From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2386 Path: news.gmane.org!not-for-mail From: Tom LEINSTER Newsgroups: gmane.science.mathematics.categories Subject: Re: Compatibility of functors with limits Date: Sun, 13 Jul 2003 19:34:48 +0200 (CEST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset="US-ASCII" X-Trace: ger.gmane.org 1241018619 3877 80.91.229.2 (29 Apr 2009 15:23:39 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:23:39 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Jul 14 16:04:52 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 14 Jul 2003 16:04:52 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19c8Xx-0001W6-00 for categories-list@mta.ca; Mon, 14 Jul 2003 15:59:37 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 14 Original-Lines: 70 Xref: news.gmane.org gmane.science.mathematics.categories:2386 Archived-At: A week ago I asked this list a question, thinking that lots of people would know the answer. Either I was wrong or those who know are keeping it to themselves, as I didn't get any answers at all. But I now understand the issue better than before, so I'd like to try re-asking my question in a different way and see if that elicits a response. I'm trying to understand a certain notion of compatibility of a functor with limits. There are of course several well-known such notions: preserves, reflects, creates. I called mine (provisionally) "respecting" limits. It is close to preservation, but not quite the same; it seems more constructive and perhaps (dare I say it?) more natural. Preservation is all very well when the domain category has all limits, or all limits of whatever type we're concerned with. But otherwise, it seems a bit suspect. For let F: A ---> B be a functor; preservation says that given a diagram D: I ---> A in A, - if D *does* admit a limit cone, then the image of that cone under F is also a limit cone, - if D *doesn't* admit a limit cone, then... well, nothing. "Respect", on the other hand, says the same as preservation in the first case, but also says something in the second case. (So respect is in general stronger than preservation.) Last time I gave a definition of respect in terms of categories of cones; that definition is appended to this mail. Here's a different way to put it: F "respects limits for D" if the canonical map \int^a (\int_i A(a, Di)) \times Fa ----> \int_i FDi is an isomorphism. Here \int^a denotes coend over a in A, \int_i is limit over i in I, and I'm working under the assumption that these (co)limits exist in the codomain category B. Now, if D does have a limit in A then the left-hand side is \int^a A(a, \int_i Di) \times Fa which by density is just F \int_i Di; so, as claimed, respect is the same as preservation in the case where the limit exists. My question was whether anyone understood "respect of limits" well, or could shed any light on it. It seems to me that, as well as being just the right thing in certain examples I've been considering, it's a very natural concept. Tom The definition of respect from last time: > Let F: A ---> B be a functor, where B is a category with (for > sake of argument) all small limits and colimits. Let I be a small > category and D: I ---> A a diagram in A; write Cone(D) for the category > of cones on/into D in A, write Cone(FD) for the category of cones on FD > in B, and write > > F_*: Cone(D) ---> Cone(FD) > > for the induced functor. Then F can be said to "respect limits for D" > if the colimit of F_* is the terminal object of Cone(FD) (that is, the > limit cone on FD).