therefore we choose basepoints even in path components that are homotopically trivial. This is necessary since, when considered as part of a cospan, these trivial components may have image in a homotopically non-trivial component.

Example 5.8

Let X and \(X_0\) be as explained in the caption to Fig. 5. Observe that \(\pi (X,X_0)\cong ({\mathbb {Z}}* {\mathbb {Z}})\sqcup \{*\}\sqcup \{*\}\), where \(\{*\}\) denotes the trivial groupoid. (Note we have omitted a choice of bracketing in the iterated coproduct.) Groupoid morphisms from \(\pi (X,X_0)\) to G are hence determined by pairs in the underlying set \(G\times G\), whose elements respectively denote the images of the equivalence classes of the loops marked \(x_1\) and \(x_2\) in the figure, so we have \({\textsf{Z}}_G^{!}(X,X_0)\cong {\mathbb {C}}(G\times G)\).

Let X be the complement in \({{\mathbb {I}}}^2\) of the embedding of \(S^1\sqcup S^1\) shown. Let \(X_0\) be the three marked points shown

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For each pair \((X,X_0)\), the vector space \({\textsf{Z}}_G^{!}(X,X_0)\) has an intrinsic basis, the maps into G. We will define a linear map assigned to each based homotopy cobordism as a matrix indexed by these bases.

Definition 5.9

Let \(i:(X,X_0) \rightarrow (M,M_0) \leftarrow (Y,Y_0):j\) be a concrete based homotopy cobordism. We define a matrix

as follows. Let \(f \in {\textsf{Z}}_G^{!}(X,X_0)\) and \(g \in {\textsf{Z}}_G^{!}(Y,Y_0)\) be basis elements, then

In other words, the right hand side is the cardinality of the set of maps h making the diagram commute. Here we are using Dirac notation: (4) is the matrix element in the column corresponding to f and the row corresponding to g.

When we have already specified the relevant cospan, we will often use \({\textsf{Z}}_G^{!}(M,M_0)\) for \({\textsf{Z}}_G^{!}\left( i:(X,X_0) \rightarrow (M,M_0) \leftarrow (Y,Y_0):j\right) \), and write the matrix elements as

where by \(h|_{\pi (X,X_0)}\) we really mean the restriction of the map h to the image \(\pi (i)(\pi (X,X_0))\).

Example 5.10

Let \(i:(X,X_0) \rightarrow (M,M_0) \leftarrow (Y,Y_0):j\) be the based homotopy cobordism represented in Fig. 6, and described in the caption. Note this is a homotopy cobordism as discussed in Examples 3.9 and 3.28. Note also that there is a finite number of marked points, and at least one in each connected component of X, Y and M.

Now \(\pi (Y,Y_0)\cong {\mathbb {Z}}\), where the isomorphism is realised by mapping the loop labelled \(y_1\) in the figure to 1. Hence a map \(g:\pi (Y,Y_0)\rightarrow G\) is uniquely determined by a choice of an element \(g_1\in G\) with \(f([y_1]_{\!\tiny \hbox {p}})=g_1\). Thus we have \({\textsf{Z}}_G^{!}(Y,Y_0)\cong {\mathbb {C}}(G)\).

Recall from Example 5.7 that \({\textsf{Z}}_G^{!}(X,X_0)\cong {\mathbb {C}}(G\times G)\), where a pair \((g_1,g_2)\) denotes the map \((g_1,g_2)([x_1]_{\!\tiny \hbox {p}})=g_1\) and \((g_1,g_2)([x_2]_{\!\tiny \hbox {p}})=g_2\).

Let x be the basepoint which is in the loop labelled \(x_1\). By Lemma 2.6, there is a bijection sending a map \(h\in {\textbf{Grpd}}(\pi (M,M_0), G)\) to a triple \((h',h(\gamma _1),h(\gamma _2)) \in {\textbf{Grpd}}(\pi (M,\{x\}), G)\times G\times G\), where \(h'\) agrees with h on \(\pi (M,\{x\})\). The space M is equivalent to the twice punctured disk, which has fundamental group isomorphic to the free product \({\mathbb {Z}}*{\mathbb {Z}}\). This isomorphism can be realised by sending the element represented by \(x_1\) to the 1 in the first copy of \({\mathbb {Z}}\) and the element represented by \(\gamma _2^{-1}x_2\gamma _2\) to the 1 in the second copy of \({\mathbb {Z}}\). Thus we can label elements in \({\textbf{Grpd}}(\pi (M,\{x\}), G)\) by elements of \(G\times G\) where \(a \in (a,b)\) corresponds to the image of \([x_1]_{\!\tiny \hbox {p}}\), and b the image of \([\gamma _2^{-1}x_2\gamma _2]_{\!\tiny \hbox {p}}\). Hence a map in \({\textbf{Grpd}}(\pi (M,M_0), G)\) is determined by a quadruple \((a,b,c,d)\in G\times G\times G\times G\) where a corresponds to the image of \(x_1\), b to the image of \(\gamma _2^{-1}x_2\gamma _2\), and c and d correspond to the images of \(\gamma _1\) and \(\gamma _2\) respectively.

Choosing basis elements \((f_1,f_2)\in {\textsf{Z}}_G^{!}(X,X_0)\) and \(g_1\in {\textsf{Z}}_G^{!}(Y,Y_0)\) the commutation condition in (4) gives conditions on allowed quadruples \((a,b,c,d)\in G\times G\times G\times G\). We have

This figure represents the concrete cofibrant cospan from Example 3.9, so M is the represented manifold, and \(X\cong S^1 \sqcup S^1\) and \(Y\cong S^1\) are the bottom and top boundary respectively, with the inclusion maps. The red points and dashed lines show a possible choice of basepoints \(M_0\) and generating paths. Let \(X_0\) and \(Y_0\) be the intersection of \(M_0\) with X and Y respectively. (Color figure online)

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Lemma 5.11

The map

is a magmoid morphism.

Proof

It is immediate from the construction that the map is well defined. Thus we only need to check that composition is preserved. Let \(i:(X,X_0) \rightarrow (M,M_0) \leftarrow (Y,Y_0):j\) and \(k:(Y,Y_0) \rightarrow (N,N_0) \leftarrow (Z,Z_0):l\) be concrete based homotopy cobordisms. Let \(f\in {\textsf{Z}}_G^{!}(X,X_0)\) and \(g\in {\textsf{Z}}_G^{!}(Z,Z_0)\) be basis elements. The matrix element corresponding to these basis elements is given by counting maps h in the following diagram.

From Definition 5.2 and Lemma 3.11, the pushout of \( (M,M_0)\xleftarrow {j}(Y,Y_0)\xrightarrow {k}(N,N_0)\) satisfies the conditions of Corollary 2.34. Hence the middle square of this diagram is a pushout. Hence each h is uniquely determined by a pair \(h_1:\pi (M,M_0) \rightarrow G\) and \(h_2:\pi (N,N_0) \rightarrow G\) such that the above diagram commutes. So we have

Now this is precisely the corresponding matrix element given by multiplying the matrices \({\mathcal {F}}_{G}^{!}(M,M_0)\) and \({\mathcal {F}}_{G}^{!}(N,N_0)\). \(\square \)

The following lemma says that \({\textsf{Z}}_G^{!}\) respects cospan homotopy equivalence.

Lemma 5.12

Suppose we have concrete homotopy cobordisms \(i:X \rightarrow M \leftarrow Y:j\) and \(i':X \rightarrow M' \leftarrow Y:j'\) which are equivalent up to cospan homotopy equivalence (as defined in Lemma 3.14). Then (by Theorem 2.31) we have homotopy equivalences \(\psi :M\rightarrow M'\) and \(\psi ':M'\rightarrow M\) which commute with the cospans. Choosing sets of baspoints \(X_0\subseteq X\), \(Y_0\subseteq Y\), \(M_0\subseteq M\) such that \(i:(X,X_0) \rightarrow (M,M_0) \leftarrow (Y,Y_0):j\) is a based homotopy cobordism, we have

where \(M_0'=\psi (M_0)\).

Proof

Let \(f\in {\textsf{Z}}_G^{!}(X,X_0)\) and \(g\in {\textsf{Z}}_G^{!}(Y,Y_0) \) be basis elements and \(h:\pi (M,M_0) \rightarrow G\) a map with \(h|_{\pi (X,X_0)}=f\) and \(h|_{\pi (Y,Y_0)}=g\). On the level of fundamental groupoids \(\psi \) and \(\psi '\) become inverse group isomorphisms making the following diagram commute.

