From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5006 Path: news.gmane.org!not-for-mail From: selinger@mathstat.dal.ca (Peter Selinger) Newsgroups: gmane.science.mathematics.categories Subject: Re: monad: (k-Set \downarrow -): Set -->Set Date: Mon, 22 Jun 2009 11:37:01 -0300 (ADT) Message-ID: Reply-To: selinger@mathstat.dal.ca (Peter Selinger) NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1245718320 9063 80.91.229.12 (23 Jun 2009 00:52:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 23 Jun 2009 00:52:00 +0000 (UTC) To: dspivak@uoregon.edu, categories@mta.ca Original-X-From: categories@mta.ca Tue Jun 23 02:51:57 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 1MIuEr-0001Fj-UF for gsmc-categories@m.gmane.org; Tue, 23 Jun 2009 02:51:54 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MItcS-0001aI-HM for categories-list@mta.ca; Mon, 22 Jun 2009 21:12:12 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5006 Archived-At: 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. > > (There is a similar monad on Cat, where we replace k-Set with k-Cat.) > > Does this monad T have a name? Has it been studied? I assume you mean to take such pairs (K,f) up to isomorphism, or else, as Peter Johnstone has already pointed out, your construction will not be well-defined. For instance, even the finite sets may form a proper class, depending on your underlying set theory. In the case where k=omega, T is the well-known finite multiset monad, which associates to each S the free commutative monoid generated by S (whose elements are also known as finite multisets in S). For other k, I would call this the "monad of multisets of size less than k". I think this works for any infinite small cardinal, not just regular ones. -- Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]