#
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.

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

TAC Home