??so what about idea of something sort of like ... ??... "getting the objects from presheaves on the site where the morphisms are just the localizations, but getting the morphisms from presheaves on the site where the morphisms are more general maps" .?? ... ???and so forth ... ???...

??any model/formula confusion here?? ... ???or something ?? ... ??and/or some sort of level confusion ... ??something about stack vs sheaf here .... ????and so forth ??? .....

??so what about... ??from toric:non-toric::"accidental topos":"accidental abelian category", also toric:non-toric::"doctrine combining topos and tag theory...":"doctrine combining abelian category and ag theory" ?? ... and so forth ...

??something about this idea as seeming to mesh nicely with ...previous idea about ... ???... ??noticing how in certain cases quasicoherent sheaves over toric variety can be thought of as ... ??abelian group objects in topos of quasicoherent sheaves ??? .... or something ... ??... ... and so forth ... ??...

??so what about some concept of "quasi-abelian category" ?? ... parallel to "quasi-topos", not "quasi-coherent" ... ???... and so forth ... ???...

??something about .... "combined doctrine" here as involved in bit about "globalization..." and "both forgetful functors preserving weakimits" (or something ...) ??? ...

??so let's consider presheaves (??or perhaps more generally i want to consider prestacks here ... ??...) on the category of (??perhaps just "finitary" in some sense...) affine schemes and "localization maps" (or something...) between them ...

???something about "globalization of module categories" (or something...) here as "accidentally" (or something) giving ag theories which just happen to also be abelian categories ...???

??so what about something about various sorts of comparison functors between these presheaves/prestacks and those on the site category where not just the localization maps are included but more general maps as well?? ... and so forth ... ???something about "getting the ag theories to match" and so forth ??? ... ??something about image factorization of geometric morphism, and so forth??? ....

??something about possible sorts of "object/morphism confusion" in connection with "site" ?? ... and so forth ... ???maybe also "morphism/covering confusion" ??? ....

??so consider commutative monoid m ... and another such k ... (??thought of as "toric base" ... or something...) ... ???and consider the constant map m -> k with value 1 ... ???so this is making it seem like a toric variety really does have a canonical basepoint??? .... ??or something ??? ....

???so .... composite (2,1)-functor _comm monoid_ -> _tag theory_ -> _cat_ ... ??something about whether it factors through _geometric theory_ -> _cat_ ??? .... ???or something ???...

??consider for example comm monoid hom 0 -> N ... ??resulting morphism of tag theories doesn't preserve terminal object ??? .... ??or something ??? ....

???but ... ???what about the glueing of toposes that we've been doing ????? ....???is it all screwed up, or what ???? .....

???????????

??what _about_ something about "artin-wraith glueing" here??? ... and so forth ... ???....

weak pullback in (2,1)-cat of categories ...

_N-set_ -> _Z-set_ <- _N-set_

"localize" ... ???

???so _is_ there something here about ... ???mostly these aren't geometric morphisms, but they _are_ in this "localize" case ????? ..... ???or something ???....

???and is it something like this in the non-toric case as well ?????..... .... hmmmmmmm ..... tensoring with a flat module .... as giving equivalence of abelian categories .... ????and so forth ????? ..... ?????? ..... hmmmmmm ..... ?????....

??so what _about_ maybe something about here... ???getting information about nice class of "stacks" (or something) giving "nice" theories .... ???? and so forth .... ???????.....

??so what _about_ tag theory of p-algebra for prop p (or something...) as topos ??? ... and so forth ... ??? ???so _is_ it a "globalization/localization fixed-point" ?? ... ... ???....

???hmmm, so what about "accidental abelian category" of an "ag pre-stack" ??? .... and so forth .... ????....

??but... ???homomorphism of comm monoids gives geometric morphism, whereas homomorphism of commutative rings doesn't give exact functor ???? ..... ????or something ??? .... ?????.....

???hmm, or _does_ homomorphism of comm monoids maybe _not_ give geometric morphism in way that i was imagining ??? ... ??stuff to check here ... again issue of parallelism between toric and non-toric cases ... ???... ??lots of confusing variance (and so forth...) questions here ... ???what about something about ... ???"affine cover of affine" stuff here ??? .... and so forth ... ???....

???something about... ??as example of (??sort of reasonably "nice", in some ways???... ???or something???...) "tag theory" that's not a topos, underlying tag theory of an ag theory ??? .... and so forth ... ???...

??so what about some sort of lowbrow version of "non-commutative geometry" (...) involving thinking pretty concretely about the spectrums of some operators that non-commute ... ?? ....

