???no unit object for intersection of basic affine opens ??? ...??? ...

??? "pro-unital" ??? .... ????...

???confusion ... day convolution wrt monoidal (or more generally pro-monoidal...) structure ... ???vs ... ??geometric morphism induced by monoid product ... ??? ... ??"essential" ??? .....

??day convolution as "extra left adjoint" of essential geometric tensor product ... ??well, in the case where there's an actual tensor product rather than just "pro-" such ... ??? ....

???doctrine where an environment is .... ???a topos t, equipped with (??the extra left-adjoint part of) an "essential tensor product" ... ???? .... ????....

(??still confusion about "theory has moduli stack over site given by ..." vs "environment has ... " .... ??? ... ????"formula has solution presheaf over site given by models" ... ???thing that "has" solution presheaf as same kind of thing occurring as object in site ... ???? .... ???? .... ...... ?????? ........ ??"theory" and "environment" as similar ...... ???????.....)

i was going to suggest ... ??restricted nature of "accidental topos" / "accidental abelian category" (as compared to tops /abelian category in general ... ??) as going against "doctrine philosophy" .... but ... not really ?? ... ??because even before you get to that issue, the "accidentalness" already makes things weird ... ??? ... ???.... ??making other alleged weirdness not so relevant ?? ... ??? ....

??tensor product of dual cones as categories enriched over fraction group ... ???....

???torus case ????? .... ??...

??non-toric analog ??? .... ??? ....

??are we getting a mismatch of some sort between :

1 ?? dual cones as "algebraic" .... ???from fan picture ???.... ...

2 ?? dual cones as "geometric" ... ???... points of accidental topos .... ????...

???"torsor" ... ????.....

???? what's really going on with idea of "pseudo-rotations" baez mentions ... ?? ...

"state" ... ... ???extra structure on ... set with a priori somewhat less structure ... ?? ....

??toric analog of "measuring failure of lexness of AG morphism between accidental abelian categories" (??some sort of "cohomology" / "derived functor" ... quasicoherent sheaves ... ???....) ... ??? "measuring failure of lexness of TAG morphism between accidental toposes" .... ???? .....

??right adjoint part of classical t-model geometric morphism ... ????....

_set_ -> t .... ???...

?? (??special sort of ??) boolean algebra object in t ????? ..... ??conceptual interpretation ?? ....

? fan as "formal colimit" ... ???some sort of "generalized fan" corresponding to "formal (2,1)-colimit" ?? ....

i want to take a stab here at describing the filteredly cocomplete category associated to a fan ...

...trying to use the "affine" case as a guide ...


the objects of the category should be the cones of the fan ...

given cones c1 and c2, a morphism from c1 to c2 should be a dual vector v such that for any x in the dual cone c1*, x+v is in the dual cone c2* .... ????...

???so for projective line, this seems to be giving hom table .... ????

N00
ZZZ
00N

???? or something ???

??really just saying that the objects are the dual cones and the morphisms the "translation maps" between them?? ... ??does closure (of fan ...) under intersection and "taking faces" lead nicely to filtered cocompleteness of the category of dual cones with "translation maps" ?

???so... the 3 models of the accidental topos of the projective line should be, from the viewpoint of the toric quasicoherent sheaves as the triples (a,b,c) with a and b N-sets and c a Z-equivariant anti-isomorphism between their tensor products with Z .... :

1 "take the underlying set of a" ...???

2 "take the underlying set of the common Z-set of a and b ..." ... ???...

3 "take the underlying set of b"


so ... are these models "representable" by a formula ??? .... ??? ....
??or maybe "tensor-representable" ??? ....????

??the two "affine" ones are representable by formula ??? ... but not the non-affine one ???? ... ???does that actually make sense??? ???not quite ??? ...

i've vaguely heard of ideas like this before ... something about self-distributivity in "foundations" ... hearing of it again now : ... wikipedia article on "laver table" and reference there to http://knot.kaist.ac.kr/2004/proceedings/DEHORNOY.pdf ...

??? .....

??so let's try to describe the "category of cones" of a fan f ... ??guided by case of f affine ?? ...

object = cone of f ...

morphism from cone c1 to c2 = ????.... ??dual vector v st ... ????...

??analogy geometric theory : artin-wraith glueing :: ag theory : ["blow-up" .... ??? ....] ???? ..... ???? ....

??"attaching fringe to bulk" ... ??toric context ... ??? fan poset ... "ideal as ring" ... ??? ...

