?? my topic today is officially (??) motivated by an ambitious big philosophical and extremely _general_ (?? perhaps write on blackboard ...) research program (?? or perhaps multiple such programs promoted by different people ... ??? ....) to develop algebraic geometry as a part of categorical logic ( ???? ..... ?? write on blackboard with subset notation ....) ..... but in order to have any chance in a short talk of actually getting to my topic, i have to begin at completely the other end, at the _particular_ end of things, and then if there's any time left over towards the end of my talk, i may be able to say a little bit about the big picture, about the ambitious and very general research programs that this (talk?) is supposed to be a part of ...
so i'm going to start with some pure topos theory, and then only towards the end of the talk will i maybe have a chance to say how it relates to algebraic geometry, and then even less of a chance to try to describe my big picture of how algebraic geometry forms a part of categorical logic ...
however, i do feel compelled to issue a warning right now about the title of my talk ... i'm not sure exactly why i chose this title; i wonder whether i may have been trying to play a trick on someone ... because my title mentions toposes, and then some algebraic geometry stuff, and then sheaves, and for people with sufficient background it's probably sensible to suspect that a talk combining those three ingredients is going to combine them in a familiar way .... whereas in fact i'm _not_ combining them in that familiar way .... so for example i'll be discussing certain toposes whose objects can be interpreted as sheaves, but the way in which the objects qualify as sheaves will be _almost_ unrelated to the way in which they form a topos ... and i'll be using those toposes, which in fact will be grothendieck toposes, for purposes of doing algebraic geometry, but the way in which i use them will be _almost_ unrelated to the way in which grothendieck used toposes in algebraic geometry ...
so here goes with a little bit of pure topos theory ...
when i was first learning topos theory, i found it to be a useful first approximation, or a useful crutch, to think of all grothendieck toposes as being presheaf toposes, or at least as being very similar to presheaf toposes ... in somewhat the same way that when you're first learning about boolean algebras, it may be useful to think of all boolean algebras as being power-sets, or at least very similar to power sets ... but eventually you want to throw away your crutch and understand how there can be grothendieck toposes which _aren't_ presheaf toposes (or boolean algebras which aren't power-sets ...) ... so a perhaps interesting question is : what's the simplest example that you can give of a grothendieck topos that's not a pre-sheaf topos? ... now this isn't a very precise question because i'm not proposing any formal way to measure the simplicity of an example ... but we can still entertain the question in a purely informal way, and i have an answer that i'm going to suggest ... but i'd be interested to hear anyone else's suggestions as well, either right now or later ... ?? so does anyone have any suggestions offhand, as to a nice simple example of a grothendieck topos that's not a pre-sheaf topos?
(?? in case anyone suggests it ... to me, sheaves over (say ...) the real line is a very _obvious_ example, but not a very _simple_ example ... ?? ...)
so let me tell you my example ....
?? but the real interest of this example, i think, is that it actually shows up in nature (so to speak) ... playing a significant role in algebraic geometry, which i'll try to describe right now ...
quasi-affine toric variety that's not affine .... ???? .....
??? hmm, that way of stating it seems to suggest that maybe we should actually even use that as our way of introducing the example, as opposed to giving the toric variety interpretation as an afterthought .... ??? .....
?? maybe even do the switch towards ag / toric varieties while still introducing question, as opposed to while giving answer ... ?? sort of re-interpretation of question... almost ...
??? maybe some slight rewriting required in places above, then ??? .... ?? algebraic geometry as not postponed so much / pure topos theory as not prolonged so much ... ??? maybe mention pure topos theory nature only of _question_, not of answer ... ???? ........ ???? ..... ?? family of examples rather than single example ?? ... ?? place / way to remark upon pleasant surprise of additional intrinsic interest of example(s) ... ??? ...
?? but then also ... ??? other nice sources of non-affine toric varieties .... for example projective instead of quasi-affine .... ????? ...... ?? hmmm, maybe use "non-affine" instead of "quasi-affine but not affine" as the main family ... ?? then mention "quasi-affine but not affine" and "projective" (?? with "... but not affine" almost redundant ?? ....) as prominent sub-families ??? .... ??? projective toric varieties as giving very interesting simple example, but quasi-affine as even simpler (if perhaps less "interesting" ??? ...) .... ????? .....
??hmm, this rough general approach as maybe suggesting ways of introducing some more of the bigger ideas ... ??? ... toric convolution as extra ingredient of tannakian stew in toric case .... ??? .... ?? ....
??... giving opening for somewhat more explicit comment about ... analogy between using "sheaf" to mean "object of topos" and using "quasicoherent sheaf" to mean "object of tensor category" ... ??? where "tensor category" presumably needs to be made somewhat more precise here .... ???? ......
??? something about simplest toric example of "quasi-affine but not affine" as perhaps even simplest not-necessarily-toric example ... ?? illustrating general phenomenon of simplest examples often being toric ??? ......
?? so ... still need further re-organizing here ... ??? .....
?? i guess that theorem should be stated ... ?? something like "subtopos of presheaf topos on commutative monoid x" = "quasi-affine toric variety in affine toric variety spec(k[x])" .... ?? where should worry a bit both about how that's stated (?? localization / open subvariety on toric variety side .... ???? ....) and whether it's actually _true_ (?? open vs more general subtopos on topos side ... ?? ....) ... ???? .... ?? actually similar concerns ... ???? .... one side of the equivalence vs the other ... open subobject (?? ...) vs more general .... ??????? .......
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