From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1518 Path: news.gmane.org!not-for-mail From: S.J.Vickers@open.ac.uk Newsgroups: gmane.science.mathematics.categories Subject: RE: Finitely presentable presheaves Date: Wed, 17 May 2000 09:04:17 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" X-Trace: ger.gmane.org 1241017894 31621 80.91.229.2 (29 Apr 2009 15:11:34 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:11:34 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed May 17 08:44:39 2000 -0300 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id IAA16540 for categories-list; Wed, 17 May 2000 08:35:44 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: Internet Mail Service (5.5.2650.21) Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 25 Xref: news.gmane.org gmane.science.mathematics.categories:1518 Archived-At: > Consider the category of presheaves C^ on a small category, C. > > Plainly, a finite colimit F of representables presheaves is finitely > presentable, in the usual sense: C^(F, -) preserves > filtered colimits. > But the converse is also true and seemingly well known: every finitely > presentable presheaf is a finite colimit of representables. > > Is this proved somewhere? I think this is obvious from the fact that the theory of presheaves over C is many sorted, essentially algebraic (a finite limit theory), with a sort for each object of C and a unary operator for each morphism. Then finitely presentable in the categorical sense is the same as finitely presentable in the algebraic sense, which is equivalent to being a finite colimit of free cyclic (i.e. one generator) algebras. In the case of presheaves, Yoneda's lemma says precisely that the free algebra on a generator of sort X (object of C) is the representable presheaf for X. I have exploited some of these facts in a paper with my PhD student Gillian Hill, "Presheaves as configured specifications". It develops a language for specifying systems by components with sharing. Steve Vickers.