Re: Prof

[Urs Schreiber <urs.schreiber@googlemail.com>]

Hi,

Mike Stay asked:

> > The bicategory of (small categories, profunctors, and natural
> > transformations), should be equivalent to the 2-category of (presheaf
> > categories, colimit-preserving functors, and natural transformations).
> > Has someone proved this?  If so, where?

Thomas Hildebrandt replied:

>  You may have a look at Prop. 4.2.4 in the PhD thesis of Gian Luca
>  Cattani from BRICS, University of Aarhus, available at
>  http://www.daimi.au.dk/~luca/thesis.html

I am guessing that the crucial statement that makes this work is the
standard fact that if a category A admits small colimits, then there
is an equivalence of categories

  Funct^cocont(PSh(C), A)  = Funct(C,A) .

In the textbook literature one can find this for instance as corollary
2.7.4, page 63 of Kashiwara-Schapira's "Categories and Sheaves".

It may be noteworthy that this statement is known to generalize from
categories to (oo,1)-categories, for instance as given in theorem
5.1.5.6 of Lurie's "Higher Topos Theory".

Colimit preserving functors between "presentable (oo,1)-categories",
i.e between localizations of (oo,1)-presheaf categories play a major
role in the theory and have some nice applications.

For instance Ben-Zvi/Francis/Nadler have recently shown that "integral
transforms" (of the Fourier-Mukai type and higher generalizations) are
precisely equivalent to colimit preserving functors between the
corresponding presentable (oo,1)-categories.

See around the highlighted box in section 4 here:
http://ncatlab.org/nlab/show/geometric+infinity-function+theory.

Best,
Urs