??_is_ this idea we already tried that conspicuously didn't seem to work (??"artin-wraith glueing for coherent sheaves" ... ???? ...), or is this a different idea ??? ...

"hopf law" ... ??? "refurbish from one-lane to two; then composite of one lane with return-trip on othe rlane as trivial" .... ???? .....

??fan of _all_ cones on Z^n ?? .... ?? ...

how to get singularity of toric variety unless built into basic affine building block ?? ... hmmm.... ??good point, i guess... so what about ... "cone over moduli stack of elliptic curves ..... ????? .....

??interaction between "stackiness" and "singularity" here ??? .... ???? ....

y^2 = x^3 ... ???? ???did this show up in our mathematica pictures ??????? .... ???...

?? does this sort of example actually fit into "fan" formalism ?? ... ??? ...

??"multi-rack" ... family of rack structures, "distributing" over each other ??? ...

??even a plain rack as having a Z-parameterized such family ??? ....???generally parameterized by a group, not necessarily abelian ....

???"strongly homogeneous space" ... ??? "generalized k-affine space" ... ??agl(k,1) ... ??? .... rack structures on x^2 ... ??? ....

some ideas derek mentions ...

?? ...

?? action of gl(n.Z) on ... ????"points" of group alg of Z^n ?? ... ???... ??? .....

?? n=1 and projective line ?? ...

?? n=2 and ... ???...

??? Z^2 as not very "cube-ish" (or even square-ish ...) ... in terms of its automorphisms ... ??? .....

??? attaching "fringes" to torus ... ??? ....

??? topos (???which one??? ...) associated to single toric variety vs topos whose objects are some sort of generalized toric varieties .... ??? ...

??how to "get morphisms between models to be localization-like" ??? ... ???analogy to for example "getting morphisms between models of object classifier to be injective by passing to [decidable object] classifier" ... ??? ... ??? .....

i wrote a mathematica program to draw toric varieties in a certain way...

this exercise suggests various ideas to me, but i'm having a bit of trouble articulating them ...

presheaves on ... ??certain subcategories of _comm monoid_^op .... ????.... ???"geometric realization" process for these ... given by co-presheaf .... ???toric variety as built out of some sort of "cubes" ... ???some sort of "oblique" (????...) morphisms ... ??? "face" ... ??? .... ??? "degeneracy" ... ???? .... ??? ... "localization" ... "local isomorphism" ... ??? ...

???open cubes vs ... ???half-open ... ??? .... ???? ..... ?? ???grothendieck topology here ??? .... ???? ...... ???....

... ???non-toric analog ... ???geometric realization of preseheaf on _affine scheme_ as space (???....) of some kind ... ??? ... given by co-presheaf ... ???examples??? ...

presheaf _comm ring_ -> _set_

co-presheaf / "model" _comm ring_^op -> _set_ .... ???"galois shapeshifter" ??? .... ??? ....

contravariant hom functor of given comm ring ... ????maybe with "topology" ???? ... ??? ......

??not quite sure how right-track this "cubic" idea is yet .... ???? .... need to think more about toric glueing ... ??? ....

??"positive" vs "negative" closed subvarieties of toric varieties ... ??...

??natural toric structure on either/both of these ??? .... ???? .... ??landmark point in each toric orbit ??? .... ??? ....

??logarithmic coordinates ?? ... ....??moduli stack of elliptic curves??? .... ??? ....

??ideal generated by difference between two monomials ... vs ideal generated by monomial ... ????.... ???..... ??or myabe particularly just : ideal generated by generator .... ??? .... ?? ....

todd suggests formulating math overflow question ... ???something like this?:

to a toric variety v can be functorially associated a grothendieck topos v# so that quasicoherent sheaves over v correspond naturally to abelian groups in v#. what papers discuss toric varieties from this viewpoint?

??confusion about preservation of products (??and/or more general limits??) by progression from dimensional theories to AG theories .... ???... ??as suggesting strongly un-"abelian" flavor of dimensional theories (despite superficial analogy??...) ?? ...

??in "toric" case, though, what then about passage to "torus", which seems rather abelian ??? ... ???? ....

???so.... punctured plane as toric variety ... fan as "boundary of quadrant" in Z^2 ... ????....

????relationship to projective line??? ....????visual relationship of fans ??? .... ??? ...

???fans with "overlay" aspect ??? .... ???relationship to "non-separated toric pre-scheme ...." ???? .... ??? .....

