From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5004 Path: news.gmane.org!not-for-mail From: Anders Kock Newsgroups: gmane.science.mathematics.categories Subject: Re: monad: (k-Set \downarrow -): Set -->Set Date: Mon, 22 Jun 2009 18:54:27 +0200 Message-ID: Reply-To: Anders Kock NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1245718159 8731 80.91.229.12 (23 Jun 2009 00:49:19 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 23 Jun 2009 00:49:19 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Tue Jun 23 02:49:17 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 1MIuCL-0000ay-HP for gsmc-categories@m.gmane.org; Tue, 23 Jun 2009 02:49:17 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MIteA-0001eU-HE for categories-list@mta.ca; Mon, 22 Jun 2009 21:13:58 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5004 Archived-At: As Peter Johnstone also emphasized in his reply, the construction which =20 David Spivak describes, namely "T(S)=3D(k-Set \downarrow {S})", is really a part of a well known= =20 "monad" on the category of categories: if S is any category, T(S) is the=20 free cocompletion of S under k-small coproducts. It is only a monad up=20 to canonical isomorphisms, because coproducts are not in general=20 strictly associative. This cocompletion "monad" under coproducts has=20 been widely studied under the name "Fam" (because T(S) is the category=20 of k-small Families of objects in S). It is an example of a KZ monad. However, replacing k-Set by k-Cat provides a monad on Cat which is not=20 KZ; David observes: "(There is a similar monad on Cat, where we replace k-Set with k-Cat.)" and Peter's reply to this: "This is correct, and it's well-known: it is the monad which freely adjoi= ns k-small coproducts to a category. " does not apply here (it slipped into the wrong place of his reply):=20 rather, David's "similar monad" is trying to provide free cocompletion=20 under colimits indexed by k-small categories, but does not, until you=20 make a category-of-fractions construction on its values. My University=20 of Chicago thesis (1967) described this way of making free cocompletions. This "similar monad" (before doing the fractions-part) has been studied=20 by Guitart, he calls it this monad DIAG. Reference: Guitart, Ren=E9,=20 Remarques sur les machines et les structures. Cahiers de Topologie et=20 G=E9om=E9trie Diff=E9rentielle Cat=E9goriques, 15 no. 2 (1974), p. 113-14= 4=20 (available electronically in NUMDAM). Anders Kock [For admin and other information see: http://www.mta.ca/~cat-dist/ ]