FOM: toposes vs. categories of sheaves

[Colin McLarty]

>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?

        Sure there are (in a reasonably loose sense of "absolutely
nothing"). But there are a whole lot of groups besides the classical
transformation groups too. There are a whole lot of Lie groups besides
those, let alone other non-Lie groups.

        The point is that Weyl very consciously uses general group theory as
far as he can to unify his treatment of the very special cases--that is why
he titled the book as he did (actually THE THEORY OF GROUPS AND QUANTUM
MECHANICS in English). The people in SLNM 753 take their tools from
elementary topos theory, and use it to explain, unify, and motivate their work.

>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".

        Yes, you do. There was even a small amount of work on sheaves of
real valued continuous functions on a topological space prior to topos
theory. (There was more on the ring of globally defined continuous functions
on a space, and on sheaves on special kinds of spaces such as analytic
manifolds.) But if you keep that distinction in mind as you read the papers
in SLNM 753 you will see they use topos theoretic methods as far as
possible, and postpone the sheaf thoeretic particulars as long as possible.  

>
>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.

        But the *classical* transformation groups are very special Lie groups.


Colin