From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1205 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Pseudo orbits Date: Sat, 28 Aug 1999 13:34:35 +1000 Message-ID: References: <4.2.0.58.19990827154835.00987a70@ichthus.syr.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" X-Trace: ger.gmane.org 1241017636 29941 80.91.229.2 (29 Apr 2009 15:07:16 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:07:16 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Sat Aug 28 12:51:54 1999 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id LAA08732 for categories-list; Sat, 28 Aug 1999 11:37:10 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Sender: street@hera.mpce.mq.edu.au (Unverified) In-Reply-To: <4.2.0.58.19990827154835.00987a70@ichthus.syr.edu> Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 34 Xref: news.gmane.org gmane.science.mathematics.categories:1205 Archived-At: Dear Gaunce Lewis, GT-colleague and all, When we regard a monoid M as a one-object category, an M-set is a functor X : M --> Set and the colimit of the functor X is the set of orbits of the M-set. What GT-colleague has is an ordered monoid which can be regarded as a one-object 2-category M, and the action F of M on the category C amounts to a 2-functor X : M --> Cat. I suspect that the construction required is the pseudocolimit of X. This kind of colimit for 2-functors was considered in the book of John Gray J.W. Gray, Formal Category Theory: Adjointness for 2-Categories, Lecture Notes in Math. 391 (Springer, 1974) and in my paper Limits indexed by category-valued 2-functors, J. Pure Appl. Algebra 8 (1976) 149-181 where I show that pseudo(co)limits and lax (co)limits are ordinary weighted (= indexed) (co)limits in the sense of enriched category theory (for the base monoidal category Cat). I suspect the condition that the identity element is initial is a red herring even though this makes it look as though canonical maps are being inverted rather than isomorphisms being introduced. Regards, Ross