Re: pasting along an adjunction

Re: pasting along an adjunction

[Urs Schreiber]

On Fri, Apr 17, 2009 at 8:06 PM, Andrew Salch <asalch@math.jhu.edu> wrote:

> In a recent paper of Connes and Consani,
> they consider the following "pasting along
> an adjunction":

[...]

> I would like to know if this "pasting along
> an adjunction" is a special case of some
> more general construction already known
> to category theory, and if basic properties
> of pasting along an adjunction have already
> been worked out and written down somewhere.

In section 2.3.1 of

"Higher Topos Theory"
http://arxiv.org/abs/math.CT/0608040

Jacob Lurie motivates the notion of "inner fibrations" and of
Cartesian fibrations of (oo,1)-categories as a generalization of this
"pasting" construction.

"Pasting" along any bifunctor C^op x D --> Set is the same as having
an inner fibration over the interval (which is an arbitrary functor
for 1-categories), and the particular "pasting" that you mention, over
hom_D(F(-),-) coming from a functor F : C \to D, gives a Cartesian
fibration over the interval (top of p. 88, leading over to section
2.4).

Best,
Urs

Re: pasting along an adjunction

[Ross Street]

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":

pasting along an adjunction

[Andrew Salch]

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": let E be a category whose object
class is the union of the object class of C and the object class of D; and
given objects X,Y of E, let the hom-set hom_E(X,Y) be defined as follows:

-if X,Y are both in the object class of C, then hom_E(X,Y) = hom_C(X,Y).

-if X,Y are both in the object class of D, then hom_E(X,Y) = hom_D(X,Y).

-if X is in the object class of C and Y is in the object class of D, then
hom_E(X,Y) = hom_C(X,GY) = hom_D(FX,Y).

-if X is in the object class of D and Y is in the object class of C, then
hom_E(X,Y) is empty.

Composition is defined in a straightforward way. When C,D are closed
symmetric monoidal categories, then E has a natural closed symmetric
monoidal structure as well. Connes and Consani use this categorical
pasting to construct schemes over F_1, "the field with one element," and I
have worked out some variations and applications of this categorical
pasting which produce other useful objects (e.g. algebraic F_1-stacks and
derived F_1-stacks, which have some useful number-theoretic as well as
homotopy-theoretic properties).

I would like to know if this "pasting along an adjunction" is a special
case of some more general construction already known to category theory,
and if basic properties of pasting along an adjunction have already been
worked out and written down somewhere.

Thanks,
Andrew S.