?? so ... given Z-filtered (nice ... ??? ...) commutative monoid x, trying to picture "rees construction" applied to it ... ??? dot (x1,j) in x X Z as answer to "is x1 born by the jth stage ... ????? ..... ??possible pos/neg confusion here ??? ... ideal power filtration .... ????? .... ?? hmm, this description makes "rees construction" sound somewhat tautological ... ??? .....

?? "convex" toric filtration ??? ..... ???? ..... ????? ..... ?? somehow stuff here reminds me of some of arnold's ideas ... ????? ..... ??? singularity ... "ray" .... "caustic" .... ??? duality ... ??? .... toric ... ???....

?? any possibility of generalizing convexness of filtration to non-toric case ??? .... ?? geometric interpretation ??? .... duality ... ??? ....

??? "toric hilbert basis theorem" .... ????? .....

??? toric case of "hilbert scheme" ..... ???? ......

?? feeling of entitlement to get nice toric variety / blow-up (... ??? ...) from poisson operad situation .... ??? so can it be pushed through somehow, or can we understand what goes wrong ??? .... ??? bit about "toric spectrum vs ordinary spectrum of toric variety" .... ???? .... ??? maybe distinction tends to vanish in "1d" case ????? ...... 1-by-1 matrix multiplication path integral as sharp (=?= toric ??? ....) .... ??????? ....... relationship to "toric" flavor of abelian class field theory ????? ..... ???? ..... transition from 1 to 2 (??...) as threshold for "non-commutativeness" and also "non-sharpness of path integral" =?= "non-toricness ??? .... ???? ..... ????? .....

(added later: ?? ...hmmm .... but there was that bit about ... ??? invertible module of comm ring r, not corresponding to any invertible action of gl(1,r) ... ????? ..... ??? ....)

??? "dynkin (vs coxeter ?? ...) kaleidoscope" (??? =?= "root system" ?? ...) as not much more than "sufficiently homogeneous (??? wrt discrete symmetry group ??? ....) toric variety" ????? ...... ????? "affine" (?? and beyond ?? ...) case ??????? ..... ???? still get perfectly good toric variety from such ????? .....

??? most obvious toric variety coming form kaleidoscope as conspicuously non-singular ??? ..... ???? vs ... ??? vague ideas about trying to relate kaleidoscopic toric variety to schubert singularities ... ???? ......

???when working on schubert singularities did we in fact venture away from the parabolic (or whatever .... ??? ....) lie subalgebra towards the rest of the weyl co-chamber ??????? ..... ??????? .......

????? so consider "N-fan" .... ??????? ...... ???? fan with extra structure ??? ..... lattice under Z, with all cones containing N+ ??? .... ?????? ...... ?? [Z = line-of-sight] perspective ...... ????? ...... wait a minute, bit of cone / cocone confusion there ... all cocones containing N+ ; all cones fitting inside a half-space .... ????? .....

?? by the way is there room for those more general "cone"s (more general than just official cones and cocones ... ??? ....) to show up here ??? ...

??? more generally, fan "respecting" another fan ??? .... respect as more general than subdivide here ??? ?? maybe still not quite general enough though ??? .... f fan .... ?? "f-fan" .... ???living on lattice mapping to lattice of f .... ????....

??? "toric etale fundamental group" ??? .....

?? so is there a classifying topos for "an object for which all restriction maps are invertible" ?? ..... ????? .......

?? "locally has enough global sections" ... ????dependence on action structure ???? ...

?? nice simple example of non-quasicoherent action .... ???? ..... ?? ?? non-universal N-equivariant map from N-set to Z-set .... ????? ..... ?? for example from N to terminal Z-set, or from initial N-set to Z .... ????? .....

?? is it clear whether concept of "quasi-coherent" is at least "site-independent" ??? .... ??? if so then does this maybe automatically imply that it can be expressed in some sort of "topos language" ... ?? .... ????? ...... ??? without necessarily being "geometric" ... ???? .... ???? .....

??? does "quasicoherent" maybe make sense only over _local_ comm ring ?? ... ??? ....

??? consider for example quasicoherent modules in ringed topos given by ... ??? space of real numbers, with ring object "constant" Z ... ???? .... ???? .....

??? identity homomorphism as localization ... ???? .....

?? hmmm .... ??? ....

?? sheaf of lattices vs of semilattices ... ????? ......

?? hmmm.... "comm ring equipped with complement to subobject given by invertibles" ... ???? conservative homomorphisms ?? .... ????? local homomorphisms as special case ???? ..... ??? anti-conservative as maybe not so easy .... ??????? .....