?"0-subvarieties" vs "1-subvarieties" of toric varieties, and different conceptual roles they play ?? .... ???1-subvariety as actual "subvariety", 0-subvariety as "fringe" ??? ... ??? .... ???but then ... ???distinction blurred or erased with "toric geometry with 0" ?? ... ??? ..... ??application to "associated graded" ??? ... ???whether ag morphism for which "associated graded" acts as geometric pullback qualifies as toric or generalized toric .... ??? ... ???...

???orbit _toric_ stack of toric variety wrt _the_ torus ??? .... ????? .... ??? .... ?? affine line as example .... ????toric quasicoherent sheaf over orbit stack as "generalized filtered set" = "Z-tree" ???? .... ?????..... ??tensor product of Z-trees ...

???quasito(r)pos of _genuine_ filtered sets ??? .... .... ???....


????cohomology of toric quasicoherent sheaf ... ??? .... ??? ???cohomology of quasicoherent sheaf over toric variety .... ??? .... frankenstein ... cover ... glueing ... ?? ....

???frankenstein _non_-doctrine ??? ...

???toric geometry as playground for ... ???..... "cohomological approach to local geometry of moduli stack" ... ??? .... "deformation theory" ... "rational homotopy theory" ... ??? .....


???role of gl(n,Z) (??? ...) in connection with fans on Z^n (...??...) vaguely reminding me of maybe similar role in connection with ... ??? ???root system ... weight diagram ... "character" .... "chern-weil" ... ???? ... ????.....

??toric structure on projective plane (for example ...) and "kaleidoscope as fan" ... ???? ....

???tendency for toric geometry (??...) to "stay under wildness threshold" ??? ....

kaleidoscope as fan .... ???? maximal torus .... ??? ....

??baez's idea to get mathematicians (?? ...) involved in ecological issues ... ???my suspicion that this is a bad idea ... ??seems like mathematicians as not the right kind of people for this ...?? instead non-mathematicians, or maybe only really bad mathematicians ... ??? ....

examples of torpos-glueing to play around with ... ??? ....

projective line ...

"affine line with doubled origin" ...

???various ways of trying to understand contrast (??...) between above two examples ??? ...

"-:-" vs "-.-." ... ????...

??? ???"-." ????? .... broken symmetry here ... ???unclear whether much of good idea at all ??? ... ??? .....

projective plane...

punctured affine plane ...

??maybe more sorts of puncturing???....

???stern-brocot ... ????.... maximal ... ideal limit .... ??? ...

[projective line]^2 ...

???pushout X pushout ...

.<-.->.
^##^##^
|##|##|
.<-.->.
|##|##|
v##v##v
.<-.->.

... ??? ...


???aut gp of N + Z ....

1 0
x +-1

... ??....

localization ... ?? at (??prime ???) ideal ??? ... "fringe" ... ???artin-wraith ??? .... ???? ....

??also examples _not_ giving torpos (in particular not topos ...???....) .... coming from "non-flat" case ... ??? .... ???? ....

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

ideas for todd ...

(also check file trimble40? ??? ... containing bunch of ideas to maybe mention in e-mail ... ????...)

??"doctrine as special case of 2-topos" ...

??in/advivisability (... ?? ......) of experimenting with varying arity not only of colimits, but also of limits ... in "topos" context ... ?? ... ??? ...... with colimits you're experimenting in the conservative direction whereas with limits in the risky direction ??? .... ???so what _are_ the dangers ??? ..... ..... ??? .... ....

???relationship between above two ideas .... ???? .....

??? general idea (pointed out by fred wilhelm in seminar the other day...) of situation where "mapping cone" of nice interesting geometric map specified up to diffeomorphism (??? ....) is nice geometrically interesting space specified up to diffeomorphism ... ??? .... ???examples given by real and complex projective planes??? .... ???? making me now try to tie this in with semi-recent speculations of mine about ... ???... information about attaching maps involved in schubert stratification encoded in root system ..... ???? .... ????....

person whose sci.math posts i remember ... david ... ???don't quite remember last name at moment... ??"anti-cantorian" ... ?? ... went off the deep end... ??maybe somewhat slowly though??.... ranted about (hard to quote accurately from memory ...) ... "math as observational science with computer as its instrument of observation" ... ??? ... ?? which rant as maybe sort of correct, but took wrong lesson from it ... "anti-cantorianness" ... ???right lesson as anti-[cult of proof ...] .... ???....

??vague memory of time i proposed debate with them ... ??interesting to try to reconstruct aspects of that ... why i begged off... (didn't get teaching job?? ...???...) ... where i was back then in my own philosophical progressions ... ??? ... ...

