??so what about idea that "toric small zariski topos" maybe isn't good terminology for what we've been using it for recently .... ????....

??but then what about analogy "quasicoherent" : "toric quasicoherent" :: "non-quasicoherent" : "toric non-quasicoherent" ??? ... ???and so forth ?? ...

???hmmm, something about ... "extremal way of cutting and pasting" and so forth ??? ??try to formalize this?? .... and so forth ... ????....

???something about ... ???start with pre-sheaf ... ????and so forth ???? .....

??maybe ... ???start with sheaf ... as object of topos ... and try to get locale from it ... ???and so forth ???? ......

???doesn't this feel like a lot of stuff that we tried which didn't seem to work ??? ... and so forth .... ???hmmm... ??maybe some stuff that we didn't try ... ????....

???something about big zariski topos of x as slice topos of big zariski topos of 1 ... ???and so forth ... ???something about how small zariski of x might be encoded in here (...) ... ??? ... and so forth ... ???...

??wait a minute; is there a problem here?:

on the one hand, we sort of think that there's this category of "affine toric varieties" that's equivalent to .... ???hmmm, i think that i was going to say something like "to the category of commutative monoids" or something like that... which is maybe not that far from the/a truth .... depending in part on what "affine toric variety" should mean exactly... might also try to develop concept of "affine toric scheme" and so forth ... anyway, also possibilities like considering just those commutative monoids that are finitely generated submonoids of free abelian groups... or something .... ??or maybe of abelian groups in general??? .... maybe lots of possibilities...

anyway... meanwhile ... on some other hand, we also sort of think that ... this same category of affine toric varieties (whatever it is...???....) should also be equivalent to ... some ("essentially (1,1)" ... or something...) (2,1)-category of certain toposes, in turn equivalent to (2,1)-category of certain filteredly-cocomplete small categories ... ??which are free filteredly-cocomplete on those commutative monoids (construed as 1-object categories) that we mentioned ??? ... ???or something???

??anyway, the potential problem bugging me here is something like ... mismatch in general between the homomorphisms between the commutative monoids, and the filtered-colimit-preserving functors between the corresponding free filteredly-cocomplete categories on them ... ???... ???something about "kleisli morphism" here ... ???....

???so what's going on here???

??shouldn't it be not that difficult to straighten this out?? ???hopefully ??... ??... ??something about ... morphisms of toric varieties (in some sense...) between affine line and punctured affine line ?????? ...... and so forth .... ????? ......

??so what _about_ how geometric morphisms here get along with day convolution wrt the (...??...) symmetric monoidal structure ??? .... and so forth ....

???hmm, so what about something about preservation of unit object here ????....

??something about ... ???maybe running into subtle questions about "property-like" here ?? .... ???and so forth ??? ....

or maybe not that subtle ... ??something about ... geometric morphisms one way or other between accidental topos and "toric small zariski topos" ... ??? ... and so forth ...

[some brief notes from a week or so ago ... copied from elsewhere ... ??actually interesting to compare to further developments here .. ??..]

??something about "cocycle" ... "kan extension" ... "structure/semantics adjointness" ... "globalization/localization" .... ??? ??something about also "formalization" here??? (??as maybe dual to "localization" in certain sense ... ????...) ...

???something about ... ??what happens to fixed points (and so forth ...) of globalization/localization adjunction when latter degenerates into something like "cohomology theory" ... ??and so forth ... ???....

??so let t be a topos ... then consider model(t) -> [t,_set_] ... ???then try to reflect back into t^op along the yoneda embedding t^op -> [t,_set_] ??? ???or something ?? ... maybe ask todd about to what extent you can get away with this .... ???......

??so what _about_ _set_ as schizophrenically both topos and filteredly cocomplete ?? ... and so forth ... ???_is_ the (...) (2,1)-adjunction here a (2,1)-equivalence ?? ... and so forth ... ???seems unlikely both ways??? (what did we mean by that??? ...???) .... ???does seem likely that _filteredly cocomplete cat_ is a reflective sub-(2,1)-cat of _topos" ??? or what??? ... and so forth .... ????....

