From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4148 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Cartesian closed without products Date: Mon, 7 Jan 2008 14:09:54 +0000 (GMT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241019753 11933 80.91.229.2 (29 Apr 2009 15:42:33 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:42:33 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Jan 7 15:19:27 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 07 Jan 2008 15:19:27 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JBxLm-0001wG-Eq for categories-list@mta.ca; Mon, 07 Jan 2008 15:09:30 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 9 Original-Lines: 26 Xref: news.gmane.org gmane.science.mathematics.categories:4148 Archived-At: A minor correction: > A small category C has finite products just when the > representables in [C^op, Set] are closed under finite > products. It is cartesian closed just when the > representables are closed under finite products and > internal hom. > > It seems natural, therefore, to consider a notion of > "cartesian closedness without finite products": categories > in which the representables are closed in [C^op, Set] under > internal hom but not necessarily under finite products. > This amounts to giving, for each pair of objects X and Y, > an object [X, Y] together with a universal natural > transformation C(-, [X, Y]) x C(-, X) -> C(-, Y). Such > categories will be closed in the sense of Eilenberg-Kelly > without necessarily being monoidal: On reflection, this bit isn't necessarily true, since EK-closed requires the unit object to be representable as well. Richard