"flatland" and pre-/history of "non-euclidean geometry" ... ??? ... ??...

?? "mathematical gnosticism" ... ??"algebra of old testament as geometry of new testament" ... ???...

??so how does [current attempt to understand "doctrine" as special case of "2-topos" ... ??as categorification of "lex theory as special case of 1-topos" ... and ... "theory as formula and environment as model" ... ???and so forth ... ???...] fit with ["moduli stack" as _of_ theory, with environment varying ... ???....] ???

??in passage from doctrine to 2-topos, arbitrary (?? weak 2-)diagram (aka "formal weak 2-colimit" ... ??...) of theories becomes honorary theory ... ???... ???"mixing colimits at different levels" and/or ... ??? "morleyfication" ... ???.... and ... ????? .......

?????"finiteness" (and ...??...) issues here ???... ...??in particular danger of needing (???...) not only non-finite colimits but also non-finite (weak ... higher ... ??) limits ??? ... ??? ...

???"individual environment can be big ... bigger than any theory ..." ...

??"intended environments" ... ??"restricted yoneda embedding" along inclusion of such ... ???... [restricted yoneda]-like "embedding" (??perhaps actually somewhat destructive ?? ...) of ... ???topos into set-valued functors on model category .... ???? .... .... ??? ...

?? sheaf : "space" :: stack : "champ" .... ???? ..... ???......

???hmmm... ??_theory_ has "moduli stack of models" ... (??decategorified analog: _formula_ has "moduli sheaf of instances" ...???...) ... ???and such moduli stacks, including those of formal weak 2-colimit theories, form a 2-topos ... ???....

???2-geometric morphism between comm-ring classifying 1-topos and AG-theory classifying 2-topos ... ??? .... ????? ..... ???? ........ ??which way, if any ??? .... ????....

??lawvere... objects of a topos as "generalized spaces" .... ???.... ??? ....

?extent to which products preserved by process taking toric variety to "fan" poset ... ??? ....

??any "toric" aspect to child's drawings ??...

??what about "fan with deficit angle" ?? ... ... ???... ??probably not quite make sense... for most part ... ???...

??so what about "non-separated toric pre-scheme" .... ??? ... ???...

??... ???all the ways of "matching" NxZ localizations of two N^2-sets ... ??as corresponding to fans of particularly simple sort ... "geometric interpretation" ... ... ?? ...

??"frankenstein ..." ... ???...

singularity ... ???...

??trying to reconcile (??...) certain approaches (??...) to "doctrine as special case of 2-topos" ....

1 : ???... "theory as 2-formula and environment as 2-model" ... ??...

2 : ??... categorification of [lex theory as special case of topos] ...

???maybe these can be reconciled fairly well if ... ??just remember ... ??2-topos has more 2-formulas than corresponding 2-doctrine does ... ...?? "syntactic (2,1)-category" .... groupoid-valued (2,1) functors on syntactic 2-category of doctrine as giving syntactic 2-category of 2-topos .... ?????only the representable (??maybe expressed in terms of weak limit preservation ... ???? ...???..) such functors preserving filtered (????...) weak colimits as being in the syntactic 2-category of the doctrine??? ....

??so maybe these two approaches really are practically the same ??? ....

what got me thinking about this today was mostly, i think, talking with alex yesterday... ???....question as to what kind of parallelism (??or "tweisted" such ??c ...) there is in ... "doctrine made up of theories, each in turn made up of formulas ..." (???...) ... ?????.....

??? ... ???working out uncategorified "lex theory as special case of topos" ... ???to what extent / when have we done this ?? .... ??? ...... ?????.....

??so... let m : x >-> y ... ???and suppose that ... ???there exists a factorization x >-> j >-> y of m st ... ??? ???pushout of some k <-< j >-> y .... ???? .... ??? ... ???....

??something about ... ??relationship between [relationship between toposes and filteredly cocomplete categories ...] and "semantics / syntax adjointness" ... ??mention to todd??...

??also something about ... ???possible problem here with ... ???something about ... ???whether model cat of topos is small-ly generated or something??? ...???what _about_ whether adamek & rosicky (...) deal with this in some interesting way ??? ...

??so what about punctured plane as toric variety??? ... its torpos ... its filteredly-cocomplete category ... as "non-free" ... ??? ... ??its fan ... ???its "toric structure" ... and so forth ... relationship to toric stacks....

??some confusion here about ... ???"slice topos" and "doctrine-combining" and "adding vs subtracting structure" ??? ... and so forth ... ???