For any such h we can obtain a map \(h'\) making the diagram commute by precomposing h with \(\pi (\psi ')\). Thus we have a set map

which has inverse given by precomposing with \(\pi (\psi )\). Thus we have that \(\langle g \,|\, {\textsf{Z}}_G^{!}(M,M_0) \,|\, f \rangle = \langle g \,|\, {\textsf{Z}}_G^{!}(M',M'_0) \,|\, f \rangle \) for all f, g. \(\square \)

5.3 Functor \({\textsf{Z}}_G:{\textrm{HomCob}}\rightarrow {\textbf{Vect}}_{{\mathbb {C}}}\)

The magmoid morphism \({\textsf{Z}}_G^{!}\) clearly depends on the sets of basepoints. Also notice that there are many ways we could obtain a based cospan from the cospan representing the identity in \({\textrm{HomCob}}\), \(\iota _0^X:X \rightarrow X\times {{\mathbb {I}}} \leftarrow X:\iota _1^X\), and in general these based cospans will not give the identity matrix under \({\textsf{Z}}_G^{!}\). In this section we add a normalisation factor, then define a map on the objects of \({\textrm{HomCob}}\) by taking a colimit over \({\textsf{Z}}_G^{!}\) for all choices of basepoints, and finally redefine the map on morphisms using the universal property of the colimit. This leads to a functor \({\textsf{Z}}_G:{\textrm{HomCob}}\rightarrow {\textbf{Vect}}_{{\mathbb {C}}}\) which does not depend on a choice of basepoints. We will find that removing the basepoint dependence will also solve the identity problem.

Varying the set of basepoints

Let \(i:(X,X_0) \rightarrow (M,M_0) \leftarrow (Y,Y_0):j\) be a concrete based homotopy cobordism. We first consider how changing the set of basepoints in the set \(M_0\) changes \({\textsf{Z}}_G^{!}(M,M_0)\).

If such a point exists, choose a point \(m\in M\setminus M_0\) such that \(i:(X,X_0) \rightarrow (M,M_0\cup \{m\}) \leftarrow (Y,Y_0):j\) is also a concrete based homotopy cobordism. By Lemma 2.6, the set of maps \(h':\pi (M,M_0\cup \{m\}) \rightarrow G \) is in bijective correspondence with the set of pairs of a map \(h:\pi (M,M_0) \rightarrow G \) and an element of G, via a bijection \(\Theta _{\gamma }\). Let \(f\in {\textsf{Z}}_G^{!}(X,X_0)\) and \(g\in {\textsf{Z}}_G^{!}(Y,Y_0)\) be basis elements. Note that for all \(h:\pi (M,M_0) \rightarrow G \) such that \(h|_{\pi (X,X_0)}=f \) and \( h|_{\pi (Y,Y_0)}=g\), the map \(h'\) obtained from a pair (h, x) using the map \(\Theta ^{-1}_\gamma \) as in the proof of Lemma 2.6 also satisfies \(h'|_{\pi (X,X_0)}=f \) and \( h'|_{\pi (Y,Y_0)}=g \). Similarly, for a map \(h':\pi (M,M_0\cup \{m\}) \rightarrow G \) satisfying \(h'|_{\pi (X,X_0)}=f \) and \( h'|_{\pi (Y,Y_0)}=g \), \(\Theta _{\gamma }(h')|_{\pi (X,X_0)}=f \) and \( \Theta _{\gamma }(h')|_{\pi (Y,Y_0)}=g\). Hence for all pairs f, g we have that \( \left\langle g \,\big |\, {\textsf{Z}}_G^{!}(M,M_0 \cup \{m\}) \,\big |\, f \right\rangle = |G| \left\langle g \,\big |\, {\textsf{Z}}_G^{!}(M,M_0) \,\big |\, f \right\rangle \), and hence that

It follows that for all \( M_0' \supseteq M_0\) we have \( {\textsf{Z}}_G^{!}(M,M_0') = |G|^{(|M_0'| - |M_0|)} {\textsf{Z}}_G^{!}(M,M_0), \) and hence

Now suppose instead that \(M_0,M_0'\subset M\) are arbitrary finite representative subsets. Then we can write

and

which together imply

and hence that

We have proven the following.

Lemma 5.13

The map \({\textsf{Z}}_G^{!!}\), assigning a linear map to a concrete based homotopy cobordism as follows

does not depend on the subset \(M_0\subseteq M\). \(\square \)

When the relevant cospan is clear, or has been given, we will refer to the image as \({\textsf{Z}}_G^{!!}(M,X_0,Y_0)\) to highlight the dependence on \(X_0\) and \(Y_0\).

In defining \({\textsf{Z}}_G^{!!}\) we have included a term counting the cardinality of \(X_0\). Some term counting basepoints in X or Y is necessary to ensure the new definition is still compatible with the composition; however we could have chosen \(\nicefrac {1}{2}(|X_0|+|Y_0|)\) for example, as is the convention in [54], and avoided the asymmetry. The reason for our convention is that it allows us to work for longer in the basis set rather than moving to the \({\mathbb {C}}\) vector space, making calculation easier. We will highlight later where this becomes relevant (Remark 5.22).

Lemma 5.14

Let \(i:(X,X_0) \rightarrow (M,M_0) \leftarrow (Y,Y_0):j\) and \(k:(Y,Y_0) \rightarrow (N,N_0) \leftarrow (Z,Z_0):l\) be concrete based homotopy cobordisms. Then

where concatenation denotes composition of linear maps, or equivalently matrix multiplication.

Proof

We have

using that, by Lemma 5.11, \({\textsf{Z}}_G^{!}\) preserves composition. \(\square \)

Basepoint independent map from \( Ob({\textrm{HomCob}})\) to \(Ob(\textbf{Vect}_{\mathbb {C}})\)

We focus here on the sets of basepoints in \(Ob(\textsf{bHomCob})\). Here we will move to using Greek subscripts to indicate varying choices of subsets, so for a space X, objects in \(\textsf{bHomCob}\) are pairs of the form \((X,X_\alpha )\). We will eventually show that we can choose just one subset to calculate the image and will then switch back to the original notation.

We proceed by constructing, for a space \(X\in Ob({\textrm{HomCob}})\), a colimit in \(\textbf{Vect}_{\mathbb {C}}\) over a diagram with vertices \({\textsf{Z}}_G^{!}(X,X_\alpha )\) for all possible choices of finite, representative \(X_\alpha \subset X\).

Proposition 5.15

Let X be a homotopically 1-finitely generated space. There is a subcategory of \({\textbf{Set}}\),

where \(Ob(\mathbf {FinSet^*}(X))\) contains all \(X_\alpha \) such that \((X,X_\alpha )\in {\pmb \chi }\) (i.e. \(X_\alpha \) finite, representative) and \(\mathbf {FinSet^*}(X)(X_\alpha ,X_\beta )\) contains the inclusion \(\iota _{\alpha \beta }:X_\alpha \rightarrow X_\beta \) if \(X_\alpha \subseteq X_\beta \), otherwise \(\mathbf {FinSet^*}(X)(X_\alpha ,X_\beta )=\varnothing \).

Proof

Note we have \(\iota _{\alpha \alpha }=1_{X_{\alpha }}:X_\alpha \rightarrow X_\alpha \) in \(\mathbf {FinSet^*}(X)(X_\alpha ,X_\alpha )\). Suppose \(X_\alpha ,X_\beta ,X_\gamma \in \mathbf {FinSet^*}(X)\), with \(X_\alpha \subseteq X_\beta \subseteq X_\gamma \), then the composition of \( \iota _{\alpha \beta }:X_\alpha \rightarrow X_\beta \) and \( \iota _{\beta \gamma }:X_\beta \rightarrow X_\gamma \) is precisely the unique morphism in \(\mathbf {FinSet^*}(X)(X_\alpha ,X_\gamma )\). (This is the only case for which we have composable morphisms.) \(\square \)

By abuse of notation, for an inclusion \(\iota _{\alpha \beta }:X_\alpha \rightarrow X_\beta \) we will also write \(\iota _{\alpha \beta }:\pi (X,X_\alpha )\rightarrow \pi (X,X_\beta )\) for the inclusion of groupoids.

Lemma 5.16

There is a contravariant functor

constructed as follows. Let \(X_\alpha , X_\beta \in Ob(\mathbf {FinSet^*}(X))\) with \(X_\beta \subseteq X_\alpha \). Let \({\mathcal {V}}_X(X_\alpha )= {\textbf{Grpd}}(\pi (X,X_\alpha ),G)\). For any \(v_\alpha \in {\mathcal {V}}_X(X_\alpha )\) we have a commuting triangle

Now let \({\mathcal {V}}_X(\iota _{\beta \alpha }:X_\beta \rightarrow X_\alpha ) = \phi _{\alpha \beta } \) where \(\phi _{\alpha \beta }:{\mathcal {V}}_X(X_\alpha ) \rightarrow {\mathcal {V}}_X(X_\beta ) \), \(v_\alpha \mapsto v_\alpha \circ \iota _{\alpha \beta }\).

