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This article is about double groupoids in higher category theory and higher dimensional algebra.

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

Double groupoid.

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

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 $\Box (H,V)$ with H,V as horizontal and vertical edge groupoids, and squares given by quadruples

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:

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

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.

References

• Brown, Ronald and C.B. Spencer: "Double groupoids and crossed modules.", Cahiers Top. Geom. Diff.. 17 (1976), 343-362.
• Brown, Ronald, 1987, "From groups to groupoids: a brief survey," Bull. London Math. Soc. 19: 113-34. Reviews the history of groupoids up to 1987, starting with the work of Brandt on quadratic forms. The downloadable version updates the many references.
• Brown, Ronald,, 2006. Topology and groupoids. Booksurge. Revised and extended edition of a book previously published in 1968 and 1988. Groupoids are introduced in the context of their topological application.
• 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.
• F. Borceux, G. Janelidze, 2001, Galois theories. Cambridge Univ. Press. Shows how generalisations of Galois theory lead to Galois groupoids.
• Cannas da Silva, A., and A. Weinstein, Geometric Models for Noncommutative Algebras. Especially Part VI.
• Golubitsky, M., Ian Stewart, 2006, "Nonlinear dynamics of networks: the groupoid formalism", Bull. Amer. Math. Soc. 43: 305-64
• Higgins, P. J., "The fundamental groupoid of a graph of groups", J. London Math. Soc. (2) 13 (1976) 145--149.
• Higgins, P. J. and Taylor, J., "The fundamental groupoid and the homotopy crossed complex of an orbit space", in Category theory (Gummersbach, 1981), Lecture Notes in Math., Volume 962. Springer, Berlin (1982), 115--122.
• Higgins, P. J., 1971. Categories and groupoids. Van Nostrand Notes in Mathematics. Republished in Reprints in Theory and Applications of Categories, No. 7 (2005) pp. 1-195; freely downloadable. Substantial introduction to category theory with special emphasis on groupoids. Presents applications of groupoids in group theory, for example to a generalisation of Grushko's theorem, and in topology, e.g. fundamental groupoid.
• Weinstein, Alan, "Groupoids: unifying internal and external symmetry -- A tour though some examples." Also available in Postscript., Notices of AMS, July 1996, pp. 744-752.