??so... ??should be pretty easy to explicitly describe the model category of the (apparent) "torpos" associated to the hopefully sort-of obvious "toric" structure on the "moduli stack of elliptic curves" ... ???then try to figure out whether it's a "generalized presheaf topos" ... ????....

so let's try looking at the flat toric quasicoherent sheaves over the projective line... in various pictures ... ??...

??_is_ this reasonably unambigous ??? .... ????.....

??we're expecting that the 3 possibilities are "NZZ", "ZZZ", and "ZZN" ?? ...???... ??does that fit with the graded action picture?? ... ???....

??confusion here about (??among other things??...) mapping out of a limit .... ???....

??what about something about ... ???whether tensor product of quasicoherent sheaves (and/or toric analog) can be thought of as "corrected" (?sheafified?? ... ??) tensor product of some more "pre-sheaf-y" underlying things... ???and whether that might be somehow relevant here... ??something about maybe getting left exact left adjoint by using _un_-corrected tensor product ... ??? ....

??anyway, the three left exact left adjoints should be given by ... tensoring the first coordinate with N, or either coordinate with Z, or the second coordinate with N ... ??? ... ????....

??so what about walking short exact sequence abelian category... ???i was going to ask whether it's a generalized module category or something, but ... ??maybe we actually already established that it's an actual module category?? ...

??so what about something about "flat quasicoherent sheaf" .... ???and "flat toric quasicoherent sheaf" ... ???or something ??? ... ???hmmmm..... ?????....

??the "something extra" that a topos model has compared to the underlying model of the corresponding "generalized presheaf topos" as some sort of "continuity" / "cohesion" ???? ??or something ??? ... ??????weird if we're suggesting that this sort of cohesion is _lacking_ in certain algebraic geometry contexts ??? ..... ??particularly if think about relationship between "cohesion" and "flatness" ??????? .... ????...

??hmm, but what about "cohesive property" vs "cohesive structure" vs "cohesive stuff" ??? ... and so forth .... ?????....

??what about the abelian category of sheaves of abelian groups over the real line??? ... ???whether it exhibits "non-toric cohesion" .... ??? ... ????.....

???blecchh, did i get this all backwards (??though a different backwards this time??) again ??? .... ???something about the something extra as some sort of _non_-cohesion ?? ....????..... ????confusion ???? ..... ???....

???maybe something about groupoid vs category here?? ... idea of moore-postnikov factorization idea as way more complicated (and/or less applicable ??? ...) in category case ???? .... ???something about "coalesce" ... ???....

hmm, ok, it really _is_ some sort of extra continuity (??"non-discontinuity" ??..... ???....) up in the space with the "less noticeable" topology, namely the discrete one ... ????... hmmm.... ????.... ???hmmm, something about "local constantness" .... hmmmmmm ..... ???????? ......

???what about something about .... ????local constantness _structure_ and/or _stuff_ ??? .... hmmm.... ????something about .... "drag/transport" ... ???...
????? .... ???...

???hmmm, what about something about ... map from smithereens to original space as maybe both epic and monic???? ... ???.. and so forth .... ????....

???what about something about "analytic continuation" and "morphism that's morally (??or do i mean _genuinely_ ?? ... ???...) mono but restricting along it is injective rather than surjective" and then maybe here "morphism that's morally epi but co-restricting along it is injective rather than surjective" ??????? ..... ???and so forth ??? .... ????.... ???some confusion here ???? ... ??something about ... ??representation ... induce ... ???and so forth ... ???....

??map that's bijective on global points .... ??? ....

???something about ... "expecting a structure but it's actually just a property" ... extendability vs extension ... ??co-extendability vs co-extension ???....

???something about "topology" and "sub-ring" .... ????.... ???something about "restricting a ring homomorphism along a (??"algebraically" ... ???..) "dense" sub-ring" ??? .... ????..... ???wait a minute, isn't "localization" in fact an example of that ??????? ..... ???? .... ????? ..... ????maybe something about "opposite" of bi-morphism as bi-morphism ???? .....

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

???something about something extra as classical complements of certain predicates ... ???? ....???...

concerning idea of extra "compatibility" in "combined doctrine" in non-toric case ... ??what about toric analog ??? .... ?? ...

??also toric analog of "reconciliation between limits and colimits at derived category level" ??? ... and so forth ... ??

??so what about ideals whose underlying modules are flat?? ... and so forth ... ???... ??something about filtration more general than ideal power filtration ... ??and so forth ... ?? ...

