From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/869 Path: news.gmane.org!not-for-mail From: Zhaohua Luo Newsgroups: gmane.science.mathematics.categories Subject: Abstract Algebraic Geometry Date: Sun, 27 Sep 1998 23:29:18 -0400 Message-ID: <360F028E.945C2B19@iswest.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 1241017266 28001 80.91.229.2 (29 Apr 2009 15:01:06 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:01:06 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Mon Sep 28 13:19:02 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id KAA17717 for categories-list; Mon, 28 Sep 1998 10:44:44 -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: 91 Xref: news.gmane.org gmane.science.mathematics.categories:869 Archived-At: The following short note (see the abstract below) A Note on Reduced Categories is available on Categorical Geometry Homepage at the following address: http://www.azd.com/reduced.html Note that this file (together with most of the other files in the homepage) can be read now by any viewer capable of graphics (the symbols are included as gif. files). Z. Luo __________________________________________________________________ A Note on Reduced Categories Zhaohua Luo Abstract: In this note we introduce the notion of a reduced object for any category A with a strict initial object 0. A pair of parallel maps f, g: X --> Z is called "disjointed" if its kernel is the initial map to X. It is called "nilpotent" if any map t: T --> X such that (tf, tg) is disjointed is initial. An object X is called "reduced" if any pair of distinct parallel maps with domain X is not nilpotent. A category A is called "reduced" if any object is reduced. One can show that any epic quotient of a reduced object is reduced. A class D of objects of A is called "uni-dense" if any non-initial object is the codomain of a map with a non-initial object in D as domain. We show that any uni-dense class D of a reduced category A is a set of generators. Other properties and criterions of reduced categories are also studied. --------------A7DB306CE63DF22C54674FBE Content-Type: text/html; charset=us-ascii Content-Transfer-Encoding: 7bit The following short note (see the abstract below)

A Note on Reduced Categories

is available on Categorical Geometry Homepage at the following address:

http://www.azd.com/reduced.html

Note that this file (together with most of the other files in the homepage) can be read now by any viewer capable of graphics (the symbols are included as gif. files).

Z. Luo
__________________________________________________________________

A Note on Reduced Categories

Zhaohua Luo

Abstract:

In this note we introduce the notion of a reduced object for any category A with a strict initial object 0. A pair of parallel maps f, g: X --> Z is called "disjointed" if its kernel is the initial map to X. It is called "nilpotent" if any map t: T --> X such that (tf,  tg) is disjointed is initial. An object X is called "reduced" if any pair of distinct parallel maps with domain X is not nilpotent. A category A is called "reduced" if any object is reduced. One can show that any epic quotient of a reduced object is reduced. A class D of objects of A is called "uni-dense" if any non-initial object is the codomain of a map with a non-initial object in D as domain. We show that any uni-dense class D of a reduced category A is a set of generators. Other properties and criterions of reduced categories are also studied.
 
 
 
 
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