?? so .... given pre-stack of categories, consider ... ?? "quasicoherent presheaves of presheaves" over (?? ...) it ... ??? and whether (?? or in what generality ...) equivalent to filteredly cocontinuous presheaves on lax colimit ... ???? ....

?? "constant" case as maybe good case to experiment with ??? ...

?? then ... ?? "generalized day convolution" .... adapted to filteredly cocontinuous case .... ???? .....

?? toric dimensional theory .... ??? orbit stack of toric variety wrt _part_ of it's torus .... ??? ....

?? kaleidoscope toric variety ... ??? as subvariety of flag variety .... ?? relationship to .... ??? grading .... by weight lattice ... ???double such grading ??? .... homogeneous coordinate algebra .... ???? .... action .... ???? base action vs fiber action ??? ..... ????? ..... ???? ..... ?? highest weight vs weight .... ????? ...... grading vs homomorphism .... ???? ....

?? reading off codimension of tangent cone in zariski tangent space "from root system" (?? by "galois-schubert" thinking ??? .... ??? .... ??? "nilradical" .... how nilpotent lie algebras relate to non-integrable distributions .... ????? ...... to what extent are we mixing up different stuff here ?? .... ???? first-order vs higher-order .... singularity .... ??? certain jargon buzword i'm still groping for here ... person's name .... ??? "levy" ??? .... ?? also "prolongation" ??? ... ??? ....) vs from kaleidoscope fan (using singular affine toric variety thinking ....) ??? .....

?? bruhat cell .... schubert variety .... "attaching map of cw-complex" ....

?? the 2 blow-downs of the g2 kaleidoscopic toric variety ... ??? non-singular one as projective plane ??? .... ??? ... other one ... ??? ... ??? "galois" / "wrap" .... ???? ...

?? sorry, did i mean to say "flag variety of projective plane" rather than "projective plane" here ??? ....

?? sorry, still not quite right .... ???? a2 kaleidoscopic toric variety, rather ?? .... ????? ....

??? generic flag pairs as dense ..... ???? ..... ...... ??? torus .... ???? ....

?? hartog kaleidoscope toric variety ... line bundles over it ... fundamental group of corresponding (truly ?) simple lie group ... ?? langlands duality ?? ... ???? ....
"galois ..." ... "wrap ..." ... ... ??? "galois-schubert correspondence" ... ??? .... cone/cocone duality ... ??? ....

"The Weierstrass zeta function was called an integral of the second kind in elliptic function theory; it is a logarithmic derivative of a theta function, and therefore has simple poles, with integer residues."

?? "logarithmic derivative" and "mellin transform" ???? ..... not clear from google hits ...


"The decomposition of a (meromorphic) elliptic function into pieces of 'three kinds' parallels the representation as (i) a constant, plus (ii) a linear combination of translates of the Weierstrass zeta function, plus (iii) a function with arbitrary poles but no residues at them.

The same type of decomposition exists in general, mutatis mutandis, though the terminology is not completely consistent. In the algebraic group (generalized Jacobian) theory the three kinds are abelian varieties, algebraic tori, and affine spaces, and the decomposition is in terms of a composition series."

hmmmm... ??? ....


"The dimension of the space of differentials of the first kind, by means of this identification, is the Hodge number h^[0,1]."

??? .....

?? possibility "partial fraction basis" stuff we've been bumping into might tie in with vaguely misremembered stuff about " ... differential of nth kind ..." and "abel-jacobi" (??? ....), via ... ??? annoying traditional integral calculus stuff ??? ....

?? langlands duality confusion showing up with kaleidoscope toric variety as quasi-schubert variety .... ?? to what extent did we take proper care in checking whether cone/cocone duality confusion might "cancel out" this ... to some extent ?? ... ???? ....

?? maybe we did ... to some extent .... ???? ....

?? derivation on field of rational functions ... partial fraction basis .... ??? ...

?? 1-form ... ???? .....

?? kaehler differential ... ????....

????? "blowing up generic point" ...... ???????? ....

?? line bundles over "projective line with only 2 points un-removed" ???? ..... ?????? ..... ?????? .....

???? "clutching" ???? ......

?? dimensional algebra .... ?????.....

?? the moment you remove even just one point, you trivialize the tautological bundle .... ???? .....

functorially blowing up everything ... birational geometry .... working with aribitrary subvarieties instead of cod 1 ..... ????..... ???? non-interference between contrasting blow-ups ???? .... toric case ???? .....

?? extra structure on toric variety associated with nice quadratic form (?? and / or symplectic structure ?? ...) on torus lattice .... ???? .....

??? elliptic curve .... ???? .....

?? higher-dim analog / cousin of diagonal matrix .... ???? as showing up with nultiplication structure constants in partial fraction basis .... 3d structure constant matrix for bilinear op ... k_[a,b,c] as nonzero only when c=a or c=b ???? .... ????

??? a,b,c here as ranging over sectors indexed by points of riemann sphere ... ???...

?? these sectors as giving canonical sum decomposition of field of meromorphic functions ??? ....

??? relationship to .... ?? "joint fractional ideal power filtration for all ideals" ...

powers here as .... ?? staying within sectors ... ????? ....

filtration vs grading .... for unadorned vector spaces ... but then with bilinear multiplication ... various sorts of non-/compatibility ... ????...

