Double groupoid
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.
A double groupoid D is a higher-dimensional groupoid involving a relationship for 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 diagram:
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 composition laws for a double groupoid D make it also describable as a groupoid internal to the category of groupoids.
Given two groupoids H and V over a set M, there is a double groupoid with H,V as horizontal and vertical edge groupoids, and squares given by quadruples
for which one assumes always that h, h′ are in H and 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:
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