Subject: Re: pasting along an adjunction Newsgroups: gmane.science.mathematics.categories Date: Saturday 18th April 2009 04:54:51 UTC (over 8 years ago) 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": |
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