Proof

We have \({\mathcal {V}}(1_{X_\alpha }:X_\alpha \rightarrow X_\alpha )=1_{{\mathcal {V}}(X_\alpha )}:{\mathcal {V}}(X_\alpha )\rightarrow {\mathcal {V}}(X_\alpha )\). Suppose \(X_\alpha ,X_\beta ,X_\gamma \in \mathbf {FinSet^*}(X)\), with \(X_\gamma \subseteq X_\beta \subseteq X_\alpha \), then \(\phi _{\beta \gamma }\circ \phi _{\alpha \beta } =\phi _{\alpha \gamma }\) since \((v_{\alpha }\circ \iota _{\beta \alpha })\circ \iota _{\gamma \beta }=v_\alpha \circ (\iota _{\alpha \gamma })\). \(\square \)

Lemma 5.17

For any space X and \(X_\beta , X_\alpha \in Ob(\mathbf {FinSet^*}(X))\) with \(X_\beta \subseteq X_\alpha \), \(\phi _{\alpha \beta }\) is a surjection.

Proof

By Lemma 2.5 for any \(v_\beta \in {\mathcal {V}}_X(X_\beta )\) we can extend to some \(v_\alpha \in {\mathcal {V}}_X(X_\alpha )\) which is equal to \(v_\beta \) on the image \(\iota _{\beta \alpha }(\pi (X,X_\beta ))\) in \(\pi (X,X_\alpha )\). \(\square \)

The colimit over \({\mathcal {V}}_X\) consists of a family of commuting triangles diagrams of the form

for each pair \(X_\beta \subseteq X_\alpha \). By abuse of notation we will use \(v_\alpha \) for both \(v_\alpha \in {\mathcal {V}}_X(X_\alpha )\) and its image in \( \sqcup _{X_\alpha }{\mathcal {V}}_X(X_\alpha )\). Hence we have

where \(\sim \) is the reflexive, symmetric and transitive closure of the relation \(v_\alpha \sim v_\beta \) if \(\phi _{\alpha \beta }(v_\alpha ) = v_\beta \). See Sect. 2.3 for more on colimits in \({\textbf{Set}}\). We use \([v_\alpha ]\) to denote the equivalence class of \(v_\alpha \) in \({\textrm{colim}}({\mathcal {V}}_X)\). Hence we have \(\phi _\alpha :{\mathcal {V}}_X(X_\alpha )\rightarrow {\textrm{colim}}({\mathcal {V}}_X)\), \(v_\alpha \mapsto [v_\alpha ]\).

Notice that this relation is certainly not itself an equivalence. For example for any \(X_\beta \subset X_\alpha \), and \(v_\beta =v_\alpha \circ \iota _{\beta \alpha }:\pi (X,X_\beta ) \rightarrow G\), then the relation says \(v_\beta =\phi _{\alpha \beta }(v_\alpha )\sim v_\alpha \) but not \(v_\alpha \sim v_\beta \) as there is no map \(\phi _{\beta \alpha }\).

Lemma 5.18

Let \({\mathcal {V}}_X:\mathbf {FinSet^*}(X)\rightarrow {\textbf{Set}}\) be as in Lemma 5.16, then we have that all maps \(\phi _\alpha :{\mathcal {V}}_X(X_\alpha )\rightarrow {\textrm{colim}}({\mathcal {V}}_X)\) are surjections.

Proof

Fix some \({\mathcal {V}}_X(X_\alpha )\). We must show that every equivalence class \([v]\in {\textrm{colim}}({\mathcal {V}}_X)\) has a representative in \({\mathcal {V}}_X(X_\alpha )\). Certainly [v] has a representative \(v_\beta \) in some \({\mathcal {V}}_X(X_\beta )\). Let \(X_\gamma = X_\alpha \cup X_\beta \) and choose \(v_\gamma \in {\mathcal {V}}_X(X_\gamma )\) with \(\phi _{\gamma \beta }(v_\gamma )=v_\beta \), which is always possible since \(\phi _{\gamma \beta }\) is a surjection by Lemma 5.17. Now \(v_\alpha =\phi _{\gamma \alpha }(v_\gamma )\) is a representative for [v] since \(v_\gamma \sim v_\beta \) and \(v_\gamma \sim v_\alpha \). \(\square \)

Lemma 5.19

Let \({\mathcal {V}}_X:\mathbf {FinSet^*}(X)\rightarrow {\textbf{Set}}\) be as in Lemma 5.16. The set \({\textrm{colim}}({\mathcal {V}}_X)\) is finite.

Proof

The groupoid \(\pi (X,X_\alpha )\) is finitely generated since X is a homotopically 1-finitely generated space. Also G is finite, hence the \({\mathcal {V}}_X(X_\alpha )\) is finite for all \(X_\alpha \), so with Lemma 5.18 we have the result. \(\square \)

It is well known that there is an adjunction between the categories \(\textbf{Vect}_{\mathbb {C}}\) and \({\textbf{Set}}\) [31]. We denote by \(\mathrm {F_{V_{\mathbb {C}}}}:{\textbf{Set}}\rightarrow \textbf{Vect}_{\mathbb {C}}\) the left adjoint which sends a set X to the free vector space over X and a function to the unique linear map between the corresponding free vector spaces which acts in the same way on basis elements. Since left adjoints preserve colimits (Theorem. 2.9), \(F_{V_{{\mathbb {C}}}}\) preserves colimits. This means we can equivalently define a map on objects in \({\textrm{HomCob}}\) terms of the vector space with basis the set \({\textrm{colim}}({\mathcal {V}}_X)\), or as \({\textrm{colim}}(F_{V_{{\mathbb {C}}}}\circ {\mathcal {V}}_X)\) in \(\textbf{Vect}_{\mathbb {C}}\). In general the colimit in \({\textbf{Set}}\) will be easier to work with.

Definition 5.20

For \(X\in \chi \) define

where \({\mathcal {V}}_X' = F_{V_{{\mathbb {C}}}}\circ {\mathcal {V}}_X \) and \({\mathcal {V}}_X:\mathbf {FinSet^*}(X)\rightarrow {\textbf{Set}}\) is as in Lemma 5.16.

Magmoid morphism \({\textsf{Z}}_G:\textsf{HomCob}\rightarrow {\textsf{Vect}_\mathbb {C}}\)

The linear map \({\textsf{Z}}_G^{!!}\) assigned to a based homotopy cobordism still depends on the basepoints in objects. Here we define a map \({\textsf{Z}}_G\) which assigns to a based homotopy cobordism, \(i:(X,X_0) \rightarrow (M,M_0) \leftarrow (Y,Y_0):j\), a linear map \( {\textsf{Z}}_G(X)\rightarrow {\textsf{Z}}_G(Y)\). We will then show that this linear map is also be independent of \(X_0\) and \(Y_0\).

Let \((X,X_\alpha ),(X,X_\beta )\in {\pmb \chi }\). In the previous section we constructed \({\mathcal {V}}_X:\mathbf {FinSet^*}(X)\rightarrow {\textbf{Set}}\) (Lemma 5.16) which sends inclusions \(\iota _{\beta \alpha }:X_\beta \rightarrow X_\alpha \) to maps \(\phi _{\alpha \beta }:{\mathcal {V}} \left( X_\alpha \right) \rightarrow {\mathcal {V}} \left( X_\beta \right) \). Notice that \(F_{V_{{\mathbb {C}}}}\circ {\mathcal {V}} (X_\alpha )={\mathcal {V}}'(X_\alpha )={\textsf{Z}}_G^{!}(X,X_\alpha )\), so we have a map \(F_{V_{{\mathbb {C}}}}(\phi _{\alpha \beta }):{\textsf{Z}}_G^{!}(X,X_\alpha )\rightarrow {\textsf{Z}}_G^{!}(X,X_\beta )\). By abuse of notation we will also use \(\phi _{\alpha \beta }\) to refer to the maps \(F_{V_{{\mathbb {C}}}}(\phi _{\alpha \beta })\). In this section we will need to vary the input space in the construction of \({\mathcal {V}}_X\), thus we add a superscript denoting the space, so we have maps

Lemma 5.21

Let \(i:X \rightarrow M \leftarrow Y:j\) be a concrete homotopy cobordism. Then for any pair \(X_\alpha , X_\beta \subseteq X\) with \(X_\beta \subseteq X_\alpha \), and concrete based homotopy cobordisms \(i:(X,X_\alpha ) \rightarrow (M,M_{\alpha \alpha ^{\prime }}) \leftarrow (Y,Y_{\alpha ^{\prime }}):j\) and \(i:(X,X_\beta ) \rightarrow (M,M_{\beta \alpha ^{\prime }}) \leftarrow (Y,Y_{\alpha ^{\prime }}):j,\) the following diagram commutes

