From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9841 Path: news.gmane.org!.POSTED.blaine.gmane.org!not-for-mail From: Newsgroups: gmane.science.mathematics.categories Subject: large integrals via manageable ends? Date: Sat, 23 Feb 2019 15:54:21 -0400 Message-ID: Reply-To: Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 8bit Injection-Info: blaine.gmane.org; posting-host="blaine.gmane.org:195.159.176.226"; logging-data="56543"; mail-complaints-to="usenet@blaine.gmane.org" To: Original-X-From: majordomo@mlist.mta.ca Sat Feb 23 21:38:43 2019 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.55]) by blaine.gmane.org with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.89) (envelope-from ) id 1gxe4I-000EYv-LP for gsmc-categories@m.gmane.org; Sat, 23 Feb 2019 21:38:42 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:45804) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1gxe40-0005xl-J0; Sat, 23 Feb 2019 16:38:24 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1gxe2g-0002Tf-4J for categories-list@mlist.mta.ca; Sat, 23 Feb 2019 16:37:02 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9841 Archived-At: With reference to the following diagram in \CAT, where \set is the category of small sets: F \A------->\B | P| | v \set it is classical that if \A is small and \B is locally small then the left Kan extension of P along F exists, call it L, and for B in \B LB=\int^A \B(FA,B) x PA. Clearly, local smallness of B can be weakened to the requirement that all \B(FA,B) are small, a condition called `admissibility' of F, by Street and Walters in their `Yoneda structures' paper. The point is that a small integral of small sets is small. However, I claim that smallness of \A can be `weakened' to local smallness of \set^{\A\op}. If we include \set in a category \SET of sets large enough to contain all the LB as above, via I:\set----->\SET then the description of L also gives a left Kan extension of IP along F. Now write p(LB) for the power set of LB and consider the following calculation: p(LB)=\SET(\int^A \B(FA,B) x PA, 2)=\int_A\SET(\B(FA,B), 2^{PA}) =\int_A\set(\B(FA,B), 2^{PA}) =\set^{\A\op}(\B(F-,B),\set(P-,2)) It shows that local smallness of \set^{\A\op} implies p(LB) is small and hence that LB is small. Thus for \set^{\A\op} locally small and F admissible, the left Kan extension of any P:\A--->\set along F exists and is given by the usual formula. This calculation was prompted by questions similar to an open problem in the paper by Freyd and Street `On the size of categories' in TAC V1. Can anybody provide pointers to similar calculations where a big integral is tamed by a manageable end? Thanks, RJ Wood [For admin and other information see: http://www.mta.ca/~cat-dist/ ]