From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1171 Path: news.gmane.org!not-for-mail From: William James Newsgroups: gmane.science.mathematics.categories Subject: Re: co-exponential question Date: Tue, 20 Jul 1999 13:57:18 -0700 Message-ID: <3794E2AC.425B@latrobe.edu.au> References: <19990716195548.73188.qmail@hotmail.com> 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 1241017614 29804 80.91.229.2 (29 Apr 2009 15:06:54 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:06:54 +0000 (UTC) Cc: categories@mta.ca To: Bill Halchin Original-X-From: cat-dist Tue Jul 20 15:08:21 1999 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id NAA07442 for categories-list; Tue, 20 Jul 1999 13:20:20 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: Mozilla 3.04 (Win16; I) Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 60 Xref: news.gmane.org gmane.science.mathematics.categories:1171 Archived-At: Bill Halchin wrote: > > This is actually a "dual" question. > Basically I want to do the dual of the construction > gives the notion of an exponential or map object. > > Suppose we have a category C with sums. Then we build the > following category from C. > > object: T+X<-----Y > > map: from T+X<-----Y to T+X'<-------Y is a C map "alpha" such > that we have the following diagram: > > I-sub-T+alpha > T+X---------------------------->T+X' > ^ ^ > \ / > \ / > \ / > \ / > \ / > \ / > \ / > \ / > \ / > \ / > \ / > \ / > \ / > \ / > \/ > Y > > Then suppose there exists a C-object called Y**T such > that T+Y**T<-------Y is the initial object of the category > just built above. What significance does Y**T have opposed > to the concept of an exponential???? If I did everything > correctly it (Y**T) should be the dual of T**Y. Forgive my ignorance: is T+Y**T<----Y to be initial because the usual construction for the exponential has the relevant arrow as terminal? Can you suggest to me a reference for that construction of exponentials? As for significance: my own research into co-exponentials is in terms of them as characteristic of lattices dual to Heyting algebras. These dual-Heyting algebras work well as algebras for a brand of paraconsistent logic (they have a complement operator which has in general that an element and its complement overlap). Alternatively, you can think of co-exponentials as productive of the interesting topological notions associated with closed sets, like boundary. My feeling is that co-exponentials count as useful in more interesting kinds of maths than turn up simply in those categories dual to toposes. William James