?? "fan spectrum" ( = (P,+) and/or (N,+) spectrum ...) vs real spectrum of toric variety ... ??? inclusion (?? ...) of former in latter ... ?? some confusion concerning ... ?? "poincare duality" schematic (?? ...) geometric interpretation of fan .... difference of toric zariski opens .... ?? may be single point in real spectrum but more in fan spectrum ... ??? because there's fewer points being _subtracted off_ in fan spectrum ???? ..... ????? ......

?? "zariski topology" here ?? .... ??? .... ??? "control" ??? .... ??? .... ...... ???? .....

??? detailed explicit look at how fan spectrum fits in real spectrum ... ???? .... ?? maybe especially in subdivision / blow-up case ... ??? .....

?? (P,+) as 1/4 of (R,*) ????? ..... ????? ...... ?? sort of ??? .... ?? but .... ?? [P^1]_(P,+) as about 1/2 of RP^1 ... ??? ..... ]0,1[, 1, ]1,infinity[ ... ????


?? cocone category of toric variety as (??maybe op- ??? ...????)lax colimit ... ???? how this relates to .... lax limit status of category of action-sheafs = category of set-valued functors on cocone category ... ??? .... ???? "borderline eckmann-hilton" .... ???? "pro-eckmann-hilton" ???? ....... day convolution .... ???? ......

?? "generalized fan" ("formal colimit" / "scheme for glueing" ...) vs "fan as spectrum" ??? ....

notes for discussion with alex today

1 ?? follow up on ... equvalence between quasicoherence and filtered cocontinuity .... ??? ....

2 ?? fan as spectrum wrt additive monoid of positives ... ?? possible subtlety ... todd ...

fan as spectrum over additive monoid of positives ?? .... ???? ?? todd's subtlety ?? ... ???? .....

??? conical wrapping up of toric variety, after all ???? .... ???? .... accidental topos ... ??? ...

?? spectrum over multiplicative monoid of unitaries ??? ....

"galois" action of gl(1,r) on add(r)-spectrum of toric variety ...... ???

torus stackiness .... ?????

kaleidoscopic toric variety ..... ???? weylotope ... ??? generalize to arbitrary toric variety ???

?? pos vs neg, vs >=1 vs <=1 .... ???? ..... ?? some confusion ???....

?? toric schubert variety .... ???? .... ??? some confusion ??? .... bruhat cell ... affine space ... zariski closure .... non-singular toric variety ... ??? .....

?? relationship between kaleidoscopic toric variety and ... ??? something that happened when i was trying to get the g2 rolling ball animation to work .... ??? "one roll away from any of the 6 special configurations" ... ????? .... ??????? ...... ???? ....

?? "super-tile" generalized chamber .... ??? corresponding co-chamber ... ??? ... as intersection of main co-chambers ... ??? ....

?? full kaleidoscopic toric variety as non-singular, but supertile-blowdowns as sometimes singular ... b2 (also g2 ...) cases as probably pretty instructive .... ??? ....

?? _does_ something about this sound vaguely familiar ?? ... ?? or am i now hallucinating a false memory about ... some aspect of schubert singularities as showing up more prominently in case of partial flag variety, as vs total ... ???? ....

?? orbit stack of kaleidoscopic toric variety under weyl group action ?? .... ?? ?? TAG-ness ??? ..... ???? .....

?? are we claiming that weyl cochamber spectrums are essentially just bruhat cells ????? ..... ????? .... ??? do dimensionality calculations work out here ?? .... ??? ....

??maybe only special bruhat cells are showing up here ??? ... ?? "allowed to use each generator only once" ???? .... ???? .... ??? have we bumped into these before ??? ...... ?? sounds plausible that we may have, but not completely sure at the moment .... ???? .....

?? would be nice to unify this with penrose diagram picture in b2 (and more general b series?? ...) case .... ???? .....

??? hmmm, blow-ups here as maybe meshing very nicely with "normal cone" intuitions .... ????? ......

?? some cone / cocone confusion (?? maybe not too bad ??? ...) here connected with toric singularities and "chess vs checkers" .... ????....