??so ... ???_are_ we claiming that ... ???the toric quasicoherent sheaves over _any_ toric pre-stack form a to(r)pos ??? ... ???or something ??? ... ???because of something about ... both (??...) forgetful functors getting along with weak 2-limits ... ???or something ??? ... and so forth .... ????....

??so .... ??consider... ??subfunctors of "[N^2,-]" : _comm monoid_ -> _set_ such as "{(x,y)|x^3=y^5}" ... and so forth ... if that makes any sense ... ???something about idea that such a sub-functor "can't participate in any reasonable cover" ... or something like that ... ??something about trying to formalize this idea ... ??? ... and so forth ... ???...

notes for discussion with todd this morning

??lots of different approaches to accidental topos of projective line ... ??or something???...

graded approach...

"globalization" approach....

???sheaves of actions" apporach ??? ... and so forth ...

??relationship between these last two, and so forth??? ...

??filteredly cocomplete category approach ?? ... and so forth ...



??some questions ... ??

???something about .... ????sheaf of monoids ... getting stack of action categories from it ... and so forth ... ???something about "quasicoherent vs non-quasicoherent" here ... and so forth ....

???something about "contrastive element of presheaf" (or _some_thing...) ... and so forth ... case of directed graph ... ???and so forth ... ???but then case of affine toric variety ... ??how to justify those certain elements ( ?? or something?? ) as qualifying as "contrastive" ... ???and so forth ... ????....

??something about issue of to what extent "being torpos" qualifies as "property" of topos ... and so forth ... ???...

something about makkai and pare vs makkai and reyes... and so forth ... ?syntax/semantics duality ... ??... and so forth... ???....

??so consider ... "czech cohomology (of space x...) with coefficients in discrete non-abelian group g" ... ???...

???something about ... ??? 1-groupoid of g-torsors over x ... ???h^1(x,g) as set of iso classes of g-torsors over x, and h^0(x,g) as (non-abelian ...) aut gp of the trivial g-torsor over x ??? (??what _about_ aut gp of non-trivial torsor, and so forth ?? ... hmmm.... "twisted" ... ??...) ???is this correct??? ... ???try case x = k(g',1) ??? ... for example g' = z; then h^1 = conjugacy classes of g, and h^0 = ... ???maybe g???

(test some stuff here in part by thinking "simplicially"... and so forth ...???...)

??this as all depending only on "1-truncation" of x??? ...?? ...

??_is_ this all on the wrong track??? ...???...

??what about maybe ... ???some relatively minor level slip involving 1 vs 2 ... ???and so forth ... ??...


???then consider "stable" case, and so forth ... ??...

???something about ???homming "positive" spectrum into "neutral" one ... ????as giving "negative" one, or what??? ... and so forth ...


???something about ... ???recently asked about "what happens when globalization degenerates into cohomology theory", or something ... ?? but then also something about... taking seriously thinking of "globalization" as "like a cohomology theory" ... and so forth ... ???...

??what _about_ "obstruction" ??? ....

??so suppose given a site ... (??not sure whether i want the grothendieck topology to be trivial?? ...)... and also a sheaf of commutative rings over it ... or possibly of commutative monoids of some other nice kind, or something ...

??then hopefully it makes sense to say something like ... ??"pass from the sheaf of commutative rings to the stack of their module categories" .... ????or something like that ???....

(????something about .... ??category-valued vs groupoid-valued stack here????? and so forth ???? .... ??level slip ??? ....)

(??something about "internal ..." here??? .... confusing ... ???.... ....???also something about ... ???topos vs 2-topos here... and so forth ... ???something about vague memory of something from n-category cafe about something like "concept internal (??or something??) to a topos which is like a stack over the (??...) corresponding site" .... and so forth ... ?? ... ??well, so what about something about simply (??...) using giraud's theorem to "cheat" here??? .... canonical (...??...) site ... ??? .... and so forth ... ???also something about ... object vs morphism ... ??and so forth ??? ... ??? .... ??so what about whether category of module objects over ring object (or something...) qualifies as "locally internal category", or something completely different, or what ... ??... ... and so forth ...)

???and then consider ... ???the groupoid of global sections of this stack ???? ... ???or something ??? .... ??any ambiguity in that??? ...???or something ???...

??anyway, somewhere around this point i seem to get very confused about "quasicoherent vs non-quasicoherent sheaves of modules ..." .... ?????and so forth ?????.....

??so consider for example a directed graph, and consider the sublattice of its subobject lattice generated by subobjects that "occur as pieces of irredundant covers" ... ??or something ???...

??perhaps any "edge piece" qualifies?? ???something about cover by the edge pieces ... ??

