categories: Re: Chu spaces vs Topoi (and maybe Stone Spaces)

[S.J.Vickers@open.ac.uk]

> Hello Cat Community,
>  
>       I have been reading some of Prof. Pratt's papers on Chu 
> spaces. I am
> just trying to understand Topos. Both seem to generalize the notion of
> topological spaces. Is there any relationship beteween Chu spaces and
> Topoi?
>  
> Regards,
>  
> Bill Halchin

In topology, from a topological space X you can make a Chu space (X, OX) in
Chu(Set,2), where OX is the set of open sets. The function from XxOX to 2 is
the "element of" relation. If f:X->Y is continuous, then the corresponding
Chu morphism is (f, f^{-1}).

The topos version of this is as follows. Let E be a topos. A point of E is a
geometric morphism from Set to E, and the points form a (large) category pt
E. Then there is a Chu space (pt E, E) in Chu(Cat, Set), with a functor P:
pt E x E -> Set defined on objects by P(x,U) = x^*(U). (If x is a point,
i.e. x: Set -> E is a geometric morphism, then it has its inverse image part
x^*: E -> Set.) If f: E -> F is a geometric morphism, then the corresponding
Chu morphism is

   ((-;f), f^*)

where (-;f) is just composition of geometric morphisms, x |-> x;f.

This apprently technical structure is better known in the topological case,
where X is a space and E = Sh X is the topos of sheaves over X. Then the Chu
space is (X, Sh X) (make X a category by using its specialization preorder
to supply the morphisms) and P is

   stalk: X x Sh X -> Set,

i.e. if x is a point and U is a sheaf then stalk(x,U) is the stalk of U at
x.

This structure is very important when dealing with toposes. On the other
hand, I'm not so sure you gain much by abstracting away from toposes and
working with Chu(Cat,Set). You lose too much of the logic itself. You see
the same when you try to use institutions (which also use Chu space ideas)
as a universal framework of logic.

There is also the problem that the conceptual points of a topos are not
adequately represented by the "global" points (geometric morphisms from Set)
mentioned above - there may not be enough of them. Instead you have to
consider "generalized points", geometric morphisms from arbitrary toposes.
Hence although the Chu space (pt E, E) is conceptually right for studying E,
it's technically wrong.

Steve Vickers.