Re: (pre-)Sheaves

[S Vickers <s.j.vickers@cs.bham.ac.uk>]

At 11:38 23/01/03 -0800, Bill Halchin wrote:
>     I am trying to get my mind around (pre-)sheaves. I have studied point
>set topology in the past, but I didn't run into the notion of local
>homeomorphism. It seems to me that every local homeomorphism is a
>homeomorphism (because in a topological space (X, T), X is always a
>neighorhood of any point in X). Am I correct?

Dear Bill,

Absolutely not.

Consider the definition of local homeomorphism for a map f: X -> 1. If f is
a local homeomorphism then every x in X has an open neighbourhood
homeomorphic to 1, and you find in fact that f is a local homeomorphism iff
X has the discrete topology.

One way to think of a local homeomorphism f: X -> Y is that the stalk map

   stalk(y) = f^{-1}({y})

is a "continuous map from Y to the class of sets". Of course, the class of
sets is not a topological space. (This isn't just a problem of size - there
really is no suitable topology.) Hence "continuity" here is a new concept
and that is what the definition of local homeomorphism captures. But
intuitively it makes some sense. The definition ensures that if y jiggles
about infinitesimally then the set stalk(y) makes no sudden jumps - no new
elements suddenly come into existence, nor any equalities between elements.

I hope you can see something of this intuition; if not, don't worry but
just stick to the definition.

The definition ensures that each stalk, as a subspace of X, has discrete
topology, so it really is a set, not a more general space. All the
non-discrete topology in X arises across the stalks, as homeomorphic copies
of bits of topology on Y. Topology within the stalks is discrete.

Topos theory is able to make some formal sense of the intuition. There is a
topos E whose class of points (not objects) is the class of sets. It is
called the "object classifier". If I (naughtily, you might think) write Y
for the topos of sheaves over Y, then geometric morphisms from Y to E are
just sheaves over Y. But it is legitimate to think of geometric morphisms
between toposes as continuous maps between their points - this certainly
works for toposes of sheaves over spaces, at least if the spaces are nice
enough (sober). Then geometric morphisms from Y to E should be thought of a
continuous maps from Y to the class of sets, as I said before.

Steve Vickers.