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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
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:



$$\begin{equation} \label{squ} \D= \vcenter{\xymatrix @=3pc {S \ar @ [r] ^{s^1} \ar @<-1ex> [r] _{t^1} \ar @ [d]^{\, t_2} \ar @<-1ex> [d]_{s_2} & H   \ar[l] \ar @ [d]^{\,t} \ar @<-1ex> [d]_s \\ V \ar [u] \ar @ [r] ^s \ar @<-1ex> [r] _t & M \ar [l] \ar[u] }} \end{equation}$$

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



$$\begin{equation} \begin{pmatrix} & h& \\[-0.9ex] v & & v'\\[-0.9ex]& h'& \end{pmatrix} \end{equation}$$

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.