#
Span, cospan, and other double categories

##
Susan Niefield

Given a double category $\mathbb D$ such that $\mathbb D_0$ has
pushouts, we characterize oplax/lax adjunctions between $\mathbb
D$ and $Cospan(\mathbb D_0)$ for which the right adjoint is
normal and restricts to the identity on $\mathbb D_0$, where
$Cospan(\mathbb D_0)$ is the double category on $\mathbb D_0$ whose
vertical morphisms are cospans. We show that such a pair exists if
and only if $\mathbb D$ has companions, conjoints, and
1-cotabulators. The right adjoints are induced by the companions and
conjoints, and the left adjoints by the 1-cotabulators. The notion
of a 1-cotabulator is a common generalization of the symmetric
algebra of a module and Artin-Wraith glueing of toposes, locales, and
topological spaces.

Keywords:
double category, lax functor, (co)span, (co)tabulator, companion,
conjoint, symmetric algebra

2010 MSC:
18D05, 18A40, 18B25, 18B30, 06D22, 18D10, 15A78

*Theory and Applications of Categories,*
Vol. 26, 2012,
No. 26, pp 729-742.

Published 2012-12-02.

http://www.tac.mta.ca/tac/volumes/26/26/26-26.dvi

http://www.tac.mta.ca/tac/volumes/26/26/26-26.ps

http://www.tac.mta.ca/tac/volumes/26/26/26-26.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/26/26-26.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/25/26-26.ps

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