From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6512 Path: news.gmane.org!not-for-mail From: Joachim Kock Newsgroups: gmane.science.mathematics.categories Subject: Re: colimits of polynomial functors Date: Wed, 02 Feb 2011 19:48:03 +0100 Message-ID: References: Reply-To: Joachim Kock NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 7BIT X-Trace: dough.gmane.org 1296693283 6978 80.91.229.12 (3 Feb 2011 00:34:43 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 3 Feb 2011 00:34:43 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Thu Feb 03 01:34:38 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Pkn9e-00043z-6N for gsmc-categories@m.gmane.org; Thu, 03 Feb 2011 01:34:34 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:51810) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Pkn9T-0005x7-8O; Wed, 02 Feb 2011 20:34:23 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Pkn9P-0000XG-LN for categories-list@mlist.mta.ca; Wed, 02 Feb 2011 20:34:20 -0400 In-reply-to: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6512 Archived-At: I just wrote, much too quickly: > At the risk of being off the point, I think that the colimits > that exist might not have been studied much because they are often > not the 'right' ones, in a sense. As an example, the polynomial > functor Set -> Set, X \mapsto X^2 (represented by 1 <- 2 -> 1 -> 1) > has two automorphisms (the identity and the twist), and if I am not > mistaken the identity functor X \mapsto X is the coequaliser of those > two in the category of polynomial functors and their strong natural > transformations (just because 1 is the equaliser of the two set auts > 2 -> 2). But the last sentence is of course pure nonsense. The equaliser of the two set auts 2 -> 2 is 0, and the conclusion is then that the constant polynomial functor X \mapsto 1 (represented by 1 <- 0 -> 1 -> 1) is the coequaliser. Sorry for the nonsense. I don't know where I had my head. I can hardly trust myself anymore, but if this second version is correct, it still illustrates the point I wanted to make, namely that the colimit is not the 'right' one. > The functor that 'ought' to be the coequaliser is of course > X \mapsto X^2/2, which is not polynomial. (For example it does not > preserve pullbacks.) Cheers, Joachim. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]