Higher fundamental groupoids for spaces. (English) Zbl 1023.55007
The author defines fundamental \(n\)-groupoids for topological spaces.
Section 1 deals with the structure of the standard path functor considering the involutive lattice structure of the interval \([0,1]\).
The functor of Moore paths, \(\mathbb{P}\), has a similar structure but an associative concatenation. Thus, Section 2 starts with the space \(\mathbb{P}X\) of Moore paths, which produces a cubical object with associative concatenations. A quotient \(\mathcal{P}X\) of the underlying set \(|\mathbb{P}X|\) gives a groupoid called the groupoid of strongly reduced paths. From this, a cubical \(\omega\)-groupoid is derived, and then, the fundamental \(\omega\)-groupoid \(\Pi_{\omega}X\). Finally, the fundamental \(n\)-groupoid is obtained by applying the ‘reflector’ of \(n\)-groupoids.
In section 3, the fundamental category of a directed topological space [see M. Grandis, Directed Homotopy Theory, I. The fundamental category, Cah. Topologie Géom. Différ. Catég., to appear. Preprint http://arXiv.org/abs/math.AT/0111048] is extended to any dimension, following a treatment parallel to the previous section.
The last section compares these constructions with the fundamental categories of (symmetric) simplicial sets, introduced in previous works by the author [see Appl. Categ. Struct. 10, 99-155 (2002; Zbl 0994.55011) and Cah. Topologie Géom. Différ. Catég. 42, 101-136 (2001; Zbl 1006.18014)].
Section 1 deals with the structure of the standard path functor considering the involutive lattice structure of the interval \([0,1]\).
The functor of Moore paths, \(\mathbb{P}\), has a similar structure but an associative concatenation. Thus, Section 2 starts with the space \(\mathbb{P}X\) of Moore paths, which produces a cubical object with associative concatenations. A quotient \(\mathcal{P}X\) of the underlying set \(|\mathbb{P}X|\) gives a groupoid called the groupoid of strongly reduced paths. From this, a cubical \(\omega\)-groupoid is derived, and then, the fundamental \(\omega\)-groupoid \(\Pi_{\omega}X\). Finally, the fundamental \(n\)-groupoid is obtained by applying the ‘reflector’ of \(n\)-groupoids.
In section 3, the fundamental category of a directed topological space [see M. Grandis, Directed Homotopy Theory, I. The fundamental category, Cah. Topologie Géom. Différ. Catég., to appear. Preprint http://arXiv.org/abs/math.AT/0111048] is extended to any dimension, following a treatment parallel to the previous section.
The last section compares these constructions with the fundamental categories of (symmetric) simplicial sets, introduced in previous works by the author [see Appl. Categ. Struct. 10, 99-155 (2002; Zbl 0994.55011) and Cah. Topologie Géom. Différ. Catég. 42, 101-136 (2001; Zbl 1006.18014)].
Reviewer: J.Remedios (La Laguna)
MSC:
| 55P99 | Homotopy theory |
| 18G55 | Nonabelian homotopical algebra (MSC2010) |
| 18D05 | Double categories, \(2\)-categories, bicategories and generalizations (MSC2010) |
| 55U10 | Simplicial sets and complexes in algebraic topology |
| 55P65 | Homotopy functors in algebraic topology |
Keywords:
homotopy theory; homotopical algebra; higher dimensional categories; higher dimensional groupoids; simplicial setsReferences:
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