??? sets and surjections ?????? ..... ??? filtered colimits .... ????? ....

??? objects vs morphisms here ???? ...... ?????

?? arbitrary comm ring as "believing itself local" ?? .... ????? .... ???? ... ....... ??? local homomorphisms ....... ????? ......

??? vs .... formal colimits built out of localizations, but with more general morphisms between them .... ????? .......

??? "local localization" ....???? ..... ???? "(??which??) 2 of 3 ..." .... ??? ....

?? having (?? boolean ... ???) complement to invertibles as seeming rather drastic ??? .... ??? anything a bit less drastic ??? ... ??? ....

??? so consider ... ?? affine plane with origin blown up .... ???? ....

?? affine plane ... k[x,y]

k[x,y:0; a,b:1]/ay=bx .... ?????....

(?? presentation as "computer program"-like ??? ...)

?? exact sequence .... ??

-> /ay=bx -> Z ... ???

?? morally qualify as "short exact" ... ??????? ..... ???? .....

?? fiber over j ... ????....

??? wait a minute .... for negative j the fiber is empty, right ???? .....

... hmmmm .....

?? need to draw a picture ..... ?????? ........



???? .....

??? short exact sequence with base N instead of Z ??? .....

??? so ... ??? consider .... X ... = .... ???? with grading given by ... ??? ??? _is_ this exactly "rees" construction ??? .... ????.... filtration .... to grading ... ????.... ??? .....

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

?? b2 kaleidoscope toric variety as projective line squared with the 4 special (...) points blown up ??? ....

(?? "torus with cross-caps at 4 special points" ....? ????

?? relate this to the flag geometry .... ??? ....

?? kaleidoscope as fan .... ???? ?? cones and cocones here vs the more general closed subsets wrt the galois connection ... ??? parabolic subalgebras vs more general galois-schubert subalgebras .... ?????? .....

maximal torus .... lie algebra vs lie group .... ??? lie group as torus but lie algebra as affine toric variety ..... ???? how to glue copies together .... ???? .....

??? exponential as transcendental ... vs algebraic .... ??? confusion ??? ....

??? "double life of young diagrams" .... ????? .......

?? inclusion of quasicoherent actions among potentially quasicoherent ... ?? as geometric morphism ... ??? ...

?? given "TAG topos", consider unit object for the TAG structure .... ???? relationship to "torus object" (???) we used to think about ??? ..... ??? ....

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

?? ... hmmm .... ???? .... ??? "core" ??? ..... ????? ..... ??? "n-affine ..." ... ???? ..... ????? .....

??? doctrine morphism ... ???? TAG doctrine to ringed topos doctrine .... ???? forcing objects in syntactic category of TAG theory to be quasicoherent .... ???? .....

??? ....

???? "abelian category equipped with commutative ring object" .... ?????? not evidently parsing yet .... ?????? ...... ?????? .....

[???actually even original toric case not parsing now ... ????? .... ??? confusion ???? ..... TAG unit in TAG topos as not topos commutative monoid .... ???? .... ????? ..... ???? .....

?? comm monoided topos as TAG topos with property .... ?????? .....

?? vs (?? ...) ... ??? maybe somehow systematically deriving TAG / topos compatibility structure (?? ...) from example of comm monoided topos ?? ....

?? blowing up point p of space x vs taking projectivized tangent bundle of x and then ..... ???? ..... ??? ....

?? collapsing each fiber except that over p ??? .... ???? .....

?? geometric morphisms between accidental topos and toric zariski locale .... ??? ....

?? retraction as not preserving filtered colimits (?? so in particular not left adjoint ... ???? ....), but section does ????? .... ????

??? whether invertible module is automatically quasicoherent .... ????? ......

?? comm monoided topos vs "TAG topos" .... ????? ......... relationships ..... ??????? ......

?? blow-up (??? ...) relationship between a2 and g2 ?? ..... ?????? ......

?? where a maximal parabolic subalgebra (for example ...) lives ... ??? "co-weight space" ??? .... cocone therein ... ???? ..... ??? dual to cone where corresponding extremal weight lies .... ???? ..... ?? more generally, higher dimensional wall of weyl chamber as cone dual to non-maximal parabolic subalgebra .... kaleidoscope-as-fan toric variety ... ??? .... conceptual / geometric interpretation ... ??? ....

notes from discussion with alex today

... we wondered whether, for example, rees algebra (??? ...) associated with ideal power filtration of certain (?? ...) ideal in cocone (?? corresponding to some infinitesimal but perhaps "irregular" neighborhood of the "basepoint" of the corresponding affine toric variety ... ??? ...) might relate to subdivision of corresponding cone in certain corresponding way ..... ????? ....... ????? .....

