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{\bf Visualizing Geometric Morphisms}

Eduardo Ochs

Departamento de Ciências da Natureza

Universidade Federal Fluminense (UFF), Brazil



Different people have different ways of remembering theorems. A person
with a very visual mind may remember a theorem in Category Theory
mainly by the shape of a diagram and the order in which its objects
are constructed. For such a person most books on Category Theory feel
as if they have lots of missing diagrams, that she has to reconstruct
if she wants to understand the subject.

The shape of a categorical diagram remains the same if we specialize
it to a particular case --- and this means that we can sometimes
remember a general diagram, and the theorems associated to it, from
the diagram of a particular case.

In this talk we will present the general technique above and one
application: reconstructing the statements, and some of the proofs, of
two factorizations of geometric morphisms between toposes described in
section A4 of [1], from particular cases that are easy to draw
explicitly --- in which our toposes are of the form $\Set^\A$, where
$\A$ is a finite category whose objects are certain points of $\Z^2$.
The tricks for visualizing sheaves on these `$\Set^\A$'s are described
in [2].



[1]: ``Sketches of an Elephant: A Topos Theory
Compendium''. P.T.\ Johnstone, Oxford, 2002.

[2]: ``Planar Heyting Algebras for Children, 3: Geometric Morphisms''.
E.\ Ochs, 20017. Preprint available at:

\noindent \url{http://angg.twu.net/math-b.html#zhas-for-children-2}.


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