??? is "invariant higher-order distribution" idea meshing nicely with blowdown aspect of "weyl co-multi-chamber toric variety" ??? insofar as both relate to schubert singularities ??? ..... ??? ?? maybe _very_ nicely ??? ..... ?? some recent remark of mine about weyl cochambers and ... well, that higher-order distribution stuff, even if i didn't mention it by that name ... ????.....

?? which sorts of "bruhat cell" really are just affine spaces ??? .... ??? some confusion here ... ?? .... ??? try to straighten out .... ???..... ???? .....

?? homotopy type of "P" aspect of algebraic variety ??? ... ????....

?? kaleidoscopic toric variety ...

?? each weyl chamber corresponds to flag in apartment .... ?? each wall between weyl chambers to projective line connecting two such ..... ?????? ....... ???? .....

?? return of frankenstein ..... ??? "tame non-linear fan" ... ???? .... ?????? .....

?? very vaguely reminds me of .... ??? bit about distributive lattice and barycentric subdivision and ..... ??????? .......

?? cocone category as 2-colimit (?? ...) of "toric structure presheaf" ... ??? .....

?? non-toric analog here as maybe complicated by having to work with structure sheaf vs structure presheaf ??? ....

?? "prime ideal" .... ?? "basic open" .... ??? ....

?? affine toric variety ... ???assume zariski tangent space at "origin" is "expected" dimension ... ?? is it true and / or easy to prove that this implies that the whole commutative monoid is completely vanilla ?? ....

?? ideal generated by non-identity elements ... ?? square of that ideal ... complement of the square in the original .... ?? "normal"-ness ...

?? "i know where we are, we're right over there ..." ... ??? .... ?? piglet level ... ??? ....

notes for discussion with alex tomorrow

1 ?? flat f-fan ... ??? ...

2 ?? theory of quasicoherent action ... ???.... and practice ...

3 ?? filtered object as Z-graded N+-invariant object ... ??? ....

4 ?? "local comm monoid" .... ??? ......

5 ?? b2 kaleidoscopic toric variety ...

6 ?? classification of non-singular affine toric varieties ... ?? completely vanilla ??

?? face of cocone as "co-ideal" ... ??? ..... ??? ideal power filtration ...

??? multiplication of ideals .... ???? ....

??? quantale .... ????? .....

?? inverting all elements on cocone face f as in some sense "localizing at f" ????

?? "(Z,N+)-filtration on object x as Z-graded N+-invariant subobject of x X Z" ??? .... ???? ....

?? "restoring unique factorization" to "square-base pyramid" /ad=bc ... ???? not fit "chemistry analogy" story so far as i can see yet ... ??? what were my ideas about this (...) ?? ... ???? ?? or did i just somehow overlook it (...) ?? .... ???? ... ?? seems would be pretty silly to overlook it ... ??? 2d vs 3d ... ????? .....

?? 2*3 = (1+5i)*(1-5i) ????? ..... ??? ...

?? .... ?? dependence on nothing more than x tensor Q+ .... ?????? .....

?? 2d intuition ... ?? intersection of all "snug free envelopes" .... ???? .... ??? messed up in 3d .... ??????? ...... ?????? .....

?? so what _does_ happen in Q(sqrt(-5)) ???? ..... ????? ..... "x tensor Q+" .... ???? .....

??? dedekind ....ideals modulo principal ideals ..... ????? .....

??? maybe it really does always (?? .... dedekind ... ??? ...) happen that the multiplicative monoid is (roughly ... essentially ... ??? ...) smug in an orthant .... ???? ...... ??? .... ??? comm monoid of principal ideals as snug in comm monoid of ideals ..... ????? ...... ??????? ..... ?????? .... or not ????? ......

?? case where "universal orthant envelope" clearly doesn't exist ???? ...... ???? ....

?? maybe ... ??? need certain degree of "non-singularness" (??? ...) before game of "unwrapping with ramification only at ..." makes sense ?? .... ????....

confusion between localization corresponding to (??basic ???) open set and "local ring" corresponding to its complement ... ???? .... ???? non-toric and toric cases .... ?????? ......

i said :

?? dimensionality aspect of delocalization poset ??? .....

