From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/819 Path: news.gmane.org!not-for-mail From: Zhaohua Luo Newsgroups: gmane.science.mathematics.categories Subject: abstract algebraic geometry Date: Mon, 13 Jul 1998 14:10:21 -0400 Message-ID: <35AA4D8D.2A40888D@iswest.com> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241017200 27488 80.91.229.2 (29 Apr 2009 15:00:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:00:00 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Mon Jul 13 21:39:31 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id TAA02104 for categories-list; Mon, 13 Jul 1998 19:55:20 -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: 85 Xref: news.gmane.org gmane.science.mathematics.categories:819 Archived-At: The following short note (see the abstract below) Atomic Categories is available on Categorical Geometry Homepage at the following address: http://www.azd.com Note that to read the special symbols on these pages requires a viewer under Win95. (thanks to Vaughan Pratt for bringing this to my attention). Please let me know if you would like to have a copy in dvi format. Z. Luo ------------------------------------------------------------------------------------- Atomic Categories Zhaohua Luo Abstract: Let C be a category with a strict initial object 0. A map is called "non-initial" if its domain is not an initial object. A non-initial object T is called "unisimple" if for any two non-initial maps f: X --> T and g: Y --> T there are non-initial maps r: R --> X and s: R --> Y such that fr = gs. We say that C is an "atomic category" if any non-initial object is the codomain of a map with a unisimple domain. Many natural (left) categories arising in geometry are atomic (such as the categories of sets, topological spaces, posets, coherent spaces, Stone spaces, schemes, local ringed spaces, etc.) In this short note we show that each atomic category carries a unique functor to the category of sets, which plays the traditional role of "underlying functor" in categorical geometry --------------84037E1FA36AB214A0902978 Content-Type: text/html; charset=us-ascii Content-Transfer-Encoding: 7bit The following short note (see the abstract below)

Atomic Categories

is available on Categorical Geometry Homepage at the following address:

Note that to read the special symbols on these pages requires a viewer under Win95. (thanks to Vaughan Pratt for bringing this to my attention). Please let me know if you would like to have a copy in dvi format.

Z. Luo
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Atomic Categories

Zhaohua Luo

Abstract:

Let C be a category with a strict initial object 0. A map is called "non-initial" if its domain is not an initial object. A non-initial object T is called "unisimple" if for any two non-initial maps f: X --> T and g: Y --> T there are non-initial maps r: R --> X and s: R --> Y such that fr = gs. We say that C is an "atomic category" if any non-initial object is the codomain of a map with a unisimple domain. Many natural (left) categories arising in geometry are atomic (such as the categories of sets, topological spaces, posets, coherent spaces, Stone spaces, schemes, local ringed spaces, etc.) In this short note we show that each atomic category carries a unique functor to the category of sets, which plays the traditional role of "underlying functor" in categorical geometry

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