Re: Sheaves of T-algebras

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

Dear Mike,

Once you have that a sheaf of T-algebras is equivalent to a T-algebra in
the category of sheaves, it's obvious: because the global sections
functor preserves finite (in fact all) limits.

But the first step is not so trivial, and in fact is not true unless you
take care over "the stalks are T-algebras". For example, if the base
space is Sierpinksi then a sheaf is just a function, with the domain and
codomain as the stalks (over the bottom and top points respectively).
You might put algebra structures on the stalks, but it won't be an
algebra in the sheaf category unless the function is a homomorphism.

You will get further issues if the base space is non-spatial, so there
aren't enough global points to provide enough stalks.

What statement of the result did you have in mind?

One sufficient condition that gets round those problems is for the
stalks and their T-algebra structure to be defined geometrically.

Regards,

Steve.



Michael Barr wrote:
> It must be known that if T is a left exact theory (same as nearly
> equational theory) and you make a sheaf of T-algebras (meaning the stalks
> are T-algebras, then the set of global sections is also a T-algebra.  Can
> anyone give me reference?
>
> Michael
>



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