???so something about ... ???how certain recent alleged straightenings-out affect "torpos" idea, and/or "frankenstein doctrine" idea??? ... and so forth ... ???maybe former survives better ??? but still ... ???something about ... ???is idea about "torus object" in torpos completely screwed up now ??? .... and so forth ....

??so in the toric case maybe we have the idea that the toric quasicoherent sheaves are the coalgebras for a lex comonad on the toric non-quasicoherent sheaves ... ??so then what about non-toric analog?? ... ??maybe monoidal comonad or something ?? ... ???something about bit about... ??relationship between strong monoidalness and adjoints between lax monoidal functors ??? ... and so forth .... ???....

so... another stab at "textbook" description of "isbell conjugation" ...

??so it looks like they're saying that there's a certain functor from small category c to [c,_set_]^op .... ???that then gets left (or something...) kan extended to give left adjoint functor from [c^op,_set] to [c,_set_]^op ... ??or something ...

so... hmm... that's equivalent to from c^op to [c,_set_] ... which there certainly is the obvious choice of such ...

"geometrically realize presheafs as "opposite co-presheafs", using as realization scheme certain hopefully obvious version of yoneda embedding ..." ???or something ...

??consider for example c = _simplex_ ...

??or maybe just 0d and 1d simplexes .... ???....

???something about "family of bi-pointed sets" or something ???

???so ... ???we want to realize the 0-simplex and 1-simplex as families of bi-pointed sets in a certain way...

???the 1-simplex as ... ??singleton family whose set has 3 elements ???

???0-simplex as singleton family whose set is singleton??... ??...

???not quite making sense yet ??? ..... ??maybe rather 0-simplex as singleton family whose set is doubleton ... seems to make sense ...

??something about geometric realization here as involving "co-glueing" rather than "glueing" of co-presheaves ??? or something ??...

??hmm, so what about something about ... ??? ??getting co-presheaf by homming given presheaf into each representable presheaf in turn ... ??and so forth ???... ??something about "bipartite" version ... "getting y-presheaf by homming given
x-presheaf into each of [y^op]-presheaf of x-presheaves" ... ???or something???

??something about ... ???homming variable thing into constant thing ... as turning colimits into limits ... ???and so forth ... ???.... ???something about stuff about ... weak limit ... and so forth ... ???....

???so what about something about?? ... whether isbell conjugation is "self-conjugate", ifykwim ... ???... ??something about whether both of the adjoints can be viewed as "spectrum", or something ??? .... and so forth ...

also ... ???what about something about "algebraic geometry" examples, and especially categorified such ...?? ... ??something about ... doctrine ... with family of favorite environments ... ????something about extent to which the two conceptual "parts" can be freely transposed in the "bipartite" case ... ????....

???something about ... ??"pairing" between commutative rings and ag theories, for example ... ??...

???something about... ???given a "pairing", using formal colimits on one side and actual colimits on the other side, vs using formal colimits on both sides ... ???or something ??? .... and so forth ... ??? (??something about trying to view former as special case of latter ... ???and so forth ...?? ...) ??what _about_ "variance twists" here, and so forth ??? ....

???something about ... ??naive "restricted yoneda embedding" / "spectrum pre-[sheaf/stack]" idea ... ???can be applied to presheaves, for example ... ???resticting along the yoneda embedding, os??? .... ???hmmm, so what _about_ all this "isbell" stuff as to do with some case of "yoneda embedding restricted along yoneda embedding" , or something ???... x-presheaves contravariantly yoneda-embed into [x-presheaves]^op-presheaves ... which can then be restricted along the yoneda embedding from x^op to x-presheaves, to give x^op-presheaves ... ???meanwhile maybe there's a sort of opposite way of "restricting the yoneda embedding along the yoneda embedding", just with ops in different places, that just gives the identity functor ??? .... ???if so then _why_, exactly ???.... ??hmmm, could yoneda lemma be construed as sayign exactly that this is the idnetity functor ??? ...???or something ??? ...


??so i'm sort of guessing that people are thinking of co-presheaves on the category of (maybe "finitary" or something...) affine schemes as sort of "generalized commutative rings", and then homming them into actual commutative rings to get presheaves on the category of affine schemes ... well, not sure that i said that exactly right yet, but in any case i'm not too impressed so far ... are any really "interesting" presheaves supposed to arise this way?? .. and so forth ... ???...