???? number-theoretic analogs of this (?? ...) .... ????? .... ??? filtration vs grading .... ??? .... carrying .... ???? ..... decimal expansion .... for rational number ... repeating .... ??? new directions ???? ..... ????? .....

first cat th conf talk ....

proj geom=dim analysis, baez scattering example, lawvere on "theory" and beck on "doctrine" (somewhat resp ...), big program, toric special case, perhaps simplest non-presheaf grothendieck topos, ab gps therein as quasicoherent sheaves over projectiv line .... ?????.....

?? millions of other things to fit in there ... tannakian philosophy ... other nice simple moduli stack examples : line, line embedded in 3-space, lie alg decomposed into n-tuple of lines, cuboquadratic algebra ... ; doctrine as 2-topos, line bundle = line object = dimension, graded comm algs of homogeneous quantities ...

?? distinguish between "synth proj geom" and "analytic/algebraic proj geom" ??? ... though cross-doctrine relationships .... ????? ...

?? impractically long, of course .... ??? ...

?? second ??? ....

?? rep th and cat th conf talk ???? ....

??? kaleidoscope fan and rolling ball animation ... ?? _if_ this really fits together .... ????..... ?? kaleidoscopic toric variety as (sort of ...) schubert variety .... ????? ...... ?? conclude with propaganda about ... ??? doctrine philosophy (or something ...) tying quasi-schubert varieties to conference topic "rep th and cat th" .... ??????? .....

?? "invariant distribution" .... ?? higher-order ???? ..... b2 ... penrose diagram ... gimbal/axis .... ???? ....

?? long root subalgebra ... ??? ...

???? "cheshire" aspect of hartog .... ???? propaganda for "proj geom = dimensional analysis" .... ???? ..... ?? excised stacky point as leaving its smile behind ???? ..... ??? smile as non-trivial line bundle ???? ..... ??? hmmm, but ... "that's funny, i could never play it before" .... ???????? ...... ?? for hartog ??? .... ?? ok in excising stacky point case ??? .....????

?? "non-linear fan" coming from ordinary fan as highly "incomplete" ???? .....???? ....

?? subdividing vs removing cones .... ???? ......

(x,y) |-> (ax+by)/x ...... ????? ...... a + b*(y/x) .... ?????

??? ....

???? so ... consider for example toric projective line .... ??? "substitute field spectrums for toruses" .... ????? ..... ???? .... "using glueing scheme but with substitute parts" .... ???? ..... ?????? ..... "model ..." ..... ????? ....


N+ ??? rational functions with no pole at 0

Z ??? rational functions

N- ??? rational functions with no pole at infinity

??????


.....

?? get actual scheme out of this ... ??? ....

?? "projective line with all poits except 0 and infinity removed" ???? ...

??? open subschemes ???? .... ?? affine such .... "can't remove poits that aren't there" ....

quasicoherent sheaves here .... ?? ....

??? pressley and segal .... ?? "factorization theorem ..." .... ????? .....

?? .... birkhoff .... ???? ....

factorization vs sum decomposition ....

??? a function which blows up only on a reducible subvariety x can be expressed as a sum indexed by the components of x of functions each of which blows up only on the corresponding component ??? .... ???? .....

(??? maybe this does indicate resolution of multiple transitivity paradox .... ?? ... not constructing any new point from two given ones .... ???? .....

?? but ... x * 1/x = 1 ..... ????? ..... )

??? prime ideal ... divisor .... ????? ......

?? product of ideals ..... ??????.....

a/b + c/d ..... ????? .....

1/x + 1/y .... ?????.....

?? seem to have some sort of silly paradox here concerning structure constants for multiplication of rational functions in "partial fraction" basis .... ???? maybe basis doesn't actually exist the way i was imagining .... ??? ....

?? multiple transitivity .... ????? .....

???schanuel's "noncommutative analytic functions" ??? .... ???? ....

?? "localization" game vs "divisor" game ??? .... ??? both involving .... ??? some sort of algebraic structure built out of meromorphic functions satisfying some sort of constraint on poles .... ????? ..... ?? interactions ??? .... ???? ....

?? "hartog" (?? ...) phenomenon, for one ... ??? ....

?? so .... consider hartog toric variety ... say for example the "checkerboard" one ... and consider .... non-trivial line bundle over it ... toric dimensional theory coming from integer tensor powers ....

??? k[a,b,c:0; d,e:1; f:2]/ac-bb,af-dd,cf-ee ..... ????

?? no wait ... ??? need generators in negative grades too ??? ... ???? .....

?? the calabi-yau manifolds as embedded in toric varieties, maybe not especially toric themselves ??? ....

?? tensor product of abelian categories, and stable monoids wrt it .... ???? ....

??? every object being flat .... ?????? .....

?? various kinds of extra structure (or ...) on quasicoherent sheafves ovr x, associated with x being toric .... interrelationships ... ???....

?? "toric quasicoherent" ...??

?? "torically equivariant" ... ?? ...

(?? above two as somewhat orthogonal ??? ... ???? ....)

?? ?? "non-linear fan" idea here ??? .... ???? .... ???? ....

?? "quasicoherent artin-wraith glueing" and ... ?? "intersections of localizations" .... pointless locales .... ??? .... ??? stacky / schubert case ..... ????? ..... also toric ... ???? ....

