Subject: categories: Abstract Algebraic Geometry

From: Zhaohua Luo <[email-masked]>
Date: Sun, 27 Sep 1998 23:29:18 -0400

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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The following short note (see the abstract below)

<P>A Note on Reduced Categories

<P>is available on Categorical Geometry Homepage at the following address:

<P><A HREF="http://www.azd.com/reduced.html">http://www.azd.com/reduced.html</A>

<P>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).

<P>Z. Luo
<BR>__________________________________________________________________

<P>A Note on Reduced Categories

<P>Zhaohua Luo

<P>Abstract:

<P>In this note we introduce the notion of a reduced object for any category
<B>A</B> with a strict initial object 0. A pair of parallel maps <I>f</I>,
<I>g</I>: <I>X</I> --> <I>Z</I> is called "<FONT COLOR="#000000">disjointed"
</FONT>if its kernel is the initial map to <I>X. </I>It is called "<FONT COLOR="#000000">nilpotent"</FONT>
if any map <I>t</I>: <I>T</I> --> <I>X</I> such that (<I>tf,</I>&nbsp;
<I>tg</I>) is disjointed is initial. An object <I>X</I> is called "<FONT COLOR="#000000">reduced"</FONT>
if any pair of distinct parallel maps with domain <I>X</I> is not nilpotent.
A category <B>A</B> is called "<FONT COLOR="#000000">reduced"</FONT> if
any object is reduced. One can show that any epic quotient of a reduced
object is reduced. A class <B>D</B> of objects of <B>A</B> is called "<FONT COLOR="#000000">uni-dense"</FONT>
if any non-initial object is the codomain of a map with a non-initial object
in <B>D</B> as domain. We show that any uni-dense class <B>D </B>of a reduced
category <B>A</B> is a set of generators. Other properties and criterions
of reduced categories are also studied.
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