?? also tried to relate something like "standard blow-up of basepoint" to something like "barycentric subdivision" .... ??? at one point i thought that syzygy might get involved, but then thought that maybe it was just a relation ..... ???? ......

?? point in cone giving grading of cocone .... ???? .......

?? classifying monoided topos for quasicoherent action .... ???? ... ???? ?? explicit description ... ??? ... non-toric version as well .... ???? ....

x comm monoid ...

y comm x-monoid ... ??? or Z-graded comm x-monoid, for example ... ??? ...

?? for example x = ... y = /ad=bc .... ??? ....

?? x-fan ... ???? ....

??? 3-place chain complex here ??? ....

?? comm monoid or Z-graded comm monoid in accidental topos... wrt TAG structure ... ???? ...... ???? ....

?? some of this stuff in paper ??? ....

notes for discussion with alex tomorrow

?? galois-schubert correspondence and toric geometry ....

?? stratification ... ??? "toric perverse sheaf" .... ????? .....

?? spirograph-flower picture of co-fan ... ???....

!!! work on paper .... ???? .......!!!! .....

?? a-series : more general dynkin (?? ...) diagrams :: certain grassmanian variety is toric : partial flag variety contains interesting toric variety .... ?????? .....

??stratification of toric variety ??? .... ??? understanding woolf's stuff ??? .....

?? "apartment structure" (?? ...) on partial flag variety / schubert variety ... ??? .... ... toric structure on toric variety ... ??? ..... "stratification" ...... ?????? ......

?? "perverse sheaf" ....

???? "toric perverse sheaf" ???????? ..... ????? .....

?? galois-schubert correspondence galois connection .... ???? toric interpretation ... ???? ..... roots .... figures .... ??? but ... ?? how did that go, interpreting the figures as weights or something .... ??? "highest weight" ... ????? .... ???? ??? annoying sign choice ... ????? ....

?? kaleidoscope as _co-fan_ ... ??? ....

?? hmmm ... ?? vague memory of ... ??? having some reasonable visual intuition about co-fan while talking to baez on his last visit here ... ?? in car and hotel room ... ?? at least, visual intuition about co-face .... ???? .... ?? actually more tactile than visual ?? .... feeling edge (?? ...) of 3d object .... ????? .....

??? given face of cocone, corresponding coface of it

??? vague intuition about "infinitesimal neighborhood of face" .... ??? or "localization" ????????? ..... ?? "the only constraints that you keep are the ones that "touch" the given face" .... ????.... ?? "that touch face f" = "for which f lies completely on border hyperplane" here ... ??? ...

?? hmmm, sort of "flower" picture of 2d (maybe higher too ???) co-fan ??? ... ?? "spirograph" ??? ... ??? dual fan ???

?? hmm, maybe "flower" (with nicely separated (??...) petals (?? = cocones ?? ...), no petal over half a circle, circle of petals closes up after just one revolution ... ????...) is more apt than "spirograph" here .... ??? hmm, or maybe "spirograph flower" is better ... hmm, actually i'm still a bit confused ... ??? ....

?? "daisy" ??? ... ??? maybe other fancy names that people who think about certain sorts of curves (??) use .... ??? ....

??spirograph flower given by "modulus as function of argument" ???.... ??? zeros of such function ... ???? ..... ?? "fourier ..." ... ???? ....

??? trying to understand toric quasicoherent sheaf in terms of spirograph flower ... ???? ....

?? so ... when you blow up a point (??) on a toric variety / subdivide co-dim 0 (??) cone in fan ..... ???? ..... the new subvariety as ... ??? maybe not precisely a toric variety .... ???? .....???? .... ???? .... ?? bit about various sorts of ideals of interest .... ???? ..... generated by toric functions vs by equations between such ... ??? .....

?? might the subarieties which don't especially seem like they "should" (?? corresponding to ideals generated by single toric functions ... ??) be nevertheless "accidentally" toric after all ?? .... ??? or "generalized toric" ... ??? ....

??? "generalized fan" ... ??? as maybe somewhat clear from discussion about "non-linear fan" ... ???? ....

??? various ways of subdividing cone, vs ... ?? ess just one way to blow up point .. ??? .... ????? .....

