From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4146 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Cartesian closed without products Date: Mon, 7 Jan 2008 11:08:55 +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 1241019752 11904 80.91.229.2 (29 Apr 2009 15:42:32 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:42:32 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Jan 7 15:19:26 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 07 Jan 2008 15:19:26 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JBxJG-0001VE-Qa for categories-list@mta.ca; Mon, 07 Jan 2008 15:06:54 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 7 Original-Lines: 35 Xref: news.gmane.org gmane.science.mathematics.categories:4146 Archived-At: Dear categorists, 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: let us call them "universally closed" for now. Obviously, any cartesian closed category is universally closed; and categorical proof theory gives us a class of non-degenerate examples built from the syntax of (classical) sequent calculi with implication but no product. The question now arises as to whether there are any non-syntactic examples of universally closed categories which are not cartesian closed. The most likely place seems to me to be domain theory, but I have been unable to track anything down. Does anyone have any pointers? Thanks, Richard.