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In mathematics, especially in higher dimensional algebra and homotopy theory, a double groupoid generalises the notion of groupoid and of category to a higher dimension.

Definition

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Double groupoid.

A double groupoid D is a higher dimensional groupoid involving a relationship that involves both `horizontal' and `vertical' groupoid structures[1]. (A double groupoid can also be considered as a generalization of certain higher dimensional groups[2].) The geometry of squares and their compositions leads to a common representation of a double groupoid in the following form:

Double Groupoid Diagram

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} \label{squ} \D= \vcenter{\xymatrix @=3pc {S \ar @<1ex> [r] ^{s^1} \ar @<-1ex> [r] _{t^1} \ar @<1ex> [d]^{\, t_2} \ar @<-1ex> [d]_{s_2} & H \ar[l] \ar @<1ex> [d]^{\,t} \ar @<-1ex> [d]_s \\ V \ar [u] \ar @<1ex> [r] ^s \ar @<-1ex> [r] _t & M \ar [l] \ar[u] }} \end{equation}}


where M is a set of `points', H and V are, respectively, `horizontal' and `vertical' groupoids, and S is a set of `squares' with two compositions. The laws for a double groupoid D make it also describable as a groupoid internal to the category of groupoids.

Given two groupoids H, V over a set M, there is a double groupoid Double groupoid square with H,V as horizontal and vertical edge groupoids, and squares given by quadruples


Vertical and Horizontal D- edges

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} \begin{pmatrix} & h& \\[-0.9ex] v & & v'\\[-0.9ex]& h'& \end{pmatrix} \end{equation}}


for which we assume always that h, h' are in H, v, v' are in V, and that the initial and final points of these edges match in M as suggested by the notation, that is for example sh = sv, th = sv',..., etc. The compositions are to be inherited from those of H,V, that is:

Double Groupoid Operations

This construction is the right adjoint to the forgetful functor which takes the double groupoid as above, to the pair of groupoids H,V over M.

Double Groupoid Category

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The category whose objects are double groupoids and whose morphisms are double groupoid homomorphisms that are double groupoid diagram (D) functors is called the double groupoid category, or the category of double groupoids.

Notes

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  1. ^ Brown, Ronald and C.B. Spencer: "Double groupoids and crossed modules.", Cahiers Top. Geom. Diff.. 17 (1976), 343-362
  2. ^ Brown, Ronald,, Higher dimensional group theory Explains how the groupoid concept has to led to higher dimensional homotopy groupoids, having applications in homotopy theory and in group cohomology

References

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