That is, the maps \({\textsf{Z}}_G^{!!}\) form a cocone over the vector spaces \({\textsf{Z}}_G^{!}(X,X_\alpha )\) and the maps \(\phi ^{X}_{\alpha \beta }\). Hence there is a unique map

Proof

First suppose \(X_\alpha =X_\beta \cup \{x\}\) for some \(x\notin X_\beta \). Let \(f \in {\textsf{Z}}_G^{!}(X,X_\alpha )\) and \( g\in {\textsf{Z}}_G^{!}(Y,Y_{\alpha ^{\prime }})\) be basis elements. We have

and

for appropriate choices \(M_{\alpha \alpha ^{\prime }}\) and \(M_{\beta \alpha ^{\prime }}\). We may choose \(M_{\alpha \alpha ^{\prime }}=M_{\beta \alpha ^{\prime }} \cup \{x\}\). There is a map from \(\left\langle g \,\big |\, {\textsf{Z}}_G^{!}(M,M_{\alpha \alpha ^{\prime }}) \,\big |\, f \right\rangle \) to \(\left\langle g \,\big |\, {\textsf{Z}}_G^{!}(M,M_{\beta \alpha ^{\prime }}) \,\big |\, \phi _{\alpha \beta }^{X}(f) \right\rangle \) given by taking the restriction of a map \(h:\pi (M,M_{\alpha \alpha ^{\prime }}) \rightarrow G\) to \(h'=h|_{\pi (M,M_{\beta \alpha ^{\prime }})}\). Note also that if \(h|_{\pi (X,X_\alpha )}=f\) and \(h|_{\pi (Y,Y_{\alpha '})}=g\), then also \(h'|_{\pi (X,X_\beta )}=\phi _{\alpha \beta }^{X}(f)\) and \(h'|_{\pi (Y,Y_{\alpha '})}=g\).

This map has an inverse given by extending any map \({\tilde{h}}:\pi (M,M_{\beta \alpha ^{\prime }}) \rightarrow G\) to a map \(\tilde{h'}:\pi (M,M_{\alpha \alpha '}) \rightarrow G\), which sends a path \(\gamma :x \rightarrow x'\) in \(\pi (X,X_\alpha )\), with \(x'\in X_\beta \), to \(f(\gamma )\), as in Lemma 2.5. If the map \({\tilde{h}}\) satisfies \({\tilde{h}}|_{\pi (X,X_\beta )}=\phi _{\alpha \beta }^{X}(f)\) and \({\tilde{h}}|_{\pi (Y,Y_{\alpha '})}=g\), then \({\tilde{h}}'|_{\pi (X,X_\alpha )}=f\) and \({\tilde{h}}'|_{\pi (Y,Y_{\alpha '})}=g\). Hence

Also \(|M_{\beta \alpha ^{\prime }}| - |X_\beta | = |M_{\alpha \alpha ^{\prime }}+1| - |X_\alpha +1|=|M_{\alpha \alpha ^{\prime }}|-|X_\alpha |\). The set \(X_\alpha \setminus X_\beta \) is finite, so we can repeat the same process for all \(\{x_1,...,x_n\}\in X_\alpha \setminus X_\beta \). \(\square \)

Remark 5.22

Notice that, had we defined the normalisation to be \(|G|^{-(M_0-\nicefrac {1}{2}(X_0+Y_0))}\), as is Yetter’s convention, the triangle in the previous Lemma would not be commutative. The fix is to redefine the maps \(\phi _{\alpha \beta }\) in such a way that they no longer send basis elements to basis elements. This complicates the picture slightly. It is straighforward to see that each choice leads to the same image on any cospan of the form \(\varnothing \rightarrow M\leftarrow \varnothing \).

Lemma 5.23

Let \(i:X \rightarrow M \leftarrow Y:j\) be a concrete homotopy cobordism. Fix a choice of \(Y_{\alpha '}\subseteq Y\) such that \((Y,Y_{\alpha '})\in {\pmb \chi }\). For each pair \(X_\alpha , X_\beta \subseteq X\) such that \((X,X_\alpha ),(X,X_\beta )\in {\pmb \chi }\) we have the following commutative diagram

The assignment

does not depend on the choice of \(Y_{\alpha '}\).

As above, where we have given a cospan we will use the notation \({\textsf{Z}}_G(M)\) for \({\textsf{Z}}_G(i:X \rightarrow M \leftarrow Y:j)\).

Proof

We show that the following diagram commutes for any pair \(Y_{\alpha ^{\prime }}, Y_{\beta ^{\prime }}\)

This implies that \(\phi _{\alpha ^{\prime }\beta ^{\prime }}^{Y}\) is a map of cocones and, by the universal property of the colimit, that \(\phi _{\alpha ^{\prime }\beta ^{\prime }}^{Y}d_{\alpha ^{\prime }}^M=d^M_{\beta ^{\prime }}\) and hence that \(\phi ^Y_{\alpha ^{\prime }} d_{\alpha ^{\prime }}^{M}=\phi ^Y_{\beta ^{\prime }}\phi _{\alpha ^{\prime }\beta ^{\prime }}^{Y}d_{\alpha ^{\prime }}^M=\phi ^Y_{\beta ^{\prime }} d^M_{\beta ^{\prime }}\).

Suppose first that \(Y_{\alpha ^{\prime }}=Y_{\beta ^{\prime }} \cup \{y\}\) for some \(y\notin Y_{\beta ^{\prime }}\) and let \(f\in {\textsf{Z}}_G^{!}(X,X_\alpha )\) and \(g\in {\textsf{Z}}_G^{!}(Y,Y_{\beta ^{\prime }})\) be basis elements. The map \(\phi ^{Y}_{\alpha ^{\prime },\beta ^{\prime }}:{\mathcal {V}}_Y(Y_{\alpha ^{\prime }})\rightarrow {\mathcal {V}}_Y(Y_{\beta ^{\prime }}) \) is surjective (by Lemma 5.17), so sends a subset of \({\mathcal {V}}_Y(Y_{\alpha ^{\prime }})\) to \(g\in {\mathcal {V}}_Y(Y_{\beta ^{\prime }})\). Thus the matrix element \(\langle f \vert \phi _{\alpha ^{\prime }\beta ^{\prime }}^{Y} {\textsf{Z}}_G^{!!}(M,X_\alpha ,Y_{\alpha ^{\prime }})\vert g \rangle \) is the sum of the matrix elements in \({\textsf{Z}}_G^{!!}(M,X_\alpha ,Y_{\alpha ^{\prime }})\) corresponding to f and each \(g'\) in the preimage \({\phi _{\alpha ^{\prime }\beta ^{\prime }}^{Y}}^{-1}(g)\). Hence we have

for an appropriate choice of \(M_{\alpha \alpha ^{\prime }}\). Following the same argument as used in the previous lemma we may choose a subset \(M_{\alpha \beta ^{\prime }}\) with \(M_{\alpha \alpha ^{\prime }}=M_{\alpha \beta ^{\prime }}\cup \{y\}\) and then

For every map \(g:\pi (Y,Y_{\beta ^{\prime }}) \rightarrow G\), there will be precisely G maps in the preimage under \(\phi _{\alpha ^{\prime }\beta ^{\prime }}^Y\), one for each choice of an element of G. This can be seen by noting that \(\phi _{\alpha ^{\prime }\beta ^{\prime }}^Y\) is the composition of the bijection \(\Theta _\gamma ^{-1}:{\textbf{Grpd}}(\pi (X,Y_{\alpha '}),G) \rightarrow {\textbf{Grpd}}(\pi (Y,Y_{\beta '}),G)\times G\) in Lemma 2.6 with the projection to the first coordinate, for some choice of \(\gamma :y\rightarrow y'\in M_{\alpha \beta '}\). Hence we have

Now suppose \(Y_{\alpha ^{\prime }} = Y_{\beta ^{\prime }} \cup \{y_1,...,y_n\}\). Then, we similarly acquire one factor of \(\vert G\vert \) and one factor \(\vert G \vert ^{-1}\) for each new point, hence \(\phi _{\alpha ^{\prime }\beta ^{\prime }}^{Y} {\textsf{Z}}_G^{!!}(M,X_\alpha ,Y_{\alpha ^{\prime }})={\textsf{Z}}_G^{!!}(M,X_\alpha ,Y_{\beta ^{\prime }})\). \(\square \)

Lemma 5.24

We have a magmoid morphism

where \({\textsf{Z}}_G\) is given in Definition 5.20 and Lemma 5.23.

Proof

Lemmas 5.13 and 5.23 give that \({\textsf{Z}}_G\) is well-defined.