??what about "geometric interpretation" of ext (and so forth....) between coherent sheaves?? ... and so forth ... ?? ... ???in terms of ... ???how "supports" (or something) of coherent sheaves intersect... ??and so forth ... ????...

???something about "intersection theory" here ???? .... ???and so forth ??? ....

??so let's try to describe some nice simple example of "non-formalness" ... ???_have_ we ever really tried this before??? ... hmmm... (??also something about "non-conicalness" ??? ... ??don't we have familiar examples where "associated graded is non-isomorphic to original filtered ..." ??? ... ???or something??? ... ...) ...

??so we want some really simple example of "ext^2" (or something...) being non-trivial ... ???...

???something about polynomials in 2 variables x,y ... or something ?? ... ???....

[a|ax=ay=0] ... free resolution ...

??trying to make this into non-trivial 2-place chain complex .... ????....

???what about "minimality" here ??? .... ...or something ...

place 0 ... < a > ...

place 1 ... < b,c,e >

place 2 ... < f >

d(b)=ax

d(c)=ay

d(e)=0

d(f)=by-cx+e ??

h^0 = < a >/< ax,ay >

h^1 = < by-cx,e >/< by-cx-e >

h^2 = < > ???

???not quite making sense to me yet ??? ....

???what about something about "yoneda _de_composition" ??? ... ???...

something about a2 kaleidoscope and non-toric morphism between toric stacks?? ...

??so, temporarily assuming that the comparison morphism from a topos t to the topos corresponding to its filteredly cocomplete category of t-models isn't always an equivalence ...

is there some nice way to recognize when a grothendieck topology gives a topos t for which the comparison morphism _is_ an equivalence?? ... ??...

??concerning idea of relationship between "pro-monoidal category" and "operad" ... and so forth ... ??what about something about bias between "operation" and "co-operation" here ??? ....

what about the idea that localization inverting f doesn't seem very "flat" at zero of f??... ??or something...

??some vague idea that "you're not supposed to look from that point because it's been excised" or something ... ?? but... ???something about module vs commutative algebra ... ???problems either way??? ... ???? ...

i'm going to try to sketch for todd here some ideas about non-toric analogs of some of the ideas that we've been trying to develop in the toric case...

so suppose that we take the topos x of "toric quasicoherent sheaves" on a "reasonable" toric variety, and then take its category c of models. then we conjecture that the comparison geometric morphism to (??or is it from??) the topos of filteredly cocontinuous set-valued functors on c is an equivalence.

then let's try working out a non-toric analog here...

the quasicoherent sheaves over a reasonable variety form an abelian category x... ??then the exact functors from x to modules over the base ring play the role of the mnodels here...

??except that here i should probably make up my mind for now as to whether i'll work with finite colimits or arbitrary ones... ?? ...

notes for next discussion with todd

??"non-toric analog" of whole bunch of stuff arising out of recent discussion about toposes of filteredly cocontinuous set-valued functors ... ??starting with "accidental abelian category" as non-toric analog of "accidental topos" ?? ... (something about accidental topos as (thus? ...) not-so-accidental ... and so forth ....)

toric and non-toric "combined doctrine" ... and so forth ... ??struggling to connect with semi-famous grothendieck topologies ... and so forth ... ??...

??something about example where toric quasicoherent sheaves over "toric pre-stack" _don't_ form a to(r)pos ?? ... and so forth ... ???....

...todd's stuff about kock-zoeberlein and co-inverter, and so forth ...

??maybe ending (or something...) with non-toric analog of "topos for which comparison geometric morphism to topos of filteredly cocontinuous set-valued functors on classical model category _isn't_ an equivalence" ...??... and so forth ...


(??no obvious concept of "deliberate abelian category" ??.....)

???hmm, something about moduli stack of elliptic curves as toric stack?? ??and looking at its accidental topos ??? .... and so forth ... ??... ???hmmm, so what _about_ whether it's of form "filt(c,_set_)" (or something...), and what this might say about "stacky analog of fan" and so forth ???....

??something about concrete geometric aspect of "toric geometry" ... 2-(maybe also 3- ??...)place chain complexes... and so forth ... ??... ??maybe also something about taking "dimension" aspect of toric dimensional theory seriously ... and so forth ... ???....

???something about issue of colimit preservation by passage from dimensional theory to ag theory ... and so forth ... ???toric analog too?? ...

check other semi-recent notes ... ??...

??so what about topos t for which comparison geometric morphism from/to (???) [t' given by filteredly cocontinuous set-valued functors on the classical model category of t] is _not_ an equivalence?? ??also what about "non-toric analog" here??