??? unified vs one-at-a-time perspective on multivariable partial fractions ... ??? .... ?? cavalieri ... 1/(x + f(y)) .... ???.... puiseux ... 1/(x^2+y^2) ... ??? ...

??? structure constants for multiplication in partial fraction basis ... ???? ...

1/(x-a) + 1/(x-b) = (2x-a-b)/((x-a)(x-b)) .... ???? ....

?? cover from fan ... ??? elements as arbitrary cones, vs... maybe just special ?? ... "highest-dim" ??? ... ???? ....

colimit as representing cocone = diagram in slice category .... ????..... ?? lax case .... ??? showing up .... "quasicoherent = filteredly cocontinuous" .... ????

?? neutral (between abelian and cartesian ...) combined doctrine .... products, coproducts, and tensor products ... ???? ..... ?? morphisms between abelian and cartesian examples here ... ????....

?? proceeding with mirror symmetry program .... ?? drawing graded commutative monoids coming from line bundles over hartog toric varieties .... fano .... anticanonical ...

?? torus action stack of toric variety ... ??? compare to "schubert stack" ... "bi-flagged building" .... ??? ...... ?? "artin-wraith ..." .... ?????....

??? module over local ring of prime ideal, tw module over quotient ring ... ????.... ?? maybe didn't say this (??? ...) quite right yet .... ???? both localizations ????? ...??????

??? vector space equipped with operator x with x-k invertible for all nonzero k ... ????.... ??? algebraic completeness here ??? ..... ???? x^2+1 being invertible ??? .... ......... ??? extensions here ????? ...... ?? ??? "functorial projective cover" (??? ...) and " ... glueing ..." ???? .... ???? .....

?? laurent polynomials vs rational functions with no negative factors of x .... ????? .......

??? explicit linear basis for field of rational functions ??? .....

??? explicit linear basis for k[x,y,z]/xy-1,xz-z-1 .... ?????

1/(x-k) .... ?? case k = 0 ??? .... ????? case k = infinity ???? ..... ??? as morally x ?????? .....

?? multi-variable case .... ????.....

?? vector space over the rational functions as much more concretely understandable than i'd realized ??? ..... invertibility as mere property .... ??? ..... ?? pressley and segal ... ??? "meromorphic vector bundle" .... ???? .....

?? concept of toric divisor as dependent only on ray-skeleton of fan ??? ..... perhaps also concept of "canonical divisor" ??? ..... ?? giving conceptual interpretation to such in general case .... ???? "serre duality dualizing object" .... derived level .... ?????? .....

?? tautological projective embedding of projective line .... composed with "tensor square" map .... ???? .....

v 2d vsp ....

p(v) -> p(v*v/2!)

p^1 -> p^2 ????......

(x,y) |-> (xx,xy,yy) .... ????? ....

??? diagonal followed by segre embedding ???? .....

?? "non-linear fan" .... ??? relationship to "cover" ????

??? nerve of caonical cover of toric variety ... ????.... ??? ....

?? "constructible sheaf" ... ???wrt cover ... ???? .... ?? such as forming nice AG theory .... ???? .... ????? ....

?? affine case .... ???? canonical cover of affine (?? non-/toric) variety ... toric case ... ???? certain comm ring obtained from comm monoid ... ???? ...."non-linear fan from linear" .... ??? ....

??? "cover" and "grothendieck topology" .... ???? .... ??? bit about .... ??? approximation .... small radius .... ???? ..... ???? ....

?? "non-linear fan" .... ??? getting one from ordinary fan, and using this to try to clarify stuff on both sides ... ??? "local comm monoid of face" .... ???? .... ??? "quasicoherent artin-wraith glueing" ?????? ..... ?? "toric big zariski topos ..." ... ???.... ?? "subdivide" ... "blow up" ... "non-/toric birational ..." ... "hartog ..." ... ???? ........

??? new kind of "toric quasicoherent sheaf" here ??? .... ?? relationship to other kind ??? .... ????.....

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

?? abelian variety with complex multiplication .... ???? where associated field extension is _non_-abelian .... ????

?? galois rep coming from equal-division points of such abelian variety .... vs case without complex multiplication at all .... vs case with complex multiplication with associated field extension abelian .... ????? ..... ????? ...

??? quadratic case ... special aspects ... ???? .....

?? certain jargon words .... ??? ?? "field of moduli" ???? .... ???? .... "equal-division points" .... ??? .... ????

?? idea of .... ?? "abelian variety that residually has complex multiplication" .... ???? .....

?? vague, possibly imaginary memory of ... ??? newton polytope and holonomic d-module ... ????? .... ?? in any case, the gelfand reference that came up looks interesting ...

??? "non-linear fan" ... ??? with just small number of "cone"s ... perhaps of low dimension .... ??? .... ?? "spectrum" of such ... ??? relationship to "frankenstein presheaf" .... ???? ... ???? .... ??? .....

?? "toric divisor" as "way of shifting the walls" ??? ....

?? "hydraulic press" and weylotope degeneration and kaleidoscope fan ..... ???? .... ?? toric and tropical .... ????? .....

?? principal toric divisor of dot in algebraic lattice as ... ??? moving the walls so that they all go through that dot ??? .... ?????.....

?? toric divisors (and/or equivalence of them ... ???) as somewhat screwed up in singular case ... ????...

