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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 == A double groupoid D is a higher-dimensional groupoid involving a relationship for both `horizontal' and `vertical' groupoid structures.〔Brown, Ronald and C.B. Spencer: "Double groupoids and crossed modules", ''Cahiers Top. Geom. Diff.''. 17 (1976), 343–362〕 (A double groupoid can also be considered as a generalization of certain higher-dimensional groups.〔Brown, Ronald, (Higher-dimensional group theory ) explains how the groupoid concept has led to higher-dimensional homotopy groupoids, having applications in homotopy theory and in group cohomology〕) 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: :: and :: 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. Other related constructions are that of a double groupoid with connection〔http://planetphysics.org/encyclopedia/DoubleGroupoidWithConnection.html Double Groupoid with Connection〕 and homotopy double groupoids.〔Brown, R., Hardie, K., Kamps, H. and T. Porter: 2002, "The homotopy double groupoid of a Hausdorff space.", ''Theory and Applications of Categories'': 10, 71–93〕 The homotopy double groupoid of a pair of pointed spaces is a key element of the proof of a two-dimensional Seifert-van Kampen Theorem, first proved by Brown and Higgins in 1978,〔Brown, R. and Higgins, P.J. "On the connection between the second relative homotopy groups of some related spaces". _Proc. London Math. Soc._ (3) (36)(1978) 193–212〕 and given an extensive treatment in the book.〔 R. Brown, P.J. Higgins, R. Sivera, (''Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids" ), EMS Tracts in Mathematics Vol. 15, 703 pages. (August 2011).〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Double groupoid」の詳細全文を読む スポンサード リンク
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