From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/679 Path: news.gmane.org!not-for-mail From: Neil Ghani(1998-03) Newsgroups: gmane.science.mathematics.categories Subject: Naturality Squares and Pullbacks Date: Tue, 24 Mar 1998 11:15:42 +0900 Message-ID: <199803240215.LAA10126@etlclu.etl.go.jp> Reply-To: nxg@cs.bham.ac.uk NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241017116 26875 80.91.229.2 (29 Apr 2009 14:58:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:58:36 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Tue Mar 24 10:51:15 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id IAA19782 for categories-list; Tue, 24 Mar 1998 08:26:42 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 16 Xref: news.gmane.org gmane.science.mathematics.categories:679 Archived-At: A natural transformation is an indexed family of arrows such that a certain diagram commutes. One could require a stronger condition, namely that the said diagram is a pullback. What would such a transformation be called? I'm sure I've seen this in the literature before but I cant remember where. Pointers? This problem arose in the context of finitary monads where T(X) is the derived operations over a set X for some signature. The naturality square for the unit turns out to be a pullback. This then implies that the unit of the monad is a monic - presumably this is a result in the literature somewhere. Again, pointers? Neil Ghani