??? equivalence relations on divisors that people talk about ... ?? why ??? ... ?? try to understand in toric context ???




filtration (??? ...) on toric divisor class group associated with .... ??? progression toric birational, toric hartog .... ???? ..... ??? any relationship to "hodge filtration" ??? .... ???? .... ??? but .... ?? "singularity" .... ????? ....???? ...... ????? "stratification" ????? ..... ?????? ......


?? explaining fan ... ?? start with nice commutative monoid, giving affine toric variety .... draw it's (P,+)-spectrum .... so, cone dual to cocone ... then explain that glueing together affine toric varieties corresponds to glueing together these cones ... into "fan" ....

?? lots of details to soft-pedal here ??? .... ?? why single ambient lattice .... ?? "localization" ... ??? ..... ??"poincare duality" aspect .... (not _too_ soft, i guess ..... glueing along open intersections vs along closed intersections ...) ????? ..... ?? indicate that not claiming things have to be this way, in greatest generality .... more like, this is one particularly nice way they can be ... one nice way to do/allow glueing ... ???? ....??? ....

?? "lefschetz principle" and "galois shapeshifter" ??? ....

?? paper "Categorical Semantics of Linear Logic For All" somewhat interesting ... ?? not completely horribly written, so far as i can tell so far ...

?? tends to suggest that although it was good for me to wonder how linear logic might relate to toric quasicoherent sheaves and "combined doctrine" and so forth, the contrasts seem more prominent than the relationships at the moment ...

?? relating cartesian to non-cartesian tensor product in linear logic ... ?? vs relating products to biproducts in (?? ...) toric alg geom ... ???? .... ??? ab gp objects vs cocomm comonoids .... ????? ....... ????? .....

??? "cocommutative comonoid(_)" as right adjoint to ... ?? .... ?? comma 2-cats here ??? .... relating to the 2-adjunction .... ???? ....

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

..."An example of a concrete model of Intuitionistic Linear Logic is the category
of cpos and strict continuous functions. A cpo is a partial order with the
property that each directed subset has a least upper bound. Note that this
entails that a cpo has a bottom element. A monotone function between cpos
is continuous if it preserves the least upper bound of any non-empty directed
subset, and it is strict if it preserves the bottom element. The symmetric
monoidal structure is given by the smash product, the internal-hom of two
objects is given by the set of strict continuous functions with the pointwise
order, and the comonad is given by the lift operation."

?? .... initial object as directed colimit ??? ..... ???? .... ?? convention ?? .... ??? ..... ?????? ..... ???? ....... ??? ......... ?? maybe "directed" vs "filtered" ?? ....

?? well, i think i see the conceptual motivation for why empty category shouldn't count as directed/filtered .... ??? no object with cocone from empty diagram ?? ...

?? not sure yet whether i see conceptual motivation for opposite convention ... ??? ...

?? what to talk about at conference ???

?? toric quasicoherent sheaves ??

?? hartog fan ??

?? kaleidoscope fan ??

?? d-building geometric theory ... ??

?? more than one conference now ....

?? category theory octoberfest ... ?? dimensional analysis = projective geometry ??? .... ??? double shibboleth ???? ....

??? curious whether anyone's run into this topos .... ??? one of simplest examples of grothendieck topos but not presheaf topos ??? .... ??? in some sense ??? ....

?? sociological theories about situations ... ?? despite not getting much of a chance to talk to anyone on either side of either situation ... ????...

"toric spectrum vs ordinary spectrum" ... confusions ... ???? hartog .... ???? .... invertible object ... ???? .....

??? vague thought about for example grothendieck topologies on comm monoid N^2 ... most as constraining some particular among (1,0) and (0,1) to act invertibly ... but then the interesting one, which i played around with as maybe being sort of like "_either_ of (1,0) or (0,1) must act invertibly ..." .... ??? .... probably intended as joke, more or less ... but seems partway true, in peculiar/annoying (? ...) way ... separated presheaf here as "on each pair of elements, either (1,0) or (0,1) acts injectively ..." ... ???? ...... but actual sheaf condition here as trickier ??? ... ???? .....

?? geometric morphism from [geometrically terminal topos and/or from presheaves over walking idempotent ... ??? .....] to accidental topos .... ???? .... ???? ....

??? "combined doctrine" ideas in alg geom (toric and non-toric) ... ??? relationship to stuff some people study with multiple binary functorial operations ... ??? ....
?? "linear logic" ... ???? .....

?? non-/"idomeneal" stuff and ... ??? cyclotomic field with non-/trivial ideal class group ... ???? ....

?????? generalizing lattice / quadratic form relationship .... ????? .... ideals .... ???? ..... ???? jugendtraum / artin reciprocity synthesis ... ???? ....

??? monte carlo approach to ideal class group ... ??? .... ??? "chebotarev" ... ??? ..... dirichlet ... ??? .....

?? so if mirror symmetry involves relating coherent sheaves (?? though at derived level ... ??? ...) over one partner to "fukaya category" of other, and if toric varieties give prominent examples somehow, then what about trying to see how _toric_ quasicoherent sheaves fit in here ????.....

wpa on mirror symmetry ...