(??"non-toric analog" of lots of things as big theme for next discussion with todd? ...)

??when is the category of filteredly cocontinuous functors on a category c no more than "ordinarily big" ??? .... and so forth ...

??so what about the category of Z-graded sets as a torpos corresponding to a "toric stack" ?? ... and so forth ... ??

??what about such alleged toric stack at "toric dimensional theory" level vs at "tag theory" level?? ... and so forth ... ????...

what about something about the way it somehow struck me as funny here for "sequence of vector spaces" (or sets, or something ...??...) to give free _co_-completion ... ???and so forth ... ???some vague intuition, possibly level-slippery, about limit vs co-limit ... and so forth ... ???....

so what about "pfaffian system" and/or dgca as something like "interlocking system of de's, one each with solution candidates being n-simplexes in space x, with face of solution as solution of previous one ..." or something ... ???subject to complications involving "syntactic vs semantic consistency" and so forth ... ???....

???so what about .... the "models" of the abelian category of quasicoherent sheaves over a nice algebraic variety ??? ... or something ... ???as analogous to the models of the topos of toric quasicoherent sheaves over a toric variety ... ???... hmmmm..... ??so what about affine case here??? .... and so forth ... ???

??so what _about_ toric variety morphisms which give geometric morphisms of accidental toposes??? ... and so forth ... mention to todd... and so forth ...

??so consider for example tag theory given by pairs of N-sets, vs one given by N^2-sets ... and so forth .... ??something about evident lack of favorite model in former case ... and so forth ...

??also something about toric dimensional theory and plain dimensional theory, and so forth ... nature (??something about "stucture on" and so forth ...) of moduli stack of models ... and so forth ...

??something about ... ???? dim th : enveloping ag th :: abelian group : its group algebra ... ???something about "group algebra" functor (valued in commutative algebras...) as preserving algebraic (weak...) colimits but not algebraic limits ... ???? thus geometric limits but not geometric colimits ... ??? ???hmm ...??seems slightly strange, since... ??i associate "dimensional theories" with "projective geometry" (more or less...) and i think of projective geometry as (of course ...??...) lying on the "good" side as far as geometric colimits are concerned ...

so then what _about_ the geometric interpretation of algebraic products of dimensional theories??? .... and so forth ...

???let's try... the algebraic product of [the usual dimensional theory of the projective line] with itself ...

... 0 0 0 *1* 2 3 4 5 ...

2 3 4 5 6 7 ....
3 4 5 6 7 8 ....
4 5 6 7 8 9 ....
5 6 7 8 9 10....
6 7 8 9 10 11....
.
.
.

??so then what about how graded modules of the above compare to pairs of graded modules.... ????... and so forth ...

??well, so what _is_ this the dimensional theory of ??? ... ??an idempotent number, together with a pair of line objects ...


???what about something about ... ???over P^1 + P^1, taking the line bundles given by "dual tautological over the first line and trivial over the second" and "trivial over the first and dual tautological over the second" ???...

???"an idempotent number z, together with a pair x,y of line objects and sections x#,x1,x2,y#,y1,y2 such that when z=0 then x#=0 and y# is invertible and y1=y2=0 whereas when z=1 then x# is invertible and x1=x2=0 and y#=0" ??? or something??? ???does that actually make any sense??? ... and so forth ...

??zx=x ... z*y1=y1 , z*y2=y2 ... (1-z)y=y , (1-z)x1=x1 , (1-z)x2=x2 ...

??but what about the idea of trying to get things to be invertible here??? ...??does that make any sense ?????.....

??hmm, maybe we left out some generating sections ... ???instead of x#, how about x+ in grade x and x- in grade -x ??... and so forth ... ???.... ???but does that screw up the numerology ??? ...??? ...

???hmmmmm..... ?????..... ??or maybe un-[screw-up] it???...

0 0 0 1 2 3 ....

0 0 0 0 1 2 3
0 0 0 0 1 2 3
0 0 0 0 1 2 3
1 1 1 1 2 3 4
2 2 2 2 3 4 5
3 3 3 3 4 5 6
.
.
.

hmmm....

??so then _is_ there maybe some sort of morita equivalence here, or something ??? .... ??maybe simply (??...) something about ... grade (g1,g2) being the direct sum of grade g1 of the first coordinate module and grade g2 of the second coordinate module ... ??seems pretty likely, i guess ... ???...