We prove \({\textsf{Z}}_G\) preserves composition. Suppose we have concrete homotopy cobordisms \(i:X \rightarrow M \leftarrow Y:j\) and \(k:Y \rightarrow N \leftarrow Z:l\). Let \(Y_0\subseteq Y\) and \(Z_0\subseteq Z\) be fixed finite representative subsets. Notice that for any finite representative subset \(X_0\subseteq X\), by Lemma 5.14, we have \({\textsf{Z}}_G^{!!}(M\sqcup N,X_0,Z_0)={\textsf{Z}}_G^{!!}(N,Y_0,Z_0){\textsf{Z}}_G^{!!}(M,X_0,Y_0)=d_0^N\phi ^Y_0{\textsf{Z}}_G^{!!}(M,X_0,Y_0)\). Thus \(d_0^N\phi ^Y_0 d_0^M:{\textsf{Z}}_G(X)\rightarrow {\textsf{Z}}_G^{!}(Z,Z_0)\) is a map commuting with the cocone given by the \({\textsf{Z}}_G^{!!}(M\sqcup N,X_0,Z_0)\), where \(X_0\) is varying. Hence by the uniqueness of the map obtained from the universal property of the colimit, we have \(d_0^N\phi ^Y_0 d_0^M=d_0^{M\sqcup _{Y}N}\), which implies that \(\phi ^Z_0d_0^N\phi ^Y_0 d_0^M=\phi ^Z_0d_0^{M\sqcup _{Y}N}\), and finally that \({\textsf{Z}}_G(N){\textsf{Z}}_G(M)={\textsf{Z}}_G(M\sqcup _{Y}N)\). \(\square \)

The functor \({\textsf{Z}}_G:{\textrm{HomCob}}\rightarrow \textbf{Vect}_{\mathbb {C}}\)

The following theorem says that \({\textsf{Z}}_G\) becomes a functor from the category \({\textrm{HomCob}}\).

Theorem 5.25

There is a functor

defined as follows.

• For a space \(X\in Ob({\textrm{HomCob}})\),

  $$\begin{aligned} {\textsf{Z}}_G(X) = {\mathbb {C}}\left( {\textrm{colim}}({\mathcal {V}}_X) \right) \end{aligned}$$

  where \({\mathcal {V}}_X\) is the diagram in \({\textbf{Set}}\) with vertices \( {\mathcal {V}}_X (X_\alpha ) = {\textbf{Grpd}}\left( \pi (X,X_\alpha ),G \right) \) for each finite representative subset \(X_\alpha \subseteq X\) and edges \(\phi _{\alpha \beta }:{\mathcal {V}}_X(X_\alpha ) \rightarrow {\mathcal {V}}_X(X_\beta )\) whenever \(X_\beta \subseteq X_\alpha \), sending each \(f\in {\mathcal {V}}_X (X_\alpha )\) to \(f\circ \iota _{\beta \alpha }\) where \(\iota _{\beta \alpha }:\pi (X,X_\beta )\rightarrow \pi (X,X_\alpha )\) is the inclusion.

• For a homotopy cobordism \(\left[ i:X \rightarrow M \leftarrow Y:j\right] _{\!\tiny \hbox {ch}}\),

  

  where \(Y_{\alpha '}\subseteq Y\) is some choice of finite representative subset and, \(\phi _{\alpha ^{\prime }}^{Y}\) and \(d^M_{\alpha ^{\prime }}\) are as in Lemma 5.23.

Proof

We have from Lemma 5.24 that \({\textsf{Z}}_G\) is a magmoid morphism so it remains only to check that \({\textsf{Z}}_G\) does not depend on a choice of representative cospan and that it preserves identities. We will need a different interpretation of the colimit to prove that \({\textsf{Z}}_G\) preserves identities, we do this in Lemma 5.37.

In Lemma 5.12 we show that \({\textsf{Z}}_G^{!}\) does not depend on the representative homotopy cobordism we choose. It thus follows that \({\textsf{Z}}_G^{!!}\) and hence \({\textsf{Z}}_G\) do not depend on a choice of representative cospan. \(\square \)

The following Lemma gives an alternative description of the image of the linear map a cospan is sent to under \({\textsf{Z}}_G\), in terms of a choice of based cospan.

Lemma 5.26

Let \(i:X \rightarrow M \leftarrow Y:j\) be a concrete homotopy cobordism, \(i:(X,X_0) \rightarrow (M,M_0) \leftarrow (Y,Y_0):j\) a choice of concrete based homotopy cobordism, and \([f]\in {\textsf{Z}}_G(X)\) and \([g]\in {\textsf{Z}}_G(Y)\) be basis elements (so [f], for example, is an equivalence class in \({\textrm{colim}}({\mathcal {V}}_X)\)), then

where \(\phi _0^{Y}:{\textsf{Z}}_G^{!}(Y,Y_0)\rightarrow {\textsf{Z}}_G(Y)\) is the map into \({\textrm{colim}}({\mathcal {V}}'_Y)\); see Definition 5.20.

Proof

We will use the notation as in (5). Since each map \(\phi _0^{Y}\) is surjective (Lemma 5.18), we can find \(d^M_{0}([f])\) by looking at \({\textsf{Z}}_G^{!!}(M,X_0,Y_0)(f)\). Hence we have

and, choosing a basis element \([g]\in {\textsf{Z}}_G(Y)\),

\(\square \)

Remark 5.27

The set of maps \(\phi _0^{-1}([g])\) contains all maps \(g':\pi (Y,Y_0)\rightarrow G\) such that \(g'\sim g\) where \(\sim \) is the equivalence relation defined by the colimit. Since we are only counting the cardinality of maps h we can rewrite the map on morphisms as

where we have removed the sum and only insist on maps h being equivalent to g on Y. In many cases, especially with the local equivalence obtained in following section, this will be the most useful formulation to use for calculations.

Example 5.28

Let \(i:X \rightarrow M \leftarrow Y:j\) be the homotopy cobordism shown in Fig. 3. Note this is a homotopy cobordism from Examples 3.9 and 3.28. Using (6), we may choose to calculate the image of \({\textsf{Z}}_G(\left[ i:X \rightarrow M \leftarrow Y:j\right] _{\!\tiny \hbox {ch}})\) using the based homotopy cobordism considered in Example 5.10. Using the results and notation from Example 5.10, we have

5.4 Map \({\textsf{Z}}_G:Ob({\textrm{HomCob}})\rightarrow Ob({\textbf{Vect}}_{{\mathbb {C}}})\) in Terms of a Local Equivalence Relation

For a general homotopically 1-finitely generated space X it is unlikely to be straightforward to calculate the colimit constructed in the previous section. Usually there will be an uncountably infinite number of choices of finite representative subsets \(X_\alpha \subseteq X\), and thus an uncountably infinite number of vertices in \({\mathcal {V}}_X\). Despite the colimit being over an infinite number of objects we proved, in Lemma 5.19, that \({\textsf{Z}}_G(X)\) is finite dimensional for all X. In this section we show that the global equivalence, given by taking the colimit over all choices of subsets, is the same as choosing a single subset and taking a local equivalence given by taking maps up to natural transformation. This will allow us to prove, in Lemma 5.37, that \({\textsf{Z}}_G\) preserves the identity. We will also need this interpretation of \({\textsf{Z}}_G\) to prove, in Sect. 5.5, that \({\textsf{Z}}_G\) is a monoidal functor.

Here we only need to work with a single space X, so with \({\mathcal {V}}_X\) as constructed in Lemma 5.16, we drop the subscript on \({\mathcal {V}}_X\), and the superscript on the \(\phi ^X\). Consider the commuting diagram,

where \(\cong \) denotes the relation obtained by taking maps up to natural isomorphism (it is straightforward to check this is an equivalence relation). The set map \(p_\alpha \) sends a groupoid map in \({\textbf{Grpd}}(\pi (X,X_\alpha ),G)\) to its equivalence class in \({}^{{\mathcal {V}}(X_\alpha )}/_{\cong }\). The map \(\hat{\phi }_\alpha :{}^{{\mathcal {V}}(X_\alpha )}/_{\cong } \rightarrow {\textrm{colim}}({\mathcal {V}})\) is the canonical map sending an equivalence class to \(\phi _\alpha \) of some representative (it remains to check this is well defined).

Theorem 5.29

For a space X, the map \(\hat{\phi }_\alpha \) is an isomorphism. Hence, for a homotopically 1-finitely generated space X

for any choice \(X_0\subset X\) of finite representative subset, where \(\cong \) denotes taking maps up to natural transformation.

Proof

We prove \(\hat{\phi }_\alpha \) is well defined and injective in Lemmas 5.30 and 5.32 respectively. Surjectivity follows directly from Lemma 5.18. \(\square \)

For a path \(s:{{\mathbb {I}}}\rightarrow X\) in X, we will also use s to denote its path equivalence class in \(\pi (X)\) and \(s{\mathop {\sim }\limits ^{p}}s'\) to mean that \(s'\in [s]_{\!\tiny \hbox {p}}\).

