Subject: categories: categorical geometry

From: Zhaohua Luo <[email-masked]>
Date: Tue, 12 Jan 1999 12:32:25 -0500

In a recent paper (TAC, Vol 4, 208-248)

On Generic Separable Objects, by Robbie Gates,

the author mentioned a well known "boolean algebraic structure of the
summands of an object in an extensive category". This reminded me a
paper I posted to my homepage last year (8/30/98, see the abstract
below), in which the same boolean structure was reconstructed (at that
time this was not "well known" to me), and was applied to define the
Pierce topology for any extensive category, extending some results of
Diers. The paper

                Pierce Topologies of Extensive Categories

is available at Categorical Geometry Homepage at the following new
(permanent, hopefully) address

             http://www.geometry.net (or http://www.azd.com)

(The new service is a little bit slow, but offers more functions than
the old one, so please be patient.)

Best wishes,

Zhaohua Luo
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Pierce Topologies of Extensive Categories

by Zhaohua Luo

Abstraction

An extensive category is a category with finite stable disjoint sums. In
this note we show that each extensive category carries a natural
subcanonical coherent Grothendieck topology defined by injections of
sums. This Grothendieck topology is induced by a strict metric topology,
which is a functor to the category of  Stone spaces. We call this metric
topology the Pierce topology of the category, as it generalizes the
classical Pierce spectrums of commutative rings. Recall that the Pierce
spectrum of a commutative ring R is the spectrum of the Boolean algebra
of idempotents of R, which is a Stone space. A theorem of R. S. Pierce
states that R can be represented as the ring of global sections of a
sheaf of commutative rings on its Pierce spectrum (called the Pierce
sheaf or representation), whose stalks are indecomposable rings (with
respect to product decomposations). Diers showed that Pierce's theorem
can be extended to any object in a locally finitely presentable category
such that the opposite of the subcategory of finitely presentable
objects is lextensive (called a locally indecomposable category). We
shall see that a weak form of Pierce representation exists for any
object in an extensive category.