??vertex qualifies if ... ???if what???

???something about the "element poset" of a presheaf ... ???and sub-poset of it consisting of just those elements that ... ??? ???or something ??

??what about idempotence or otherwise of functorial process on posets (or something...) here???... and so forth ... ???...

??hmm, bit consider for example representable presheaf on _comm monoid_ ... or something ... ???maybe something about considering also how presheaf may fit into other presheaves, in trying to decide what the "good" pieces are??? ... and so forth ... ????...

??so what _about_ getting locale from presheaf, by ... ????....

??together with concept of "basic open" ??? ...

???something about getting _site_ from ... ???presheaf over site category, or something ?? ... ??or maybe even sheaf over site category, or something??? ...

??but the key idea ... no idea yet as to to what extent it makes sense ... ??something about ... "universal way of expressing as colimit of representables" ... ???or something ??? ... and so forth ... ??something about "pro- ..." ... ??...or something ...

??is "generalized" (in a certain sense) toric algebraic geometry essentially just "affinely relativized" such?? ...??or something ??...

???something about ... ???plenty of morphisms of affine toric varieties (for example) that are like "inclusion of clised sub-variety ... ??...

???so what about certain kinds of geometric colimits of affine varieties, and/or of their associated ag theories, and so forth ... ???something about ones which seem "unproblematic" ?? ... or something ... ??something about "pair of polynomials with same nth order truncation", and so forth ... ???something about "pair of [vector space with linear operator]s, together with isomorphism between their kernels", and so forth ...

??is there something about ... ???certain kinds of geometric colimits involving "glueing together closed pieces" (or something...) tending to be _un_problematic, so that attention is more focused on some problematic aspects of "glueing together open pieces" ??? .... ???and so forth ?? ...

??what about sheaves over site given by ... ??something like category of "spaces" (or something ...) and "local homeomorphisms" (or something ...) and grothendieck topology given by ... ??perhaps obvious ?? ... ???and so forth ...???..

??vs something about ... ??using some concept of "local isomorphism" to specify the grotehndieck topology ... ??in maybe somehwat obvious way .. ??given that category has sums, or something?? ... and so forth ... ??...

??hmmm... ??some funny stuff going on here??? ... ???something about "giraud's theorem" and so forth ??? ... ??but with something about freyd's (?...) stuff about "non-topos which is locally a topos" and so forth ... ???something about only allowing hausdorff spaces here (and so forth ...) ... ???something about lawvere's "big vs small topos" ideas... ???and so forth ??? ....

??something about ... ???"borrowing colimits from the objects whose spectrums (??wrt some "pairing" or something???...) are being taken", vs ... "deciding that some given colimits (for example of affine schemes or something ...??...) are broken because of not being preserved by certain spectrum process" ... ????and so forth ???

???confusion here about "pairing" vs "functor along which to restrict yoneda embedding" ??? ..

???something about terminology "restricted yoneda embedding" for restricting _value_ functors of yoneda embedding ....

???something about ... ???"spectrum" here (...) as like "yoneda embedding" ... ???something about pretty severe constraint to ask for yoneda embedding to preserve colimits .... ????something about "point-like objects" ???? .... ???something about "field" and "spectrum" thereof .... ????something about "voodoo" ... ??? ... "generic point" ... ...and so forth ...

????something about _various_ special cases of "spectrum" (and so forth ...) idea where one or other of the (...) categories involved is a presheaf topos ... and so forth ... ???something about "isbell conjugation" and _limits_ of presheaves ... and so forth ... ????...

???...limits of commutative monoids .... ??... (2,1)-limits of tag theories ... and so forth ... ???....

???something about "relative" (or something...) geometric morphisms between slice toposes ... and big zariski topos of x as slice topos of big zariski topos of 1 ... and so forth ... ???....

???so what _about_ "open cover" vs "colimit" ??? .... and so forth ... ???....

???in catalog of "toric" toposes, include those related to "orbit stack of torus action" ?? ... and so forth ...

???so... ??seems like there _is_ this standard functor model(t)^op -> t ... ??and maybe sometimes it's full and faithful, but not always... ??something about needing "enough degeneracy morphisms" (or something...) to get full-and-faithfulness ... ??...

??something about "isbell conjugation" here ??? ...

??something about ... ??projective variety as lacking in globally defined functions ... ??and so forth ... ?? ....

??what about a geometric colimit such as ... "two lines intersecting transversally at a point" ?? ... and so forth ... ???something about "glueing of quasicoherent sheaves" here, and so forth ??? ....

??hmm, so what about something about "xy=0" and "_generalized_ toric variety" ??? ... and so forth ... ????....