From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4866 Path: news.gmane.org!not-for-mail From: Urs Schreiber Newsgroups: gmane.science.mathematics.categories Subject: Re: Prof Date: Fri, 22 May 2009 16:20:48 +0200 Message-ID: Reply-To: Urs Schreiber NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1243179649 24887 80.91.229.12 (24 May 2009 15:40:49 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 24 May 2009 15:40:49 +0000 (UTC) To: Thomas Hildebrandt , categories Original-X-From: categories@mta.ca Sun May 24 17:40:42 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1M8FoQ-0000ui-RN for gsmc-categories@m.gmane.org; Sun, 24 May 2009 17:40:34 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1M8FBM-0004a1-8Q for categories-list@mta.ca; Sun, 24 May 2009 12:00:12 -0300 X-PMX-Version: 5.4.6.354141, Antispam-Engine: 2.6.1.350677, Antispam-Data: 2009.5.22.140733 X-PerlMx-Spam: Gauge=IIIIIII, Probability=8%, Report='BODY_SIZE_1600_1699 0, BODY_SIZE_2000_LESS 0, BODY_SIZE_5000_LESS 0, BODY_SIZE_7000_LESS 0, WEBMAIL_SOURCE 0, __BOUNCE_CHALLENGE_SUBJ 0, __C230066_P1_5 0, __CP_URI_IN_BODY 0, __CT 0, __CTE 0, __CT_TEXT_PLAIN 0, __FRAUD_419_WEBMAIL 0, __FRAUD_419_WEBMAIL_FROM 0, __HAS_MSGID 0, __HELO_GMAIL 0, __MIME_TEXT_ONLY 0, __MIME_VERSION 0, __PHISH_SPEAR_STRUCTURE_1 0, __RDNS_GMAIL 0, __SANE_MSGID 0, __TO_MALFORMED_2 0' Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4866 Archived-At: 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