FOM: toposes vs. categories of sheaves
Stephen G Simpson
simpson at math.psu.edu
Wed Jan 14 12:44:19 EST 1998
Colin Mclarty writes:
> >Categories of sheaves are toposes, but the notion of topos is much
> >more general.
>
> This is like saying Hermann Weyl's GROUP THEORY AND QUANTUM
> MECHANICS contains no group theory, since it only concerns the
> classical transformation groups and the abstract group concept is
> much more general.
Huh? This strikes me as a misleading analogy.
I'm no expert on toposes, so correct me if I'm wrong, but I'm pretty
sure that the notion of topos is *much much much* more general than
the category of sheaves on a topological space or even a poset. For
instance, I seem to remember that if you start with any group G acting
on any space X, there is a topos of invariant sheaves. So arbitrary
groups come up right away. And this is just one example. Aren't
there lots of other toposes that have absolutely nothing to do with
sheaves?
In any case, I think there is a big conceptual difference between
"topos" and "sheaves on a topological space". The first is some kind
of very abstract theory of functions; the second is a set-theoretic
construct of a kind that is familiar in analysis and topology. You
need to distinguish between "topos-theoretic foundations of the real
number system" and "sheaves of real-valued continuous functions on a
topological space".
By contrast, a lot of group theory is concerned with subgroups of
classical transformation groups, and *any* group can be viewed as a
transformation group. So it may not be inappropriate for Weyl to
speak of "group theory" in the abstract, even while focusing on
classical transformation groups.
-- Steve
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