* Re: Prof
@ 2009-05-21 19:55 Thomas Hildebrandt
0 siblings, 0 replies; 3+ messages in thread
From: Thomas Hildebrandt @ 2009-05-21 19:55 UTC (permalink / raw)
To: Mike Stay, categories
Mike Stay wrote:
> 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?
>
> Thanks!
>
Dear Mike,
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
Best
Thomas Hildebrandt
IT University of Copenhagen
www.itu.dk
^ permalink raw reply [flat|nested] 3+ messages in thread
* Re: Prof
@ 2009-05-22 14:20 Urs Schreiber
0 siblings, 0 replies; 3+ messages in thread
From: Urs Schreiber @ 2009-05-22 14:20 UTC (permalink / raw)
To: Thomas Hildebrandt, categories
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
^ permalink raw reply [flat|nested] 3+ messages in thread
* Prof
@ 2009-05-20 22:26 Mike Stay
0 siblings, 0 replies; 3+ messages in thread
From: Mike Stay @ 2009-05-20 22:26 UTC (permalink / raw)
To: categories
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?
Thanks!
--
Mike Stay - metaweta@gmail.com
http://math.ucr.edu/~mike
http://reperiendi.wordpress.com
^ permalink raw reply [flat|nested] 3+ messages in thread
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