??so does diaconescu's theorem (or something...) imply (or something...) that the classical model category of a presheaf topos is the _free_ filteredly cocomplete category on the opposite of the site?? ... ??something about this as maybe giving clear examples of free filteredly cocomplete category on a small category being non-small??? ... ??and also maybe supporting idea about ... ??finite colimits beck-distributing over filtered ones ?? ...???or something ?? ...

??so given (a,b,c,d,e) with a and e N-sets, c a Z-set, b "N-equivariant from a to c along N included into Z as the negatives" and d "N-equivariant from e to c along N included into Z as the positives" ...

??something about ... ????comonad here .... ??"replace c by the pullback of ..." ... ????or something ?? ... ??not quite right??...

???maybe also... ???replace a and e by pullbacks ... ????or something ???....

??so what _about_ trying to get a better understanding of "classically invisible" grothendieck topologies?? ... and so forth ...

??so _do_ we have any really good examples so far ?? ...

???what _about_ something about "booleanness" here ?? ... and so forth ....

??something about ... ???classical invisibleness of other things besides grothendieck topologies ?? ... and so forth ....

??hmm, so what about taking the slice topos before imposing the grothendieck topology, vs vice versa?? ... and so forth ...

???something about the topology as simply removing one model ... as usual ... ???

??so maybe it's important to take the slice topos first ... ??sort of because ... the model property corresponding to the grothendieck topology essentially refers to the "Z-frame" structure ... ?? ??or something ?? ...

??so what _about_ condition on N^2-set of pair (x,y) with xb=ya coming from unique element z with (x,y)=(za,zb) ?? ... ???as _not_ a sheaf condition ??? ... ???or what ???... ??hmm, or maybe it _is_ a sheaf condition ....

??ok, so what _about_ the models of the accidental topos of the projective line??

??also... ??what about model(t)^op -> t ... given by ... ???or something ??


for example consider t := the object classifier ...

a model here is just a set m ...

which (??contravariantly??) gives a functor from _finset_ to _set_, namely "x |-> x^m" ... ??...

??have i been getting terminology "cone" (in fan of toric variety) a bit mixed up ??? ... ??or something ?? ...

???hmmm, so what about something about ... ??filteredly-cocomplete category x ... ???yoneda embedding x^op -> [x,_set_] ... ??but then composed with functor [x,_set_] -> [x,_set_]_filtered-colimit-preserving which is right adjoint part of "surjective geometric morphism" ... something about "cofree coalgebra of comonad", or something ??... ???_is_ this the way it goes ???....

??something about ... ???comonad on _set_^2 ... (a,b) |-> (aXb,aXb) ... ???and so forth ???... ???something about having left adjoint monad (c+d,c+d) <- (c,d) ??? or something ???.... ????some level slip about "idempotence" here ??? ???"idempotence" of factorization system (and so forth ...) vs ... ???idempotence of monad or comonad associated with one of the adjunctions that an adjunction is factored into ?? ... and so forth ... ??... ???so what _about_ the "model objects" in a topos ??? ... that is, simply the image of the embedding (??or something...??) model(t)^op -> t ... ???...

???so what _about_ something about "ab=cd" as adjoint orbit of gl(2), or something ??? .... and so forth ... ????.....

??in light of recent partial straightening out of relationship between toric variety accidental topos and filteredly-cocomplete category associated to its fan, it looks to me at the moment like maybe "frankenstein doctrine" idea isn't really working ... ??or something?? ...

??but what about maybe something about ... relatively free cocompletion of only finitely cocomplete ag theory... ??as maybe thought of as theory of doctrine including something about filtered colimits ... ??or do i mean filtered limits here ???... hmm, probably lots of confusion here, but ... ???...

??something about "compactness" issues here, and so forth ??? ... ??quasiprojective vs projective and so forth ??? ....


???hmmm, so am i maybe now catching another big mistake (or another part of one same big mistake...) ... ??something about ... i wrote to todd ... :

"given a filteredly cocomplete category x and the corresponding topos x# of filtered-colimit-preserving set-valued functors on it, we have the yoneda embedding from x^op into x#, and i'm trying to tell you (in a particular case where we have a reasonable concrete description of x#, and would like to obtain a similarly reasonable concrete decription of x) how to reverse-engineer x from x# by finding x^op as a certain 3-object full subcategory of x#."