Mirror symmetry, in a class of models of toric varieties with zero first Chern class Calabi-Yau manifolds and positive first Chern class (Fano varieties) was proven by Kentaro Hori and Cumrun Vafa.[4] Their approach is as follows. A sigma model whose target space is a toric variety may be described by an abelian gauge theory with charged chiral multiplets. Mirror symmetry then replaces these charged chiral multiplets with uncharged twisted chiral multiplets whose vacuum expectation values are FI terms. Instantons in the dual theory are now vortices whose action is given by the exponential of the FI term. These vortices each have precisely 2 fermion zeromodes, and so the sole correction to the superpotential is given by a single vortex. The nonperturbative corrections to the dual superpotential may then be found by simply summing the exponentials of the FI terms. Therefore mirror symmetry allows one to find the full nonperturbative solutions to the theory.

??? ....

?? toric quasicoherent sheaves over projective line ... as not just actions of single comm monoid, or even of presheaf of such .... but then .... ??? when loop periodicity and/or tail length are constrained ..... ????? ..... ???? ...

?? non-toric analog of above ??? .... ???? ....

?? toric correspondence ... ??? ... ??? doctrinal approach ??? ...???? ....

??? flat vs non-flat ??? .... ??? also in non-toric case ???... ???? abelian cats as forming an abelian-ish 2-cat .... ???? .....

?? reconstruct original level from correspondence level ... ??? .... ?? in special situations ... ?? ... ??? toric and / or non-toric ??? .....

?? spectrum of cohomology ring of flag variety ... "sophisticated fiber" ... grading ... tangent cone ..... ????? ..... ???? ...

?? tangent bundle of torus as torus cross affine line ... which carries toric structure, but .... ??? not in natural way ??? .... ??? make this precise ???.... ???? .....

?? bump .... section on riemann-roch ...

??? "linear equivalence of divisors as interesting only in complete case" ?????? .....

?? "adele" / "repartition" .... "geometric" case .... ???

?? :

Example 12.22. This example shows that the Hurwitz genus formula fails
when there is wild ramification. Suppose that the characteristic of k is p > 0.

... ??? ....

Conversely, given a linear system L, one may try to construct a projective
embedding of X (or at least a rational map into projective space) which realizes
L as the linear system cut out by hyperplanes. Sometimes this cannot be done,
since a linear system may be empty. However, if L is "large enough," it may
always be done—see Proposition 13.8 for an illustration of how this may be
accomplished. Hence divisors and linear systems are at the heart of the problem
of constructing embeddings of X into projective space. It becomes essential to
understand the projective space |P| more precisely, or equivalently, the vector
space L(D) = {/ £ F\(f) > -D}. When X is a curve, the Riemann-Roch
Theorem is a formula for its dimension.

????? ....

Let us mention that the concepts of divisors and linear equivalence is not
special to curves. Let X be a variety which is nonsingular in codimension
one, and let F be its function field. A Weil divisor on X is defined to be an
element of the free abelian group generated by the irreducible subvarieties of
codimension one.

???? .... hmmmm .... "hartog ..." .... ???

?? line bundle over torus ..... ????? .....

?? divisor class group .... ????? .....

?? every ideal in the laurent polynomials is principal .... ???? ....

???? ....

??? AG morphism from reps of Z/2 (...) to modules of laurent polynomials .... ????

??? dimensional core .... ????? .... ????? ........

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

??? real torus here as .... ?? not having non-trivial loops .... ??????? ...... ??? ??? non-trivial Z/2-torsor ???? ..... ??????? ..... ????? .....

?? naive visual intuition for line bundle over punctured complex ("argand") plane ... "branch cut" .... ?????? ...... ???? ..... ?? holomorphicness as property .... ???? ...... ???? .....

?? tangent bundle of total space of vector bundle as .... ??? tangent bundle of base space ... pull the vector bundle back along the projection from the tangent bundle .... double the fiber ... ???? .....

?? apply in case where vector bundle = (?? co- ??)tangent bundle of projective line ?? ...

vector field lie algebra of projective line ... ???? ....

?? progression birational, hartog, ... ???? ....

?? intersection co-/homology (...???...) for hartog varieties ... ??? ...

?? accidental topos as etale groupoid .... ???? ..... how does this work out ... ???....

??? functoriality of "hartog theory of nice commutative monoid" .... ??? .... ?? morphisms between hartog theories .... ?? ...

?? complex spectrum of toric variety using "n-tuple coset" approach ... imitating real case .... ????? .....

??? hartog fan canonical line bundle .... ???? .....

?? affine somewhat generalized toric variety with homogeneous coordinate monoid z/23 .... ???? hard to believe that toric ideal class group here is non-trivial .... ????? .....

??? hartog environment ??????? ..... ????? .......

??? "downward decorrelation" vs kummer's chemistry analogy .... ??? kummer's chemistry analogy as "downward decorrelation without a net" ???? .... ???? .... ??? explcitly getting hartog theory of nice commutative monoid .... ???? .....

??? some confusion about .... in certain "algebraic geometry" (... ?? ...) contexts, what sort of sub-things (?? ...) correspond to grothendieck topologies ... ???? ....

open vs closed .... ??? ....

?? in TAG topos context (?? just open ??? .... ???? .... ??? "localization" .... flat .... ???? ....), vs .... AG context ??? .... serre's theorem, but also .... ????? .... ?? "quasicoherent artin-wraith ..." .... ???? ....... ????? ..... ?????? ......

?? .... vs .... ????? ...........

