toposes of toric quasicoherent sheaves

this talk is part of my "algebraic geometry for category-theorists" program, which is also what my talk at the conference that we just went to was about ... so that talk that i gave is a useful prerequisite for this talk ...

a big part of "algebraic geometry for category-theorists" is to convince category-theorists that, probably without realizing it, they've been engaging in certain prototypically algebraic-geometric activities for their whole category-theoretic lives ...

when i say "probably without realizing it", i'm extrapolating from my own experience ... my general method as to assume that everyone else made the same stupid mistakes that i did, and then try to offer gentle correction of those mistakes (?? bit about impulse to teach arising from constantly walking around in a daze muttering to yourself "if only i'd known _that_ when i was just starting out" ...) .... which can really confuse people if they never made those mistakes and/or never even had the chance to make them ... ??? ....

and of course it's good in a way when that (?? ...) sort of thing happens, that you find out that you already know something that you didn't know you knew .... because then you don't have to go through the bother of learning something really new; instead you just start taking credit for something you already knew .... ??? ....

so when is this alleged time in your category-theoretic lives that you've been engaging in a prototypically algebraico-geometric activity without realizing it? it's not when you might naively have guessed- that's why you probably didn't realize it ... instead, the time that i'm talking about is whenever you find yourself studying some kind of "structured categories" and thinking of them as "theories" of some kind of "logic" ... (?? and so forth .... thinking of the homomorphisms between such structured categories as "interpretations" or "models of the domain theory in the environment provided by the codomain theory" ...)

?? examples, with audience participation ..... ??? ....

?? bit of imagined history .... lawvere and theories, beck and doctrines .... ?? relatively contemporary ideas about some sort of 2-toposes .... ??? ..... where "my" doctrines fit in there ... "categorified gabriel-ulmer duality" ... categorification of "lex theories as special toposes" .... ???? ....

anyway that's a very general preamble, but today i'm hoping to actually get into interesting specifics ... describing in some detail a little part of this program .... so what i'm actually hoping to talk about today is how this (?? ...) big general tannakian philosophy / program specializes to a particular part of / topic in algebraic geometry known as _"toric varieties"_ ...

?? status of toric geometry as "easy toy example" in algebraic geometry ... ??? ....

?? actual (?? ...) toruses involved, vs "based on sets instead of vector spaces / abelian groups" .... latter approach as leading, perhaps not _too_ surprisingly, to involvement of toposes ....

???? so in fact, in addition to the whole big conceptual justification that i've just tried to give here, i have a secret entirely (?? ...) different justification for what i'm planning to talk about .... this alternative justification is that the examples of toposes that are going to turn up here happen to be what i consider somewhat amusing (?? ...) and educational examples of toposes .... they're what i informally (?? ...) might consider to be "the simplest examples of grothendieck toposes that aren't presheaf toposes" .....

?? so perhaps i should ask ... more audience participation .... your opinions about what sort of topos that might be ...

??? lots of different ways of thinking about toposes .... ?? might be interesting to hear about alternative ways ... ?? .....

?? filteredly cocontinuous .... ????? ....

??? various contrasting analogies (?? ...) from "topos" to ["abelian category" or some related concept ...] .... ??? ....

??? trying to formulate compatibility conditions ... ??? "combined doctrine" ... ??? exploit audience expertise ... ??? ....

?? bicommutative bialgebra .... ???? ......