Sheaves on Set^D, where D is a finite poset

Sheaves on Set^D, where D is a finite poset

[Eduardo Ochs]

Hello list,

I am preparing a series of seminars about sheaves on toposes of the
from Set^D, where D is a finite poset, for a mixed audience containing
non-logicians, non-categorists, non-topologists, non-algebraists, and
even non-mathematicians (e.g. physicists), and to keep everything very
elementary I am using lots of explicit examples, taking D to be posets
like "Vee" and "Kite" (my heroes!), depicted below...

                {1,2,3}
                 ^   ^
                /     \
   1     2   {1,3}   {2,3}
    \   /       ^     ^
     v v         \   /
      3           {3}
                   ^
                   |
                   {}

I am basically taking Harold Simmons's "The point-free approach to
sheafification", J.L.Bell's "Toposes and Local Set Theories", Fourman
and Scott's "Sheaves and Logic", and chapter A4 of the Elephant, and
reworking some of the main theorems into that setting... Actually (and
I only dare to confess this because I am far away enough, so no one
will going to be able to beat me up with a big stick!8-) I am trying
to find _finite_ examples that are expressive enough, and then see how
much of the theory I can explain clearly by starting from these
examples and then generalizing...

Anyway: I am having to discover lots of the nice properties of toposes
of the form Set^D by myself - the topologies involved are finite, so
they're even "better" than Alexandroff topologies... - because I have
never seen those categories being explicitly discussed in detail
anywhere.

So: anyone has any papers, theses, notes, slides, etc, to recommend?

   Thanks in advance,
     Eduardo Ochs
     eduardoochs@gmail.com
     http://angg.twu.net/


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