Lemma 5.30

Let \(v_\alpha ,v_\alpha ' \in {\mathcal {V}}(X_\alpha )\) be two groupoid maps such that \(p_\alpha (v_\alpha ) = p_\alpha (v_\alpha ')\). Then \(\phi _\alpha (v_\alpha ) = \phi _\alpha (v_\alpha ')\).

Proof

There exists a subset \(X_{{\tilde{\alpha }}}\subseteq X_\alpha \) containing precisely one basepoint in each path-connected component, and maps \({\tilde{v}}_\alpha , {\tilde{v}}_\alpha ':\pi (X,X_{{\tilde{\alpha }}})\rightarrow G\) such that \(\phi _{\alpha {\tilde{\alpha }}}(v_\alpha )={\tilde{v}}_\alpha \) and \(\phi _{\alpha {\tilde{\alpha }}}(v_\alpha ')={\tilde{v}}_\alpha '\). We will show that \({\tilde{v}}_\alpha \) and \({\tilde{v}}_\alpha '\) are equivalent in the colimit, implying \(v_\alpha \sim v_\alpha '\). The idea of this proof is illustrated by the following diagram.

We use the morphisms in the natural transformation connecting \(v_\alpha \) and \(v_\alpha '\) to extend the map \({\tilde{v}}_\alpha \) to a map \(v_\gamma :\pi (X,X_\gamma )\rightarrow G\), where \(X_\gamma \) is a larger set of basepoints. We also trivially extend the map \({\tilde{v}}_\alpha '\) to \(v_\gamma ':\pi (X,X_\gamma )\rightarrow G\), and show that these extensions have the same image under some \(\phi _{\gamma \beta }\), and therefore are equivalent in the colimit.

The set \(X_{{\tilde{\alpha }}}\) is finite so we can write \(X_{{\tilde{\alpha }}}=\{x_1,...,x_N\}\). Recall that \(X_{{\tilde{\alpha }}}\) contains one point in each path-component, thus all morphisms in \(\pi (X,X_{{\tilde{\alpha }}})\) are represented by loops. Since \(v_\alpha \) and \(v_\alpha '\) are related by a natural transformation, for all points \(x_n \in X_{{\tilde{\alpha }}}\) and for all equivalence classes of loops \(s:x_n \rightarrow x_n\), the below square commutes.

Recall that the image of \(v_\alpha \) and \(v_{\alpha '}\) is a groupoid with one object, so the image on points is always the same. Hence the two maps must be the same on any path-components that have no non-trivial paths.

Choose another set of points \(X_\beta =\{y_1,...,y_N\}\) as follows. If there are no non-trivial loops based at \(x_n\) then \(y_n=x_n\), otherwise choose \(y_n \ne x_n\) and, for each n, choose a path \(t_n:x_n\rightarrow y_n\), with \(t_n\) the constant path if \(x_n=y_n\). This is always possible since a non-trivial loop based at \(x_n\) must contain some \(y_n\ne x_n\).

Let \(X_\gamma =X_{{\tilde{\alpha }}} \cup X_\beta \). We define a map \(v_\gamma :\pi (X,X_\gamma ) \rightarrow G\) as follows. Let \(v_\gamma |_{\pi (X,X_{{\tilde{\alpha }}})}= {\tilde{v}}_{\alpha }\), and \(v_\gamma (t_n) = \eta _{x_n}\) unless \(t_n\) is the constant path, in which case \(v_\gamma (t_n)=1_G\). By Lemma 2.5 this completely defines \(v_\gamma \). Notice \(\phi _{\gamma {\tilde{\alpha }}}(v_\gamma )={\tilde{v}}_{\alpha }\), hence \({\tilde{v}}_{\alpha } \sim v_\gamma \).

Define another map \(v_\gamma ':\pi (X,X_\gamma ) \rightarrow G\) by \(v_\gamma '|_{\pi (X,X_{{\tilde{\alpha }}})}={\tilde{v}}_{\alpha }'\) and \(v_\gamma '(t_n)=1_G\). We have \(\phi _{\gamma {\tilde{\alpha }}}(v_\gamma ')={\tilde{v}}_{\alpha }'\) and so \({\tilde{v}}_\alpha ' \sim v_\gamma '\).

Now we check that \(\phi _{\gamma \beta }(v_\gamma )=\phi _{\gamma \beta }(v'_\gamma )\), hence \(v_\gamma \sim v_\gamma '\). Since \(X_\beta \) has only one point in each path-connected component we only need to check that \(v_\gamma \) and \(v_{\gamma '}\) agree on loops. For any trivial \(s:x_n \rightarrow x_n\) with \(y_n=x_n\), we have \(v_\gamma (s)=1_G={\tilde{v}}_\alpha (s)={\tilde{v}}'_\alpha (s)\).

Now suppose \(s:y_n \rightarrow y_n\) is any class of loops with \(y_n\ne x_n\),

and similarly,

Hence \(\phi _{\gamma \beta }(v_\gamma )=\phi _{\gamma \beta }(v_\gamma ')\) so \(v_\gamma \sim v_\gamma '\) and \({\tilde{v}}_{\alpha } \sim {\tilde{v}}_{\alpha }'\). \(\square \)

The following Lemma is well known, and straightforward to prove. We write the proof here only as an opportunity us to fix maps exhibiting the equivalence which will be useful in subsequent proofs.

Lemma 5.31

For any finite representative subset \(X_\alpha \) of a space X, \(\pi (X,X_\alpha )\) and \(\pi (X)\) are equivalent as categories. \(\square \)

Proof

We have an inclusion \(\iota _\alpha :\pi (X,X_\alpha )\rightarrow \pi (X)\). We define explicitly a map \(r_\alpha :\pi (X)\rightarrow \pi (X,X_\alpha )\) as follows. For each \(x\in X{\setminus } X_\alpha \), choose a point \(y_x \in X_\alpha \) in the same path-connected component as x, and a path \(t_x:x \rightarrow y_x\). If \(x\in X_\alpha \) choose \(y_x=x\) and \(t_x\) the trivial path. Now define \( r_\alpha (x)= y_x \) and, for a path \(s:x\rightarrow x'\) in \(\pi (X)\), \( r_\alpha (s)=t_{x'} s t_{x}^{-1} \) The composition \(r_\alpha \iota _\alpha \) is equal to \(1_{\pi (X,X_\alpha )}\) and a natural transformation \(\eta :1_{\pi (X)} \rightarrow \iota _\alpha r_\alpha \) is given by \( \eta _{x}= t_x \). \(\square \)

Lemma 5.32

Let \(v_\alpha ,v_\alpha ' \in {\mathcal {V}}(X_\alpha )\) be two maps such that \(\phi _\alpha (v_\alpha )= \phi _\alpha (v_\alpha ')\). Then \(p_\alpha (v_\alpha ) = p_\alpha (v_\alpha ')\).

Proof

The maps \(v_\alpha \) and \(v_\alpha '\) being equivalent in the colimit means there is some finite sequence of relations \(v_\alpha = v_0 \sim v_1 \sim ... \sim v_N = v_\alpha '\), where \(v_n\ne v_{n+1}\), and of maps \(v_n:\pi (X,X_n) \rightarrow G\) such that for each pair \(v_n,v_{n+1}\) we have one of the following two diagrams:

Then we have a commuting diagram of the following form

where we let \(X_{n,n+1}\) be the larger of \(X_n\) and \(X_{n+1}\) and one of \(\iota '_n\) and \(\iota '_{n+1}\) is a strict inclusion, and the other is the identity. The middle arrow is either \(v_n\) or \(v_{n+1}\). Consider the (non-commuting) diagram below, where the maps r are as constructed in the proof of Lemma 5.31.

We will show there is a natural transformation \(v_0 r_0\) to \(v_N r_N\). Since \(X_0=X_N=X_\alpha \), \(\iota _0=\iota _N\) and hence \(v_0 r_0\cong v_N r_N\) implies \(v_0 r_0\iota _0 \cong v_N r_N \iota _N\). This implies \(v_0 \cong v_N\) since \(r_\beta \iota _\beta =1_{\pi (X,X_\beta )}\) for all finite representative \(X_\beta \subseteq X\) by Lemma 5.31.