???so ... _is_ this screwed up ??? .... because of something about ... ??yoneda embedding here as maybe not actually landing in x# ???? .... ???or what???...

??well, so we should really test this example of "N-torsors", i think ... ??though there could be danger of extra-special coincidences of some kind here ... ??...

???so, a functor _N-torsor_ -> _set_ consists of ... ???an N-set, and a Z-set, and an N-equivariant map from the N-set to the Z-set ?? ... ???is that correct ????.... ??and the functor is filtered-colimit-preserving precisely in case the N-equivariant map is the comparison map from the N-set to its tensor product over N with Z ?? .... ???is that correct ??? ....

(??hmmm... ??so what _about_ relationship to "quasicoherent vs non-quasicoherent" and so forth ???? ...)

??so anyway ... ??we want to test whether each value of the yoneda embedding _N-torsor_^op -> [_N-torsor_,_set_] (contravariantly assigning to an object x the covariant functor "homming from x") is filtered-colimit-preserving ...

??so two cases to check ... homming from the N-torsor N, and homming from the N-torsor Z ...

so let's try homming from the N-torsor N .... ??seems like N represents concept of "element" ...

??something about ... ??in case of free filteredly-cocomplete category, of course the "generating objects" should get taken to connected projectives by the contravariant yoneda embedding ... ???what we're seeing here being part of that... ??or something ???....

and just as of course, the _non_-generating objects should get taken to _non_-[connected projective]s, right ??...

in any case, let's check "homming from Z" here to make sure about what's going on ... seems like it's _not_ going to preserve filtered colimits ...

so... "homming from Z" takes N to the empty set, but takes Z to Z ... and Z is _not_ the tensor product of the empty set over N with Z ...

so yeah, it seems clear that that message that i sent to todd was screwed up ...

so then what _about_ how to try to straighten out the situation?? well for one thing... instead of trying to recover a filteredly cocomplete category as a certain subcategory of the topos of filtered-colimit-preserving set-valued functors on it, why not simply recover it as the model category of that topos ??? .... ??to what extent does that "fix" various problems / confusion ???... and so forth ....???? ....

??maybe something about "isbell duality" (??or something ???) here ?? ... ?? ....

?????some further (?????....) confusion here ???? ..... ?????something about .... ????hadn't we pretty much convinced ourselves that the difference between the quasicoherent and the non-quasicoherent sheaves (in the toric case ...) was ... something about ... ??the quasicoherent ones as being sheaves wrt some (??further?? ... ??or something?? ...???) grothendieck topology?? (and then i was going to say: whereas it wasn't until after that that we caught the mistake about preserving filtered colimits as not being a sheaf condition; thus contaminiation by that mistake ... and so forth ... ) or maybe no, that's not quite what we'd convinced ourselves of ... ???rather maybe just the bit about "lax glueing vs strong glueing" or something ... ???also various other ways of thinking about it; would probably be good to go back and try to synthesize them all together, or something ... ???... ???so maybe now we're more or less claiming that the strong glueing arises from the lax one by imposing the further filtered-colimit-preservation property ... ???or something??? .... though hmmm, then why don't i rememebr anyone trying to express quasicoherence as something like a filtered-colimit-preservation property (or something ... from a certain point of view ... toric vs non-toric case here ...) .. ??? ... and so forth ... ???

??so ... might it be that the topos of toric quasicoherent sheaves arises from the topos of toric non-quasicoherent sheaves as the coalgebras for a nice comonad?? ... ???or something ?? ... ???if so then what about various possible nice conceptual interpretations here??? ... and so forth ... ???and again maybe something about "isbell conjugation" (and so forth ...) ... ??? .... ??...

??so what about something toric structure on quasicoherent sheaf as maybe some nice sort of lifting of torus action, or something??? ... and so forth ...

???so... ??something about ... ??finite weak coproduct of filteredly-cocomplete categories as "working at the underlying category level" ?? ... ???or something??? ... ???relationship to "beck distibrutivity" ?? ... ... and so forth ... ??...