???"hartog" fan / theory (...) of (?? nice?? ...) commutative monoid .... ???? .... ?? especially non-invertible morphisms in theory ?? ... ???? ....

?? ... kummer ...???? ....

?? strictification (of theory ... ?? ...) here ??? ....

??relationship between stuff about "TAG toposes" we've been studying semi-recently, and ... ??? stuff todd's mentioned about .... ???? "faithful representation of theory of symmetric monoidal closed categories ..." ?? .... ????

?? toric divisor class group of 2d fan consisting entirely of just 3 rays (plus origin ...) .... ??? ....

??? tensored with Q gives 1-dimensional vsp ?? ...

??? n-dim toric kummeresque situation ... ??? n-dim fan consisting entirely of just n rays (plus origin) ???? ..... ????? .... ??? (TAG-ish ....) grothendieck topology on affine thing here ???? .... ???? explicit description ... ???.... ?? "lizard tail can regrow from any 2 independent successors" ???..... ????? ..... ???? "n-tuple coset" aspect here ?????.... "clutching" .... "extra dot ..." ... boiwn, pressley and segal .... ???? ....

???? relationship to n-tuple coset aspect of "glueing (P,*)-spectrum copies together to get (R,*)- or (C,*)-spectrum" ??? ..... ???? level slip ??? ... ???



?? ray in fan give line bundle / divisor .... ?????? but higher-dimensional cone in fan as maybe somehow responsible for multi-dimensional family of line bundles / divisors ??? .... ???? ..... ???? ....

?? "covering morphism" between toric varieties ... ??? problematicness of using self-duality trick on both domain and co-domain .... ??? ....

??? extent to which ("double" ...) cone falls apart when apex is removed .... ????? .....

?? so ... ?? total space of certain line bundle over projective line ... ???? "2nd" ... ??? canonical, or antocanonical, maybe ??? .... as .... ????non-singular quasi-affine, with Z/2 divisor class group?? the non-trivial line bundle being ... ??? hopefully somewhat easy to grasp, visually and so forth .... ????? .....

???? trying to understand this ("kummeresque") divisor class group fan-wise ... ???? ....

??? some confusion about this .... ???? cocone category .... riding on bike path that day .... south of fairmount park turnoff .... modified version ... "ignoring negligible ... " .... ???? ..... ???? different perspective now ??? .... but what about that old perspective ??? ....

co-/tangent bundle vs co-/sp (...) bundle over projective line .... double cone shape ....

??? extent to which line falls apart when origin is removed ... ???? .... ?? hmmm, very different question?? ... different dimensioj ... ?? ...

?? pulling back from base space the spinor bundle ... ??? ....

?? interesting (?? "kummeresque" .... fluorine radical ....) _quasi_-affine toric divisor class group ...

?? any secret missing point in kummer's original context ?? ...

?? some confusion (maybe not bad ... repairable ... ?? ...) here between removing bulk and removing single special point ... ??? recently we claimed that removing bulk leads to coarse classification ... ?? whereas here we're just removing special singular point ...

?? removing singular point .... to get divisors to behave better ... ?? seems to make good conceptual sense ??? ....

??? trying to extract actual toric variety (??? ...) from number-theoretic situation ... ??? .... ??? "localization" ?? .... ???? ....

?? concept of divisor as getting weird in "singular" case ?? .... which is affecting toric stuff here ... ??? blatantly cutting out singular point .... cone apex for example ... and "assisted ideal class group" .... ????? .....

???? "derived" approach here ??? ..... ????? ......

?? "hodge theory" for toric varieties ... ?? also "variation of hodge structure", whatever that means? ... ?? semi-mysterious remarks in connection with mirror symmetry ... ??? .....

?? idea of automating to some extent incorporation of zero (...) in toric spectrum ... ????? .....

?? "degeneration" .... ???? .... ?? "natural" degeneration ?? ....

notes for discussion with chris rogers

mirror symmetry ... toric divisor ... ??? ....

real spectrum of toric variety ....??? ...

notes for next discussion with alex

1 ... ???? general question about quasicoherent action of presheaf of commutative monoids .... filteredly cocontinuous ... ???? .... ??? do we actually have conjecture fairly straight here ??? ..... thought we did, but ... ????? .....

1.2 .... ??? situation where quasicoherent is automatic .... "promoting filtered oclimit to retract" .... ???? .... ??? loops and tails ... coarsening ... ????.....

2 ?? toric divisors .... ???? .... mirror symmetry ???? ..... ???? trying to fit toric quasicoherent sheaves in here ... ???? .... trying to understand :

"A sigma model whose target space is a toric variety may be described by an abelian gauge theory with charged chiral multiplets." ....

... canonical toric divisor ... ??? .....

?? hodge theory for toric varieties ...

2.2 ?? ... hartog fan .... ??? ....

3 toric correspondence ... ??? ....

4 ?? complex spectrum of toric variety using "n-tuple coset" approach ... imitating real case .... ????? .....

?? also "wrapping morphisms" between toric varieties ... ??? ....

5 logistics ... ???

travel arrangements ... paper ... ??? need more meetings for this ... ??? ... research proposal ....

?? tendency for single-ray toric divisors to be severely non-ample .... ??? .... ?? vs non-toric case ?? ....