We show all triangles in the diagram commute up to natural transformation. The bottom triangles commute exactly by the construction explained in the first part of the proof. Notice that \(\iota _n' r_n = r_{n,n+1} \iota _n r_n\), where \(\iota _n:\pi (X,X_n)\rightarrow \pi (X)\) is the inclusion. By Lemma 5.31 we have that \( r_{n,n+1} \iota _n r_n \simeq r_{n,n+1}\). \(\square \)

Example 5.33

Let \(X=S^1\sqcup S^1\). Then, letting \(X_0\subset X\) be a subset with precisely one point in each connected component, \({\textbf{Grpd}}(\pi (X,X_0),G)=G\times G\) as discussed in Example 5.7. Taking maps up to natural transformation corresponds to pairs of conjugacy classes of elements of G, so we have \({\textsf{Z}}_G(X)={\mathbb {C}}(G/G \times G/G)\).

Example 5.34

Consider again Example 5.28, which in turn refers to Examples 3.9 and 3.28. The equivalence class \([g_1]\) in the basis of \({\textsf{Z}}_G(Y)\) consists of all maps sending \(S^1\) to something in the conjugacy class of \(g_1\). This allows us to refine the result of Example 5.28 as follows.

Example 5.35

Let X be the complement of the embedding of two circles shown, and explained in the caption of, Fig. 5. Then, letting \(X_0\subset X\) be the subset shown, we have \({\textbf{Grpd}}(\pi (X,X_0),G)=G\times G\) as discussed in Example 5.8. Since all objects are mapped to the unique object in G, taking maps up to natural transformation means taking maps up to conjugation by elements of G at each basepoint, hence in this case maps are labelled by pairs of elements of G, up to simultaneous conjugation, so we have \({\textsf{Z}}_G(X)={\mathbb {C}}((G \times G)/G)\).

This figure represents the concrete cofibrant cospan from Example 3.8. The red points and lines show a possible choice of basepoints \(M_0\) and paths. It can be seen that the equivalence classes of the marked paths generate \(\pi (M,M_0)\). We can see X and Y are homotopically 1-finitely generated by considering the intersection of the marked points and paths with X and Y respectively. Thus it is a concrete homotopy cobordism. (Color figure online)

Full size image

Example 5.36

Consider the concrete homotopy cobordism shown in Fig. 2. This represents a manifold M, which is the complement of the marked subset in \({{\mathbb {I}}}^3\), and X and Y are given by the bottom and top boundary respectively. This becomes a cospan with the inclusion maps. This is in fact a homotopy cobordism from Examples 3.8 and 3.29.

We calculate \({\textsf{Z}}_G\big (\left[ i:X \rightarrow M \leftarrow j:Y\right] _{\!\tiny \hbox {ch}}\big )\). We choose to use the based homotopy cobordism shown in Fig. 7 for calculation. The set \(M_0\) consists of all marked points and \(X_0\) and \(Y_0\) consist of the intersection of \(M_0\) with X and Y respectively.

We have from Example 5.34 that basis elements in \({\textsf{Z}}_G(X)\) are given by equivalence classes \([(f_1,f_2)]\) where \(f_1,f_2\in G\) and [] denotes simultaneous conjugation by the same element of G.

Basis elements in \({\textsf{Z}}_G(Y)\) are given by elements of G taken up to conjugation, denoted \([g_1]\).

Let \(x\in X\) be the basepoint which is in the connected component of X homotopy equivalent to the punctured disk, and \(x'\in X\) some choice of basepoint in the other connected component. It follows from Lemma 2.6 that there is a bijection sending a map \(h\in {\textbf{Grpd}}(\pi (M,M_0), G)\) to a quadruple \((h', h(\gamma _1), h(\gamma _2), h(\gamma _3))\in {\textbf{Grpd}}(\pi (M,\{x,x'\})\times G\times G\times G\), where \(h'\) is the restriction of h to \(\pi (M,\{x,x'\})\). Now \(\pi (M,\{x,x'\})\) is the disjoint union of the groupoids \(\pi (M_1,\{x\})\) and \(\pi (M_2,\{x'\})\) where \(M_1\) is the path connected component of M containing x, and \(M_2\) is the path connected component containing \(x'\). The group \(\pi (M_2,\{x'\})\) is trivial, so there is one unique map into G. The group \(\pi (M_1,\{x\})\) is equivalent to the twice punctured disk (see Example 3.29), which has fundamental group isomorphic to the free product \({\mathbb {Z}}*{\mathbb {Z}}\). This isomorphism can be realised by sending the loop \(x_1\) to the 1 in the first copy of \({\mathbb {Z}}\) and \(x_2\) to the 1 in the second copy of \({\mathbb {Z}}\). Thus we can label elements in \({\textbf{Grpd}}(\pi (M_1,\{x\}), G)\) by elements of \(G\times G\) where \(g_1 \in (g_1,g_2)\) corresponds to the image of \(x_1\), and \(g_2\) the image of \(x_2\). Hence a map in \({\textbf{Grpd}}(\pi (M,M_0), G)\) is determined by a five tuple \((a,b,c,d,e)\in G\times G\times G\times G\times G\) where a corresponds to the image of \(x_1\), b to the image of \(x_2\), and c, d and e correspond to the images of \(\gamma _1\), \(\gamma _2\) and \(\gamma _3\) respectively. Hence we have

We now use Theorem 5.29 to prove that identities are preserved.

Lemma 5.37

The identity homotopy cobordism \(\left[ \iota _0^X:X \rightarrow X\times {{\mathbb {I}}} \leftarrow X:\iota _1^X\right] _{\!\tiny \hbox {ch}}\) for a space X is mapped to the identity matrix by \({\textsf{Z}}_G\).

Proof

We will show that the matrix element

is 1 if \(\left[ f\right] =\left[ g\right] \) and 0 otherwise.

First note, there is an isomorphism

given by sending a representative path to the pair containing the classes of each projection (see 6.4.4 in [8]). Hence we have that \(\left\langle f \; \big |\; {\textsf{Z}}_G^{!}(X\times {{\mathbb {I}}}) \;\big |\; g \right\rangle \) is given by the cardinality of the set of maps

such that

where \(e_i\) denotes the constant path at the point i. Any pair in the product space can be written as a composition of pairs with only one non-identity component. The morphisms of \(\pi \left( {{\mathbb {I}}},\{0,1\}\right) \) are generated by the equivalence class of the path \(\textrm{id}_{{\mathbb {I}}}:{{\mathbb {I}}}\rightarrow {{\mathbb {I}}}\). Thus a map h is completely defined by specifying its action on pairs of the form \((e_{x_j},\textrm{id}_{{\mathbb {I}}})\). Let \(s:x_0\rightarrow x_1\) be a path in X with \(x_0,x_1\in X_0\). Notice that

Hence such an h exists if and only if the \(h(e_{x_i},\textrm{id}_{{\mathbb {I}}})\) form a natural transformation from f to g. By Theorem 5.29 this means the matrix element corresponding to f and g is zero unless \([f]=[g]\).

Now we consider the matrix element

A map h is a defined by a choice of \(h(e_{x_0},\textrm{id})\in G\) for each \(x_0\in X\), and all choices define a natural transformation. Using the definition of \({\textsf{Z}}_G\) from Lemma 5.26, we must sum over all \(\left\langle f \; \big |\; {\textsf{Z}}_G^{!}(X\times {{\mathbb {I}}}) \;\big |\; f' \right\rangle \) with \(f'\sim f\) equivalent in the colimit, which, by Theorem 5.29, means there is a natural transformation f to \(f'\). Hence all choices of \(h(e_{x_0},\textrm{id})=g\in G\) will contribute to the sum, and there are \(|G|^{|X_0|}\) such choices. Taking account of the normalisation, the matrix element is 1. \(\square \)

To complete this section, we give our final alternative way to define the map on objects.

Theorem 5.38

Let X be a homotopically 1-finitely generated space. Then

where \(\cong \) denotes the relation given by taking the set of groupoid maps up to natural transformation.

Proof

We have from Theorem 5.29 that \({\textsf{Z}}_G(X)\cong {\mathbb {C}}({\textbf{Grpd}}(\pi (X, X_0),G)/\cong )\), for some finite representative set \(X_0\).

We have from Lemma 5.31 that \(\pi (X,X_0)\) and \(\pi (X)\) are equivalent as categories. Let \(\iota _0:\pi (X,X_0)\rightarrow X\) and \(r_0:\pi (X)\rightarrow \pi (X,X_0)\) be as in the proof of Lemma 5.31.

Let \([f]\in {\textbf{Grpd}}(\pi (X),G)/\cong \), then there is a map

We show this map is well defined. Suppose \(f'\in [f]\), so there is natural transformation, say \(\eta \), from f to \(f'\). Then \(f'\circ \iota _0\sim f\circ \iota _0\) using the restriction of \(\eta \) to \(x\in X_0\).

