Re: pasting along an adjunction

[Ross Street <street@ics.mq.edu.au>]

Dear Andrew

There is a bicategory Mod whose objects are categories and whose
morphisms are "modules" (also called bimodules, profunctors and
distributors). A module from B to A is a functor m : A^op x B --> Set.
Modules m : B --> A and n : C --> B are composed using a tensor-product-
over-B-like process: see Lawvere's paper:
	<http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html>.

Every functor g : B --> A gives a module g_* : B --> A taking (a,b) to
the set A(a,gb) (which you would write hom_A(a,gb) ).

The bicategory Mod has lax colimits (which we call collages because of
their gluing- and pasting-like nature).

Each single module m : B --> A can be regarded as a diagram in Mod.
The collage C of that diagram is the category whose objects are
disjointly
those of A and of B, morphisms between objects of A are as in A,
morphisms
between objects of B are as in B, there are no morphisms b --> a, while
C(a,b) = m(a,b). There are fully faithful functors i : A --> C and
j : B --> C
and such cospans A --> C <-- B are precisely the codiscrete cofibrations
in Cat. This was important in my paper in Cahiers:
<http://www.numdam.org:80/numdam-bin/feuilleter?id=CTGDC_1980__21_2>

Also see
	<http://www.tac.mta.ca/tac/reprints/articles/4/tr4.pdf>
which relates to stacks.

Your case is the collage of the module g_*. It doesn't matter whether g
has an adjoint or not (that simply allows the module to be expressed in
two different ways).

Regards,
Ross

On 18/04/2009, at 4:06 AM, Andrew Salch wrote:

> Let C,D be categories, let F be a functor from C to D, and let G be
> right
> adjoint to F. In a recent paper of Connes and Consani, they
> consider the
> following "pasting along an adjunction":