From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/768 Path: news.gmane.org!not-for-mail From: Susan Niefield Newsgroups: gmane.science.mathematics.categories Subject: preprint available Date: Wed, 24 Jun 1998 10:49:40 -0400 (EDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017170 27300 80.91.229.2 (29 Apr 2009 14:59:30 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:59:30 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Wed Jun 24 16:30:33 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA12232 for categories-list; Wed, 24 Jun 1998 15:03:19 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 28 Xref: news.gmane.org gmane.science.mathematics.categories:768 Archived-At: The following reprint is available at http://www1.union.edu/~niefiels/ESU.ps http://www1.union.edu/~niefiels/ESU.dvi EXPONENTIABILITY AND SINGLE UNIVERSES by Marta BUNGE and Susan NIEFIELD ABSTRACT - The search for suitable single universes for opposite or dual pairs of notions (such as those of discrete fibration and discrete opfibration, or of open and closed inclusions, or of functions and distributions on a Grothendieck topos) leads naturally to exponentiability. Using exponentiability techniques, such as model-generated categories and glueing, we settle a standing conjecture and an open problem. The conjecture, due to F. Lamarche, states that for a small category B, the category of unique factorization liftings (also known as discrete Conduche fibrations) over B is a topos. We also construct the smallest topos containing the local homeomorphisms (functions) and the complete spreads (distributions) over any given topos satisfying a certain condition (true of presheaf toposes). This solves a problem posed by F. W. Lawvere. Along the way, we introduce two new sorts of geometric morphisms, characterize locally closed inclusions in Cat, and investigate new features of generalized coverings in topos theory, such as branched coverings, cuts, and complete spreads.