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.
http://www.tac.mta.ca/tac/volumes/16/18/16-18.dvi
http://www.tac.mta.ca/tac/volumes/16/18/16-18.ps
http://www.tac.mta.ca/tac/volumes/16/18/16-18.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/18/16-18.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/18/16-18.ps