??even so, i'm still confused... still seems like... ???progression from dimensional theory to ag theory can't make up its mind (or my mind... or something...) as to whether it preserves geometric sums ... ???or something ??? ...

??maybe it _is_ my mind rather than its own? ... because it's making me think that i should have the same confusion in lots of other contexts as well ... ??or something ...

hmmm, lots of confusion here ... ???"progression from dimensional theory to ag theory" ... as left adjoint part of "doctrine interpretation" ... ??? but could also think of "moduli stack of theory" as stand-in for theory, in which case ... ??well, something about (2,1)-category of such moduli stacks as opposite of corresponding (2,1)-category of theories, so ... ???from this viewpoint what was left adjoint seems like right adjoint ???....

something about ... ???in general, 2 distinct ways of getting right adjoint g from left adjoint f... namely, by taking g = right adjoint of f (??in this context something about "underlying poorer environment" right adjoint to "free richer theory" ...), but then also by taking g = "f^op" ... ???or something ???? ...as above ... ?? ...

(??what _about_ how these relate in case of adjunction coming from morphism of locally presentable categories?? .... and so forth ... ... ????.... hmmm, not particularly "the same" ??? ... ??or something???)

(??something about idea "propositional doctrine" ... ??supposed to mean a doctrine whopse theories are "propositional", sort of ?? ... ??with syntactic and semantic (2,1)-categories being actually just 1-categories.... ??or something ???....)

??anyway... ??so are we saying something like that... the right adjoint (2,1)-functor taking the moduli stack of a dimensional theory to the moduli stack of the resulting ag theory has the extra property not ordinarily expected of a right adjoint that it preserves sums ?? ...???or something ???

(??as opposed to something about whether the right adjoint (2,1)-functor taking an ag environment to its poorer underlying dimensional environment in turn has a right adjont??? ....???also what _about_ something about ... "the (...) decategorified analog that doesn't seem to work" ... ???something about ... nevertheless, there's an example nearby (or something ... ???...) of a functor with both adjoints, namely ... the inclusion functor from _ab gp_ to _comm monoid_
... ???or something??? ....)

??then what _about_ whether it preserves more general (weak...) colimits, and whether it in turn has a right adjoint ??? .... and so forth ... ????....


???then also similar questions about "de-toricization" ?? ...

??any analogy between "basepoint" of toric variety and "degenerate models" of dimensional theories ??? ...???and so forth ???? ....

??what about something about toric geometry and stern-brocot tree??? ... ??...

weird ... :

??aspects of... ??geometric interpretation of geometric weak colimits of toric stacks ?? ... os ... asf os... ????... ???something about for example trying to g;ue together two lines intersecting at a point, but instead of such singularity (os...) seems to give something sort-of like "direct sum" ... ???asf ?? ???....

???sa ... ???weak colimits of filteredly cocomplete categories??? ... and so forth ... ??...


counterexample ... :

???so... in toric/non-toric case ... ???sa ... trying to contrive example of pre-stack (os...) whose quasicoherent sheaves (defined via "globalization" os... asf os...) aren't a topos/abelian category ... os... by ... sa... ???trying to get non-flat homs involved, os??? .... asf os... ????...

???wa sa ... ????whether flat comm-ring homs are closed under composition ?? ... os... asf os...



prometheus ... :

modern frankenstein ... ???something about filtered-colimit preserving modules of a ringoid, os, asf os????... ???but what about sa where tensor product is coming from ???? .... asf .... ????....



singularity ... :


??so w_a_ singularities of toric varieties??? .... asf .... ???....

??so what about ... given functor f:x->y, considering functor _presheaf(x)_ -> _presheaf(y)_ given by ... "taking formal colimit of x-objects to formal colimit of their y-object f-values" ?? ... or something ... seems obvious ... ???...

??relationship to ... ???forgetful (2,1)-functor taking theory of "combined doctrine" to that of uncombined ... ???and so forth ... ???...

??so what about construeing "combined doctrine" here as simply weak pullback of (2,1)-categories of syntactic theories ?? ... or something ... ??...

???so what about "torically local toric localization of toric variety" ??? ... and so forth ...

??what about "point of toric variety over abelian group (or something...)" as maybe analogous to "point of variety over field" ?? ... and so forth ... ???

??so consider geometric morphism induced by functor including localization morphisms among more general morphisms between fp commutative rings ??? ... and so forth ...

???more generally, what about image factorization of geometric morphism induced by functor??? ... and so forth ...

???hmm, isn't it something like ... ???full-and-faithful corresponds to "injective" here ??? ... or something ... ???....

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

??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?? ...