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In article <ARA.93Jan10125554@camelot.ai.mit.edu> ara@zurich.ai.mit.edu (Allan
Adler) writes:
>
>Does the category of sites have products and if so how does one construct
them?
>
>It follows from material I found in Johnstone's book Topos Theory
>that the category of Grothendieck toposes has products but I don't know
>if the same is true for sites in general.
>

You don't say what your morphisms are between sites (nor between toposes for
that matter).

One sensible approach is to define the morphisms between sites so that they
correspond to geometric morphisms between the corresponding sheaf toposes. Then
the categories of sites and of Grothendieck toposes are equivalent, so of
course the category of sites has (finite) products.

Details will depend on exactly how you define site - e.g. is it an arbitrary
category with Grothendieck topology, or a category with finite limits? But what
it boils down to is that the site is some kind of presentation by generators
and relations for the topos, so a morphism is an interpretation of one site in
the sheaf topos of the other.

If you are considering sites based on arbitrary categories, then I guess the
product is going to be a "disjoint union" of sites. (Remember that geometric
morphisms in a sense go backwards, so limits look like colimits.)

Steve Vickers.