This \(\phi \) has inverse \(\phi '\), where \(\phi '(g)=g\circ r_0\). Again this map is well defined, this time if \(\eta \) is a natural transformation in \({\textbf{Grpd}}(\pi (X, X_0),G)/\cong \), then the maps \(\eta _{r_0(x)}\) give the required natural transformation. \(\square \)

5.5 Monoidal Functor \({\textsf{Z}}_G:{\textrm{HomCob}}\rightarrow \textbf{Vect}_{\mathbb {C}}\)

We now show that the functor \({\textsf{Z}}_G\) is symmetric monoidal when considering \({\textrm{HomCob}}\) with the monoidal structure described in Sect. 3.3.1 and \(\textbf{Vect}_{\mathbb {C}}\) with the monoidal structure described in Sect. 2.1.1. We will need the following lemma.

Lemma 5.39

Let X and Y be homotopically 1-finitely generated spaces. There is a bijection

where \({\mathcal {V}}_X\) is as in Lemma 5.16

Proof

For any subsets \(X_\alpha \subseteq X\) and \(Y_{\alpha '} \subseteq Y\), and points \(x\in X_\alpha \) and \(y\in Y_{\alpha '}\), we have that \(\pi (X\sqcup Y,X_{\alpha }\sqcup Y_{\alpha '})(x,y)\) is empty. Thus there is an isomorphism of groupoids \(\pi (X\sqcup Y,X_\alpha \sqcup Y_{\alpha '})\xrightarrow {\sim }\pi (X,X_\alpha )\sqcup \pi (Y,Y_{\alpha '})\) and we have a bijection \({\textbf{Grpd}}(\pi (X\sqcup Y,X_\alpha \sqcup Y_{\alpha '}),G)\xrightarrow {\sim } {\textbf{Grpd}}(\pi (X,X_{\alpha }),G)\times {\textbf{Grpd}}(\pi (Y,Y_{\alpha '}),G)\) sending a map to the appropriate pair of restrictions. Equivalently we have a bijection \({\mathcal {V}}_{X\sqcup Y}(X_\alpha \sqcup Y_{\alpha '})\xrightarrow {\sim }{\mathcal {V}}_X(X_\alpha )\times {\mathcal {V}}_Y(Y_{\alpha '})\). Thus \({\textrm{colim}}({\mathcal {V}}_{X\sqcup Y})\) is isomorphic to the colimit over the diagram with vertices of the form \({\mathcal {V}}_X(X_\alpha )\times {\mathcal {V}}_Y(Y_{\alpha '})\) and maps of the form \((\phi _{\alpha \beta }^X,\phi _{\alpha '\beta '}^Y)\), which we denote \({\textrm{colim}}({\mathcal {V}}_{X\sqcup Y})'\). We construct a bijection between \({\textrm{colim}}({\mathcal {V}}_{X\sqcup Y})'\) and \({\textrm{colim}}({\mathcal {V}}_X)\times {\textrm{colim}}({\mathcal {V}}_Y)\).

Suppose \([(f,g)]=[(f',g')]\) in \({\textrm{colim}}({\mathcal {V}}_{X\sqcup Y})'\) with \((f,g)\in {\mathcal {V}}_X(X_\alpha )\times {\mathcal {V}}_Y(Y_{\alpha '})\) and \((f',g')\in {\mathcal {V}}_X(X_\beta )\times {\mathcal {V}}_Y(Y_{\beta '})\). By the construction of the colimit, there exists a sequence of product sets \({\mathcal {V}}_X(X_0)\times {\mathcal {V}}_Y(Y_0),...,{\mathcal {V}}_X(X_n)\times {\mathcal {V}}_Y(Y_n)\) with \({\mathcal {V}}_X(X_0)\times {\mathcal {V}}_Y(Y_0)={\mathcal {V}}_X(X_\alpha )\times {\mathcal {V}}_Y(Y_{\alpha '})\) and \({\mathcal {V}}_X(X_n)\times {\mathcal {V}}_Y(Y_n)={\mathcal {V}}_X(X_{\beta })\times {\mathcal {V}}_Y(Y_{\beta '})\), and a sequence of maps \(\phi _0,...,\phi _{n-1}\) connecting (f, g) and \((f',g')\) where each \(\phi _i\) is either a map \( {\mathcal {V}}_X(X_n)\times {\mathcal {V}}_Y(Y_n)\rightarrow {\mathcal {V}}_X(X_{n+1})\times {\mathcal {V}}_Y(Y_{n+1})\) or a map \({\mathcal {V}}_X(X_{n+1})\times {\mathcal {V}}_Y(Y_{n+1})\rightarrow {\mathcal {V}}_X(X_n)\times {\mathcal {V}}_Y(Y_n)\). The projections of this sequence of maps give sequences of maps connecting f and \(f'\) in \({\textrm{colim}}({\mathcal {V}}_X)\) and g and \(g'\) in \({\textrm{colim}}({\mathcal {V}}_Y)\). Thus there is a well defined map

It is easy to see this map is a surjection. To see that it is an injection, suppose now that \([f]=[f']\) in \({\textrm{colim}}({\mathcal {V}}_X)\) and \([g]=[g']\) in \({\textrm{colim}}({\mathcal {V}}_Y)\) then there are sequences \(\phi ^f_0,...,\phi ^f_n\) and \(\phi ^g_0,...,\phi ^g_n\) as in the proof of well definedness. Now the sequence given by \((\phi ^f_0,\textrm{id}),...,(\phi ^f_n,\textrm{id}),(\textrm{id},\phi ^g_0),...(\textrm{id},\phi ^g_1)\) is a sequence connecting (f, g) and \((f',g')\) in \({\textrm{colim}}({\mathcal {V}}_{X\sqcup Y})'\). \(\square \)

Lemma 5.40

The functor \({\textsf{Z}}_G:{\textrm{HomCob}}\rightarrow \textbf{Vect}_{\mathbb {C}}\) endowed with \(({\textsf{Z}}_G)_0=1_{\mathbb {C}}:{\mathbb {C}}\rightarrow {\mathbb {C}}\) and natural transformations

which acts on basis elements as

with \(\kappa \) as in Lemma 5.39, is strong monoidal.

(Here \(\textbf{Vect}_{\mathbb {C}}\) has the monoidal structure from Sect. 2.1.1 and the monoidal structure on \({\textrm{HomCob}}\) is as in Sect. 3.3.1.)

Proof

Notice \({\textrm{colim}}({\mathcal {V}}_{\varnothing })=\varnothing \) as \({\mathcal {V}}_{\varnothing }\) has just one vertex, the empty set, and no maps. Hence \({\textsf{Z}}_G(\varnothing )={\mathbb {C}}\) so \(({\textsf{Z}}_G)_0\) is well defined.

The vector space \({\textsf{Z}}_G(X) \otimes _{{\mathbb {C}}} {\textsf{Z}}_G(Y)\) has a basis isomorphic to \({\textrm{colim}}({\mathcal {V}}_X)\times {\textrm{colim}}({\mathcal {V}}_Y)\). Thus the map \(({\textsf{Z}}_G)_2(X,Y)\) is the linear extension of \(\kappa ^{-1}\), hence an isomorphism by Lemma 5.39.

The only complication in checking the associativity relation is understanding the image of the associator, \({\textsf{Z}}_G(\alpha _{X,Y,Z})\). On basis elements \({\textsf{Z}}_G(\alpha _{X,Y,Z})((f\otimes _{{\mathbb {C}}}g)\otimes _{{\mathbb {C}}}h)=f\otimes _{{\mathbb {C}}}(g\otimes _{{\mathbb {C}}} h)\). Similarly we can check the unitality relations using that, on basis elements, we have \({\textsf{Z}}_G(\lambda _{X})(\varnothing \otimes _{{\mathbb {C}}} f)= f\) and \({\textsf{Z}}_G(\rho _{X})(f\otimes _{{\mathbb {C}}} \varnothing )= f\). The proofs of each identity are similar to the proof of Lemma 5.37, that the identity is preserved, so we don’t write out the details here. \(\square \)

Lemma 5.41

The monoidal functor \({\textsf{Z}}_G:{\textrm{HomCob}}\rightarrow \textbf{Vect}_{\mathbb {C}}\) is symmetric monoidal.

Proof

As in the previous proof it is straightforward to check the relevant identity. \(\square \)

Lemma 5.42

The functor

where \(\textrm{Cob}_{n}\) is as in Proposition 3.34, is a topological quantum field theory for all \(n\in {\mathbb {N}}\), i.e. is a symmetric monoidal functor.

Proof

We have from Propositions 3.34 and 3.37 that \(\textrm{Cob}_{n}\) is a symmetric monoidal functor into \({\textrm{HomCob}}\) and from Theorem 5.25 and Lemma 5.41 that \({\textsf{Z}}_G\) is a symmetric monoidal functor \({\textrm{HomCob}}\rightarrow \textbf{Vect}_{\mathbb {C}}\). \(\square \)