From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/592 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Re: Combining monads Date: Thu, 15 Jan 1998 17:10:14 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017059 26405 80.91.229.2 (29 Apr 2009 14:57:39 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:57:39 +0000 (UTC) To: categories Original-X-From: cat-dist Thu Jan 15 17:11:25 1998 Original-Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id RAA01978; Thu, 15 Jan 1998 17:10:14 -0400 (AST) Original-Lines: 59 Xref: news.gmane.org gmane.science.mathematics.categories:592 Archived-At: Date: Thu, 15 Jan 1998 15:44:30 +1100 (EST) From: Steve Lack > Date: Wed, 14 Jan 1998 15:10:43 -0400 (AST) > From: categories > > Date: Wed, 14 Jan 1998 16:21:51 +0000 (GMT) > From: Tom Leinster > > > Is the pullback of a monadic functor along a monadic functor > necessarily monadic? > Is the diagonal of the pullback square monadic? > Does this work if your restrict yourself to, say, finitary monadic > functors? > > (E.g. it works for finitary monads on Set: the theory of sets with > both ring and lattice structure (not interacting in any particular > way) comes from a monad.) > > Thanks, > Tom Leinster > > Let K be a complete and cocomplete category, and Mnd(K) the category of monads on K and strict morphisms of monads. If T and S are monads on K which preserve (alpha-)filtered colimits (for a regular cardinal alpha), then (i)the coproduct T+S exists in Mnd(K) (ii)this coproduct is ``algebraic'', meaning that the diagonal of the pullback square K^S | | v K^T-->K is the forgetful functor K^(T+S)-->K (iii)the projections K^(T+S)-->K^T and K^(T+S)-->K^S are monadic. Much can be done without completeness, but the proofs become a bit harder. See the paper G.M.Kelly, A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on, Bull. Austral. Math. Soc. 22(1980):1--83 for a survey of many such results. In fact if K is locally finitely presentable then the category Mnd_f(K) of finitary monads on K and strict morphisms of monads is itself locally finitely presentable; for this see my paper ``On the monadicity of finitary monads'', to appear in JPAA, but in the meantime available at http://www.maths.usyd.edu.au:8000/res/Catecomb/Lack/1997-29.html. Regards, Steve.