Paths in double categories

R. J. MacG. Dawson, R. Paré, and D. A. Pronk

Two constructions of paths in double categories are studied, providing algebraic versions of the homotopy groupoid of a space. Universal properties of these constructions are presented. The first is seen as the codomain of the universal oplax morphism of double categories and the second, which is a quotient of the first, gives the universal normal oplax morphism. Normality forces an equivalence relation on cells, a special case of which was seen before in the free adjoint construction. These constructions are the object part of 2-comonads which are shown to be oplax idempotent. The coalgebras for these comonads turn out to be Leinster's fc-multicategories, with representable identities in the second case.

Keywords: double categories, oplax double categories, paths, localisation

2000 MSC: 18A40, 18C20, 18D05

Theory and Applications of Categories, Vol. 16, 2006, No. 18, pp 460-521.

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