From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4997 Path: news.gmane.org!not-for-mail From: David Spivak Newsgroups: gmane.science.mathematics.categories Subject: monad: (k-Set \downarrow -): Set -->Set Date: Fri, 19 Jun 2009 15:33:03 -0700 Message-ID: Reply-To: David Spivak NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1245608524 8747 80.91.229.12 (21 Jun 2009 18:22:04 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 21 Jun 2009 18:22:04 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Sun Jun 21 20:22:01 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 1MIRg0-0006LD-Kg for gsmc-categories@m.gmane.org; Sun, 21 Jun 2009 20:22:00 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MIR1v-0006TC-LN for categories-list@mta.ca; Sun, 21 Jun 2009 14:40:35 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4997 Archived-At: 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? Thank you, David [For admin and other information see: http://www.mta.ca/~cat-dist/ ]