From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1928 Path: news.gmane.org!not-for-mail From: Carsten Fuhrmann Newsgroups: gmane.science.mathematics.categories Subject: Re: Kleisli and colimits Date: Wed, 25 Apr 2001 17:04:34 +0100 Organization: School of Computer Science, The University of Birmingham, U.K. Message-ID: <3AE6F592.C9298FD8@cs.bham.ac.uk> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018209 1234 80.91.229.2 (29 Apr 2009 15:16:49 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:16:49 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Apr 25 14:13:19 2001 -0300 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3PGdRr01165 for categories-list; Wed, 25 Apr 2001 13:39:27 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: Mozilla 4.77 [en] (X11; U; SunOS 5.7 sun4u) X-Accept-Language: en Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 26 Original-Lines: 32 Xref: news.gmane.org gmane.science.mathematics.categories:1928 Archived-At: The construction of the Kleisli category can be turned into a left adjoint functor whose domain is the category of Mnd monads (on arbitrary categories) and monad morphisms. However, the codomain of this functor is not Cat, but a category AbsKl whose objects I call "abstract Kleisli categories". An abstract Kleisli category is a category K together with a functor L:K->K, a natural transformation \epsilon: L->Id, and a (not generally natural) transformation \theta:Id->L, such that (1) \theta_L is a natural transformation (2) L\theta o \theta = \theta_L o \theta (3) \epsilon o \theta = id (3) L\epsilon o \theta_L = id A morphism K->K' of AbsKl is a functor that preserves the solutions of the non-naturality square \theta o f = Lf o \theta (I) The Kleisli construction forms a functor Mnd->AbsKl. Its right adjoint sends an abstract Kleisli category K to the evident monad on the subcategory given by the solutions of Equation (I). The counit of this adjunction is in fact an iso, so AbsKl is a full reflective subcategory of Mnd. The full subcategory of Mnd which is equivalent to Abskl is given by those monads for which every component of the unit is a regular mono. Cheers, Carsten