?? for example, delocalizations of rational functions in 2 variables .... ???? .... ??? analog of "2d (??limiting case??) fan containing all rays" ?? .... ??? ....

[end quote]

hmmm ... ?? local rings of irreducible subvarieties of various dimensions .... ????? .... ??? "multi-local" rings of reducible subvarieties ??? .... .... ????? ....

??? toric analog ?????? ..... ????? .... ??? "local commutative monoid" .... ???"prime ideal" in commutative monoid ?? .... (?? visual intuition about cocone face associated to cone face ... ????) .... ?? seems like this (...) is probably well-known; in particular that stuff i bumped into once that i only vaguely remember .... ?????..... ??? whole bunch of things to try with this .... ??? for example toric analogs of topos stuff occurring in ordinary algebraic geometry ?? ... tierney "spectrum" stuff ... ??? .... ??? ....

???any possibility of .... polytope : toric de-/localization :: "q-deformed polytope" : some kind of non-toric de-/localization ??? .... ???? .....

?? toric line bundles (???? ....) over kaleidoscopic toric variety vs line bundles over flag variety ??? ..... ??? ..... ???? .....

??? relationship between "local" and "nilpotent" here ????? .... ????? ......

??? aspects of "filtered" which live in toric context ... ??? vs other aspects ?? .... ?? "associated graded ..." ... ???

?? dimensionality aspect of delocalization poset ??? .....

?? for example, delocalizations of rational functions in 2 variables .... ???? .... ??? analog of "2d (??limiting case??) fan containing all rays" ?? .... ??? ....

??? "glueing together two copies of affine line along generic point of both" .... ????? .......

?? confusion about spaces as objects of toposes, vs ... ?? as objects of opposite category of model category ???? ..... ???? ..... hmmm..... ???? "formal colimits of affine schemes with only the localizations as the glueing arrows" ... ??? ... vs .... ??? tierney (and so forth...) 's stuff about "spectrum ..." ... ?? involving "local morphisms" (=?= conservative morphisms = anti-localizations) between local rings ..... ????? .... ??? idea of localizing a local ring as peculiar ???? ...... ??? "local localization" ... ???

??? maybe not so much peculiar as impossible ??? .... ??? any paradox here ???? ..... ?? localizing an element in the maximal ideal as ... ??? hmm, so _does_ this always cause complete collapse ??? .... ??? _is_ this pretty much same issue we got really confused about once ???? ..... ?? inverting a nilpotent as drastic, but what about non-nilpotent elements of the maximal ideal ????? ......

?? formal colimits of affine schemes with only the anti-localizations as the gluring arrows ????? ..... ???? localness of ring in this context ??? .... ??? vs in some other context ... ????? ....... ?????? .......

?? analogy between ... ??? localization between commutative rings, and (localization,universal) pair between (comm ring, module) pairs ... ??? ..... ?? ... object vs morphism ... ????

?? co- / simplicial topos ... ???? ..... ??? ..........

theory of truth value -> theory of decidable truth value ... ???? classically property, geometrically structure .... ???? ....

?? unit object in accidental topos ... ??? ....

?? co-monad on accidental topos ... ???? .....

?? quasicoherent module in big zariski ringed topos .... ???? .....

?? "moduli stack" (?? ...) of toric dimensional theory ... ??? as generalization of ... kernel of ab gp hom ...... ????? ..... ??? toric rees construction ?? .... ?? singularness of point in (...) toric rees construction ... ??? geometric vs algebraic (??? ...) picture here .... ???? .....

?? "flat" N-fan ?? .... ?? nice pictorial interpretation ?? ??also interpetation in terms of accidental topos ?? .... ??? ....

?? fibers .... ????? .....

?? flat =?= "completely avoiding subdivision" ????? ...... ???? ......

??? "flat" vs "open" ???? .... ?????? .....

?? relationship between accidental topos and toric zariski locale here .... ???? ....

?? in special case of torus, for example ??? ...

??? actually .... ??? fair amount of confusion here .... ????? ......

?? don't forget case of torus morphism where (in lattice picture ...) cokernel has torsion ... ??? ....