abstract algebraic geometry

[Zhaohua Luo <zack@iswest.com>]

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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
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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).



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