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