?? "correspondence" ... "spectrum" .... ????? .....

?? various ideas about "correspondence" .... ?? one being concept of simply "adjunction" ????? ...... ???? ..... (??? this as similar / related to idea of "sesquiherent sheaf" as correspondence ... ??? ...) ?? but ... ??? well for one thing, consider trying to apply that interpretation of "correspondence" to the case of "pure stuff" ... ag (?? ...) theory of g-torsors for some g some nice "algebraic group" .... ???? ..... ?? say work over field of complex numbers or something like that, maybe .... then underlying cocomplete algebroid of the theory probably becomes something like just a "2-vector space" in (roughly ...) kapranov-vovoedsky sense ... ?? which would make the "correspondences" here rather "bland" ... though this sort of blandness can it seems still be the setting for interesting stuff ... thinking here of vaughan jones / kapranov-vovoedsky / "lowbrow mackay correspondence" style "categorified matrix algebra" stuff ... ?? but ... ?? two extremes here, each annoying in their own way ... ?? ... correspondence category mysteriously knows too much, vs knows too little ..... ???? ....

??? "spectrum" ... ??? "mysterious spectroscopic data" .... (?? "reductionism" .... ?? ...) ... vs "moduli stack" ... "world of possibilities" ... ??? .....

?? idea that "correspondence" should be sensible "logical/geometric" object .... ??? span (?? of groupoids vs of ... ??? ...) idea vs "sesquicoherent sheaf" idea vs "sesquiherent cosheaf" idea .... ???? which of these have good "logical" / "geometric" / "conceptual" status .... ??? maybe first two but not last one ???? ..... ???? .... ??? ...

??? derived level reconciliation between sesquicoherent sheaf and sesquiherent sheaf ??? ....

??? "sensible logical / geometric / conceptual status of (?? specific version of .... ??) correspondence concept" vs (?? ....) .... ?? "doctrine" interpretation of correspondence ??? .... ??? correspondence as simply theory interpretation wrt poorer doctrine .... ??? how this relates to logical / geometric idea .... ?? "hecke operator" .... ??? "orientation" .... ??? .....



??? paper ... ?? arranging certain part of section 1 (?? ...) in something like "dictionary correspondence" form ... ??? so get to mention stuff like ... ??? well, that alleged bit about "affine corresponding to totally distributive" ... ?? in connection with grothendieck topologies, or maybe with toposes ... ??? .... ??? ... also, double negation topology as corresponding to (?? the ?? ...) torus .... ??? ....

?? hmmm .... ?? stating (?? "dictionary correspondence" style ?? ...) stuff in terms of grothendieck topologies vs in terms of geometric morphisms .... latter viewpoint as not completely more general (?? without a bit of going out of way ... ??? ...) in that ... ?? interesting if _all_ grothendieck topologies on toposes with toric interpretation participate in that interpretation ... ?? closedness of sub-2-category here under taking arbitrary sheaf subtoposes ???? .... ??? raising issue of other closedness properties of the (?? ...) subcategory ... ?? ... ?? also (?? thus ?? ...) raising issue of "reflectiveness" of subcategory ... ??? .... and / or of similar issues ... "birkhoff theorem" - style theorems ... ?? closure of semantic category (??) under certain processes / relationships as giving information about nature of corresponding (?? ...) syntactic category ... ??? ... ??? hmmm, extent to which we've already encountered this idea of relating this "birkhoff"-style stuff to "doctrine" and / or "generalized gabriel-ulmer duality" (?? ... ??? level slip ?? ... ???? ..... ?? anyway "daulity" of some kind ... perhaps various kinds .... ??? ...) ideas .... ???? makkai-??? .... .... ???? ..... ???? ....

?? perhaps part of what i meant to be trying to say here was ... ??? ... birkhoffian "closedness" properties of semantic (?? ...) categories as maybe conceptually "dual" (?? or almost so ??? ... ?? some funny aspects ... ?? ... maybe some effortful decategorification ... ?? ...) to abstract syntactic operations (realized (somewhat ... ?? ...) concretely on syntactic categories of theories ...) defining doctrines ... ?? .... .... ??? "concreteness" (??? .... ??? ...) of semantic categories ... ??? anything dual on syntactic side ???? ..... ???? single- versus multi-environment semantics here .... ??? .... single- case and (?? "generalized gabriel-ulmer" ?? ...) "duality" ... ??? ..... ??? .....

??? anyway ... ?? so what about actually trying to take some sort of appropriate "birkhoffian" hull of the toric varieties among the toposes ??? .... ???? ..... ?? if it's not already some such sort of hull, which it likely isn't ... ??? ....

cockett .... ???? ..... partial maps .... ??? .... ?? certain (?? "accidentally" ??? ...) nice things about the toric case .... ?? dense opens and analytic continuation .... ?????? .... ?? connectedness .... ??? (??? new discrete sums ??? ... ????) ... ?? extent to which these might be unavailable in more "adult" context ... ??? ...

"birational toric geometry" as maybe a sort of "homotopy theory" ??? ... ?? model category ?? ... ... ??? .... 2-category .... ??? .... ?? torus as fibrant (?? ...) object or something ???? .....

???? (bad ... ?? ...) idea of regarding general grothendieck topos theory as "noncommutative toric geometry" .... ???? .....