(for martin)

hi...

i'd like to try to state some questions here that i'm interested in ...

we've already talked about many questions of the general form:

for some specific algebraic-geometric theory t, can we give a nice description of its universal property; that is, of what it's the "classifying space" for, or of what it's the "moduli space" of?

(where "algebraic-geometric theory" is my terminology for "symmetric monoidal cocomplete k-linear category"; sometimes i use "finitely cocomplete" instead of "cocomplete" but for now i'll stick with "cocomplete".)

thus for example we've talked a lot about the case of t = quasicoherent sheaves over P^n, and we've explored possible answers in that case such as "t is the theory of a line object L equipped with a good embedding into the direct sum of n+1 copies of the unit object".

but the new questions that i'm interested in (actually i've been thinking about them for a while, but i don't think that i've gotten a chance to explain them to you very well yet) are the same kind of questions, except dealing with so-called "geometric theories" instead of "algebraic-geometric theories". and just as "algebraic-geometric theory" is a synonym for "symmetric monoidal cocomplete k-linear category", "geometric theory" is a synonym for "grothendieck topos".

(roughly speaking, a grothendieck topos is a category which has all finite limits and all small colimits, and where the finite limits "distribute over" the colimits in the same way that they do in the category of sets.)

thus for example, at there's a brief discussion of many different ways of associating a topos to a scheme:

"More exotic examples, and the raison d'être of topos theory, come from algebraic geometry. To a scheme and even a stack one may associate an étale topos, an fppf topos, a Nisnevich topos..."

my main idea here is that when we create a topos from a scheme (or stack) in this way, the universal property of the resulting topos (or "geometric theory") should be very directly related to the universal property of the algebraic-geometric theory of quasicoherent sheaves over x.

thus for example consider ...

??something about "strictly local ring" and/or "henselian ..." or something ?? ...