From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5001 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: monad: (k-Set \downarrow -): Set -->Set Date: Sun, 21 Jun 2009 22:38:30 +0100 (BST) Message-ID: Reply-To: "Prof. Peter Johnstone" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1245678824 11342 80.91.229.12 (22 Jun 2009 13:53:44 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 22 Jun 2009 13:53:44 +0000 (UTC) To: David Spivak , categories@mta.ca Original-X-From: categories@mta.ca Mon Jun 22 15:53:41 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 1MIjxs-000547-3O for gsmc-categories@m.gmane.org; Mon, 22 Jun 2009 15:53:40 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MIjL0-0000VP-L2 for categories-list@mta.ca; Mon, 22 Jun 2009 10:13:30 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5001 Archived-At: On Fri, 19 Jun 2009, David Spivak wrote: > Dear Categorists, > > Does anyone know a name for the monad described below and/or whether > it has been studied? > > Let k-Set denote the category of k-small sets (for some small regular > cardinal k). For a set S, we denote by > > T(S)=(k-Set \downarrow {S}) > > the set whose elements are pairs (K,f), where K is a k-small set and > f:K-->S is a function. This construction is functorial in S. I > claim that the endo-functor T: Set -->Set is a monad. The identity > transformation S-->T(S) is given by "singleton set" and the > multiplication transformation TT(S)-->T(S) is given by Grothendieck > construction. > I don't think this construction works at the level of sets rather than categories. The problem is that k-Set is a category, not a set, so T(S) also has a category structure, and you can't simply "forget" this. If you do, then you have the problem "*which* singleton set?" for the unit (i.e., which singleton set do you choose as the domain of the functions 1 --> S which you identify with elements of S?), and whichever choice you make you are going to run into problems verifying the monad identities. > (There is a similar monad on Cat, where we replace k-Set with k-Cat.) > This is correct, and it's well-known: it is the monad which freely adjoins k-small coproducts to a category. Peter Johnstone [For admin and other information see: http://www.mta.ca/~cat-dist/ ]