?? for each co-fan lattice vector, consider "minimal divisor for which it gives holomorphic section" ... ??? ... ??? is this simply "principal divisor" ??? ..... ??? ... certain confusions here ... ??? morphism vs isomorphism .... ???? .....

??? "wrt a jointly ample dimensional subcategory of line bundles, every quasicoherent sheaf has enough twisted sections" ... ??? by definition ??? ..... ????? ..... of ... ???? ....

??? "riemann-roch" and vague idea that .... ?? must be _something_ to compensate for severe non-ampleness ... ??? ....

??? detailed picture of 2-ray toric divisor groups ... with "principal" subgp indicated .... singular cases ....????

?? danger of thinking that i can prove that non-trivial affine toric divisor class groups should exist ... ??? by "lattice-dependence" .... ????? .....

?? in fact .... ???? confusion... ??? paradox .... ???

??? maybe .... ????? idea that .... toric "ideal" theory might be more motivated by progression to AG context than by remaining in TAG context .... ???? ..... ???bit about TAG-ish spectrum vs AG-ish spectrum of toric variety .... ???? dimensional theory case .... multi-dimensional theory case .... ????? ...... ??? todd's idea about number fields and integral group rings ...... ????? .... maybe pure toric ideal class group remains trivial while ordinary ideal class group of comm monoid ring may be interesting .... ?????? ...... ..... ?????? ...... ???? maybe idea of kummmer/chemistry/chess/checkers idea fitting with toric geometry makes perfect sense, but _not_ in "pure TAG / toric" context; rather in "toric transferred to AG" context .... ???? .... ?? this all (...) as very tied in with how this whole "toric divisor" stuff works out .... ???? .....

?? role of zero in toric birational geometry and ... ??? "assisted toric ideal class group" .... ???? .... ???? .....

??? given an element of the co-fan lattice .... consider .... translating by it .... as an attempted map from .... ???? the unit toric quasicoherent sheaf to .... invertible quasicoherent sheaf obtained by ..... ?????

???? relationship to ..... ????morphisms in cocone category as translations .... ???? .....

??? morphisms between cocones vs morphisms between line bundles ..... ????? ....

?? case we ran into of ... ?? Z^2-graded comm monoid with "non-convex decategorification" .... ???? ..... ?? "L-shaped" ... ?? .... vs toric .... ???? .....

?? "loops and tails" ... ??? been somewhat overlooking case of "infinite loop" (so to speak ... not actually loop at all .... ???? .....) .... ?? relevant to "toric divisor" stuff ... ??? ....

??? toric baez-kim .... ????.....

?? toric divisor as "standard of positiveness" ...

?? toric divisor gp as free abelian on rays ??? .... ??? kaleidoscope case .... ???? ....

???? given ray r, create Z-graded comm monoid where grade j is .... ???? things meeting positiveness-standard r_j .... ???? ......

??? toric zeta function .... ????? relationship to some of that stuff about funny sorts of categorified zeta functions ..... ??????? ..... hmmmm..... 2-spectrum .... ?????? .....

?? role of zero in "toric birational geometry" ... ??? .... homomorphism into "toric field" taking zero as value .... ???? .....

??? "place of toric field" ... ??? ..... "archimedeanness" and/or other types of non-/pathology ... ????? ....

??? underlying TAG theory of AG theory of r-modules, vs TAG theory of [underlying monoid](r)-actions, for comm rig r .... ????? .... ??? ..... spectrums ... ??? ..... wrt such various things ... ??? ... ???? .....

?? funny "poincare dual" mismatch between through-the-looking-glass aspect of kaleidoscope and of real spectrum of toric variety ... ??? ....

?? "toric chern class" ....

?? toric blow-up ... ?? "divisorization" .... (?? vs generalized blow-up .... ???? .....universal property ... ?? ....) ... ?? arbitrary dimension cone going to ray .... ?????? ..... ???? ....

?? so how screwed up was my visual intuition about toric divisors and so forth in 2d case ?? ... ??? usual sorts of 2d duality flips ?? .... 1d, 2d, 3d ... ??? ....

??? toric blowup and "diamond-cutting" .... ???????? .....

... morphisms between divisors ...

?? should all be relatively easy to work out ??? ....

?? recent discussion about "toric rees ..." ... ?? ... ?? "filtered vs generalized filtered" .... ???? .... accidental quasitopos ... ??? ....

?? when filtered colimits become retracts (?? ...) then filteredly cocontinuous functors become just functors .... ????? ....

??? then quasicoherence becoming automatic ... ???

?? (0,0) as retract of both (1,0) and (0,1) ... ??? ...

???? quantaloid .... ???? ?? stable ?? ....

?? toric line bundle over 2d toric variety ....

??? assign to each 1d cone between 2d cones an element of the dual lattice .... ????? .....

??? 0d cone .... ?????? .....

??? "divisor" ??? ....

??? toric birational geometry" ....

?? "toric holomorphic structure on standard toric meromorphic line bundle" ... ??? ....

?? divisor here as something like "sense of flatness" ??? ....

??? "effective divisor" ... ???? .....

?? "picard stack" ... ???? ....

??? getting Z-graded commutative monoid from divisor here ... ??? ....

?? riemann-roch ... ???? ....

