From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/828 Path: news.gmane.org!not-for-mail From: Zhaohua Luo Newsgroups: gmane.science.mathematics.categories Subject: abstract algebraic geometry Date: Fri, 17 Jul 1998 15:28:29 -0400 Message-ID: <35AFA5DD.8323F00C@iswest.com> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="------------980E00C6D8F26CAA5DFAE228" X-Trace: ger.gmane.org 1241017214 27658 80.91.229.2 (29 Apr 2009 15:00:14 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:00:14 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Sat Jul 18 12:19:12 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id KAA22154 for categories-list; Sat, 18 Jul 1998 10:19:41 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: Mozilla 4.05 [en] (Win95; I) Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 89 Xref: news.gmane.org gmane.science.mathematics.categories:828 Archived-At: --------------980E00C6D8F26CAA5DFAE228 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit The following short note (see the abstract below) Uniform Functors (html) is available on Categorical Geometry Homepage at the following address: http://www.azd.com (the dvi version is under preparation) Zhaohua (Zack) Luo ------------------------------------------------------------------------------------- Uniform Functors Zhaohua Luo Abstract: In a previous note [atomic categories] we introduced the notion of an atomic category, and showed that each atomic category C carries a canonical functor to the category of sets, called the unifunctor of C. We also introduced the notion of a uniform functor between atomic categories. In this note we give an intrinsic definition of a uniform functor between any two categories with strict initials. Roughly speaking a functor is uniform if it induces isomorphisms between the complete boolean algebras of normal sieves on the objects. We show that any uniform functor to the category of sets is unique up to equivalence. A functor between Grothendieck toposes is uniform iff it induces an isomorphism between the complete boolean algebras of complemented subobjects. Since any unifunctor is uniform, this implies that a Grothendieck topos is atomic iff the complete boolean algebra of complemented subobjects of each object is atomic (or equivalently, there is a uniform functor to the category of sets). --------------980E00C6D8F26CAA5DFAE228 Content-Type: text/html; charset=us-ascii Content-Transfer-Encoding: 7bit The following short note (see the abstract below)

Uniform Functors (html)

is available on Categorical Geometry Homepage at the following address:

(the dvi version is under preparation)

Zhaohua (Zack) Luo
-------------------------------------------------------------------------------------
Uniform Functors

Zhaohua Luo

Abstract:

In a previous note [atomic categories] we introduced the notion of an atomic category, and showed that each atomic category C carries a canonical functor to the category of sets, called the unifunctor of C. We also introduced the notion of a uniform functor between atomic categories. In this note we give an intrinsic definition of a uniform functor between any two categories with strict initials. Roughly speaking a functor is uniform if it induces isomorphisms between the complete boolean algebras of normal sieves on the objects. We show that any uniform functor to the category of sets is unique up to equivalence. A functor between Grothendieck toposes is uniform iff it induces an isomorphism between the complete boolean algebras of complemented subobjects. Since any unifunctor is uniform, this implies that a Grothendieck topos is atomic iff the complete boolean algebra of complemented subobjects of each object is atomic (or equivalently, there is a uniform functor to the category of sets).

--------------980E00C6D8F26CAA5DFAE228--