??so what _about_ various sorts of relationships between (...small...) zariski (??basic??? ... and so forth ...) open subspaces and "sheaves" of various kinds??? ... and so forth ... ???...

??something about "hartog ..." here ... ???...

??even if we may have slightly straightened out one confusion going on here ... ??maybe others ... ???....

??something about ... ???assigning to open subspace u "walking u-section" ...

??assigning to u "walking u-equation" ... ???.... ???relationship to "skyscraper sheaf" or something ??? ....

??assigning to u quasicoherent sheaf given by .... ????something about "localization ..." ?? ... ???how _does_ this relate to other stuff here ?? ...

.... ???? ....

??containment of zariski opens induces morphism between "localization" quasicoherent sheafs _contra_variantly ?? ... ??or something ?? ... ???what _about_ something about ... ????homming walking section into structure sheaf, and so forth ??? ..... ???something about ... ??non-quasicoherent sheaf ??? .... and so forth ... ???....

??try to make this more intelligible ... ???....


locally finitely presentable ???_free_(??) filteredly-cocomplete ???.... ??something about ... ??trying to straighten out ... something about stuff that todd pointed out about ... ???locally finitely presentable categories and so forth, in connection with topos associated to filteredly-cocomplete category ...


??sa property-ishness of topos being torpos ... ??... ???something about whether arbitrary geometric morphism between torposes gives tag morphism ... ???and so forth ??? ... ???hmmm, what about even (...) in "affine" case ??? ....

???so what about something about ??some sort of "generalized gabriel-ulmer duality" (os ...) stuff going on here (...), in light of ... ??maybe somewhat straightening out bit about getting topos from filteredly-cocomplete category ... ??? os... asf os...


???mention to todd idea that ... ??"slice topos" approach to articulating geometric theory embodied by accidental topos of toric variety ... that we thought about a bit but didn't get too far with ... should now fit somewhat nicely with stuff about .... filteredly cocomplete category with objects corresponding to cones of toric variety's fan ... and so forth .... ???...

??hmm, have i been falling into a slight glitch here??? ... something like ... ???given a filteredly cocomplete category, you want to somewhat systematically look for a topos with it as classical model category... or something like that ... so... ??that means that the site category should actually be the opposite filteredly __complete category ... ??is this really correct?? ...seems like it shoul be ... ??does this maybe help to straighten out any (??especially "variance" ...) confusions that we were having ?? ... ??something about "frankenstein doctrine" and so forth ??... ??something about "pre-sheaf vs co-presheaf" ??? .... ???and so forth ??....

??actually i'm pretty confused right now; not sure which (...) way it should go ... ??something about idea that ... ???in formulating a grothendieck topology, doesn't it make more sense to use _colimits_ in the site category ???? .... ???or something??? ....

... really try to straighten this out ... ??...

??well, i'm trying to straighten it out but getting _really_ confused ... ??something about ... ??seemingly very opposite (or worse...) ideas about ... ???relationship between open set and sheaf (or something ... ???) associated to it ... ???? ...... ????????? .....

??something about sheaves and weak limits and so forth ... ??? ....

???hmm, so what about the possibility that the main big confusion here is that rather than considering sheaves for a grothendieck topology here, what we really want here is something about ... ???filtered _co_-limit-preserving set-valued functors .... ????or something ???? ..... and so forth .... ????......

notes for next discussion with todd

filteredly cocomplete (??small ?? ...) category and special grothendieck topology on it... ??and how to recover this site from its topos...

(hmmm, that might be screwed up .... maybe instead of sheaves for grothendieck topology we want set-valued functors preserving filtered colimits ... or something ... ??anyway, maybe with this one minor/major little change, the basic ideas can still go through ... ???....)

free vs non-free filteredly cocomplete category... ??something about the above topos as just a presheaf topos in the free case ... ??...

motivation for above in terms of accidental topos of toric variety... and so forth ...

??maybe something about "toric model" idea ??? .... and so forth ...

??maybe warning about ... ???not reallty getting to the "interesting" topos stuff yet .... ???want to get to catalog, but ... ??....

??idea of collaborating with alex?? ...