From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6254 Path: news.gmane.org!not-for-mail From: Jiri Velebil Newsgroups: gmane.science.mathematics.categories Subject: Re: quotients and finitary functors Date: Wed, 29 Sep 2010 07:32:19 +0200 (CEST) Message-ID: References: Reply-To: Jiri Velebil NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: dough.gmane.org 1285802444 9040 80.91.229.12 (29 Sep 2010 23:20:44 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 29 Sep 2010 23:20:44 +0000 (UTC) Cc: categories@mta.ca To: Ondrej Rypacek Original-X-From: majordomo@mlist.mta.ca Thu Sep 30 01:20:41 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P15x0-00055o-Ai for gsmc-categories@m.gmane.org; Thu, 30 Sep 2010 01:20:38 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:37940) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P15w2-0003HO-6P; Wed, 29 Sep 2010 20:19:38 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P15vx-0001bV-H1 for categories-list@mlist.mta.ca; Wed, 29 Sep 2010 20:19:33 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6254 Archived-At: Dear Ondrej, The answer is yes, for every category \A for which it makes good sense to say a ``finitary endofunctor''. Hence, take any locally finitely presentable category \A and denote by J : \A_\fp --> \A the full inclusion representing finitely presentable objects in \A. Then J exhibits \A as a free cocompletion of \A under filtered colimits, hence there is an equivalence (1) finitary endofunctors of \A ~ [\A_\fp,\A] Now, if you consider another functor, namely E : |\A_\fp| --> \A (where |\A_\fp| is the underlying discrete category of \A_\fp), then restriction-along-E provides you with a monadic functor U: [\A_\fp,\A] ---> [|\A_\fp|,\A] By [KP] (reference below), the category [|\A_\fp|,\A] can be seen as the category of finitary signatures on \A. The left adjoint F to U assigns a polynomial functor H_\Sigma to the signature \Sigma. Monadicity of U then implies that every finitary functor H (using the equivalence of categories (1) above) can be expressed as a coequalizer of the form F(\Gamma) ---> F(\Sigma) ---> H ---> and this is the ``quotient'' you asked about. All of the above can be proved slightly more generally by replacing ``locally finitely presentable'' by ``locally \lambda-presentable''. [KP] G.M.Kelly and A.J.Power, Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads, Journal of Pure and Applied Algebra Volume 89, Issues 1-2, 8 October 1993, Pages 163-179, doi:10.1016/0022-4049(93)90092-8 Hope it helped, Jirka On Tue, 28 Sep 2010, Ondrej Rypacek wrote: > Dear All, > It is well known to me that a set functor is finitary iff it is a > quotient of a polynomial functor (for some finitary signature). > > Are there any similar results for categories other than Set (toposes) ? > > > Thanks, > Ondrej [For admin and other information see: http://www.mta.ca/~cat-dist/ ]