?? blowing up a higher dim (??? good ???) subvariety on a toric variety ... ???? .... ?? subdividing higher-dim cocone in cofan ... ????.... ??? ....

subdivision, stackiness, poisson blow-up ... ?????.....

??? "subdivision" ... "blow-up" .... "explosion" ... ???? ....

??? comparison between two ideas about "toric morphism" ... 1 : comm monoid hom and / or TAG theory morphism ... ??? .... vs 2 : "torue-equivariant map riding torus morphism" ... ??? trying to compare in detail ... ?? nice simple examples ?? ....

???functor from _comm monoid_ to _dimensional algebra_, assigning to commutative monoid x the dimensional algebra [monoid-alg(x) graded by fraction-group(x)] ... ????

?? extending this in nice way to functor from _TAG theory_ to _AG theory_ ??? .... ???? ..... ??? extra objects, corresponding to .... ??? .... fraction-group of "center" ??? .... ???? ..... ???? .....

?? comparison to leaving out the grading ... ???

?? toric (?? ...) aspect of "blowing up "comm alg" sub-stack (?? ...) of "assoc alg" stack" ... ???? .... ??????????????? ..... poisson alg .... ???????...... ??? level slips and strange loops ... ???? .....

????level slips and strange loops ??????????? .......

?? dim 1 face (?? "ray" ??) of cone dual to cod 1 face of cocone ... ??? visualization ... ???... ?? "nail down the points on the cod 1 face; in fact mod out by them, and map the residual ray to the standard ray ..." ???

?? dim 2 face of cone dual to cod 2 face of cocone ... ??? visualization ... ??
?? "nail down the points on the cod 2 face; in fact mod out by them, and map the residual dim 2 "silhouette" (?? ...) wedge to the standard ray ..." ??? ....

?? cocone co-face _not_ as just "omitting corresponding cocone face" ... ?? do we really still believe that cocone co-face has corresponding cocone face ??? ... ?? well, maybe yes ... ??? ..... ?? some confusion ... ???....

?? can't just take an extreme ray of a cone and then take the convex hull of the other extreme rays ..." ... ??? at least, not without ... ??? expecting silly things to happen ?? ... ???? .....

?? corresponding to cocone face is cocone co-face ??? ...

?? dual to cone face also corresponding to cocone face ???

?? tensoring cocone with truth-value comm monoid t ... ?? then homming it into t also, and comparing that to combinatorial face structure of cone ... ???

?? ["subdivision morphism" between toric varieties] approached this way ... ???...

??? "dual to cocone face is cone face where the inequality it represents is the pullback of the fundamental inequality" ??? .....

"fundamental inequality" 0 <= p ???? ....

?? ... "fenchel ... " ??? ....

?? discussion with todd ... me at first thinking that cone / cocone duality was just about "N as schizophrenic between comm monoid and comm monoid" (???? ....) ... ?? but then todd pointing out what (?? ...) was wrong with that ... so then i suggested "0 <= f(x) galois connection between x and f" idea .... ?? possibility of distinction between these ideas as relating to some sort of "cohomology for comm monoids" ?? ... ????? ......

?? "folded schizophrenia" ... ??? torn between two identical copies of self ... ??? ....

??? but now ... "dual to cocone face is cone face where the inequality it represents is the pullback of the fundamental inequality" vs ... ?? contrasting idea ?? ... ... truth-values ... ????? ..... ??? or maybe not contrasting ?? ?? but ... i want to say / understand here about ... ?? how truth-values are fitting in here ????? .... ?? "zero on cocone face and "tribal infinity" off it" ?? ...

?? so what _was_ the anomaly (?? ...) that todd pointed out ??? ... ?? corresponding cone as not always precisely hom[cocone,N] ?? .... ?? example ?? ...

??? "boundary" fan of cone ??? .... ???? ..... ?? "child's drawing" and "through the looking-glass" ??? ..... "polytope" vs "combinatorial manifold" ... ?? ... "morse theory" ... ??? "removing stacky point to obtain projective space" ... ???? ...

?? through the looking-glass ... adjoining cones in fan, glued together along mutual face ... ???? ....

?? anti-fundamental inequality .... ????? ..... subdivision morphism ... ??? ....

?? case where adjoining cones have same "affine hull" ??? .... vs other case ... ??? .....

?? adjoining highest-dimension cones ... ??? ....

?? "toric morphism" ... ??? riding torus morphism ...?? ... equivariance ... ??? stackiness .... ????? ..... "sleeping with one eye open" ... ?????......