From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1906 Path: news.gmane.org!not-for-mail From: Tom Leinster Newsgroups: gmane.science.mathematics.categories Subject: Re: Looking for adjoints Date: Mon, 2 Apr 2001 22:47:04 +0100 (BST) Message-ID: 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 1241018196 1127 80.91.229.2 (29 Apr 2009 15:16:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:16:36 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Apr 3 15:23:46 2001 -0300 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f33Had607827 for categories-list; Tue, 3 Apr 2001 14:36:39 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: ELM [version 2.5 PL3] X-Scanner: exiscan *14kCAA-0002TC-00*DMkDUwjrCXU* http://duncanthrax.net/exiscan/ Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 4 Original-Lines: 44 Xref: news.gmane.org gmane.science.mathematics.categories:1906 Archived-At: Here's an adjunction from which various basic results in category theory can be read off. (Useful, but somewhat inward-looking...) Fix a small category C, and consider the forgetful functor U: [C^op, Set] ---> [ob C, Set]. This has a left adjoint F, which can easily be written down explicitly (and whose existence is also guaranteed because it's a Kan extension). Hence U preserves limits - and this is part of what's meant by the statement that limits are computed pointwise in a presheaf category. Moreover, the adjunction is monadic, from which it follows that (a) U creates limits (which is the rest of what's meant by the "computed pointwise" slogan), and (b) every presheaf is the colimit of representables (using the fact that every algebra for a monad is a coequalizer of free algebras). Dually, U has a right adjoint, so the dual results also hold. Tom > From: Jean-Pierre Marquis > To: > > I would like to have a large pool of examples of adjoint functors in as many > different fields of mathematics as possible. I am looking for the "nicest", > in whatever sense you can think of this expression (e.g. unexpected, their > existence is equivalent to a classical theorem, etc), cases in various > fields. > > References or examples anyone? (Besides the standard ones found in Mac > Lane, etc.) > > Thank you, > Jean-Pierre Marquis > > >