??? hmmm.... ??? keep on running into various situations where good analog in TAG (...) of AG ccncept of "ideal" (....) seems to be something like "face-flavored" version of ideal ... ????? though maybe not really so various after all ?? ... ?? tightly linked ??? ..... "blow-up" ... "ideal power filtration" .... "idea;/divisor class group" ..... ??????? .....

???? ideal class group as affine case of divisor class group ... toric .... ?? "face" ... ???? ....

??? kummer's chemistry analogy ?? ... ?? "face" .... ????? .....

???? principal toric divisor ...... ?????????? ......

??? lattice-dependence of toric divisor-class group .... ???? ......

?? "downward decorrelation" .... ???? ....

?? morphism vs isomorphism here ... ???? .....

?? invertible ideal .... invertible and/or non-invertible morphism between such ... toric and/or non-toric case ... ?? .... ??? principal ... effective .... ??? .... ???? .....

"generic" 1-cone fan as breaking symmetry vs torus ... ????? ... ?? tangent bundle .... ??? .... ???? toricness thereof ... ??? .....

?? tensor product of actions of a semilattice ...

walking element ...

?? "family of pointed sets" ...

j+1 tensor k+1 ... ???? as j*k + 1 ??? ....

??? cartesian product of sets as non-cartesian wrt partial maps ??? .... then extend by coproducts .... ?????.....

?? tempted to also try to get sum of the sets involved here , but ... ??? ....

??? connected projectives .... ???? .....

??? ab gp objects ...

?? relationship between "single-cone" fan and "complete" fan ??? ..... ??? in "truth-value" case ??? .... ??? ....

?? semilatticeization of cocone as face semilattice ... ???? ...

?? does concept of "kaehler differential" (??? ...) live at toric level ???? .... ????

???maybe no ??? "derivation" ... ??? "leibniz" ... ???? ....

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

a*b = c

d = 1 ... ??? ....

???? but .... ????? ideal power filtration of diagonal ...... ????? .....

??? hmmm .... ?? but .... any good cocone interpretation here ??? .... ??? ....

?? hmmm .... "toric ideal" ..... ????? ....... ???? hmmm, as hidden ???? .... ?????

a = o .... vs b = c .... ????? b/c = 1 ..... ????? ....... ?????? .....

??? the ideals for which there's a nice pictorial "toric" interpretation of ideal power filtration ( .... ???? ....) as the "a=0" ones rather than the "b=c" (or "b=1" ... ???) ones .... ??? .....

??? diagonal as not a face .... ????? .....

?? possi-probabiity distribution approach in mastermind, for example ?? ....

?? "game theory" .... ???? "comparing options" ... ??? ...

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

1 tangent / toric .... co-/tangent space of toric variety at point ... .... fano .... polytope .... ????? ......

2 level .... eckmann-hilton .... ????????? ...... loops and tails .... ????? x^a * x^b = x^a .... ??? ..... ??? category enriched over _semilattice_ ... vs semilattuce, also vs (??TAG??) topos ... ???? weak limits of syntactic categories, vs colimits in semantic categories ... ????? ....

3 ???flag .... "russian approach" ... direct sum of all irreps as multi-homogeneous coordinate algebra of flag variety .... ???? ... ???? relationship to toric stuff .... ???? ??? map ???? .... .... ???vs direct sum of all twisted tensor squares as coordinate alg of alg group itself .... ???? .... ...

4 comm monoid with "-1" that's not just free .... ????....

5 generalized lagrange extrapolation ....

?? is tangent bundle of toric variety "toric" ??? .... ???? ....

?? fiber vs stalk ... ??? quasicoherent artin-wraith .... ??? ....

?? map from kaleidoscopic toric variety to projective line associated with particular mirror .... ?????? ......

?? how filtered colimit (?? maybe just colimit of sequence ??) "gets promoted" to retract ??? ..... ???? colimit of sequence where all the arrows are the same idempotent ... ????? .....

"derived" / "higher" "stable" (??????) colimits .... ????? and relationship between filtered colimit and stable colimit ... ???? ....

?? extremal eckmann-hilton ... frankenstein ... ??? ... morita / karoubi .... ???? ....

?? .... ??? situation where .... shortage of "affine" stuff .... or existence of non-affine stuff, glued together from affine ... ??? but then progression (?? or degeneration ??? ... ????) to situation where after all everything qualifies as affine .... ????? what's happening in such situation ???? ..... ???? more objects ... ??? .... more morphisms ... ????? ..... ???? .... ???? ....

??? x^4 = x^2 example .... ???? try to work out .... ???? (and / or maybe x^3 = x^1) .... ????? .....

filtered colimits becoming retracts ... more retracts ... ???? .....

period 2 loops and length 1 tails .... ????? ..... ??? connected projectives over projective line???? ???projective line looks affine here ???? ..... ????? ....

connected .... ??? unordered pair of ordered pairs of sets ... ?? or single ordered pair of sets ... ???? .....???? .....

??? cocone .... retract ... ???? .....

?? quasicoherent presheaf of actions of presheaf of any kind of reasonable monoids as equivalent to filteredly cocontinuous presheaf on lax (???) colimit .... ?????? .....

in particular thus form topos ... ??? ....

?? quasicoherence as "local" property ???? ..... ????_any_ affine cover, vs maximal cover .... ????? ....

?? applying (?? ...) law "x^m = x^n" to toric geometry .... ???? for example m=3, n=1 .... ??? real spectrum .... ???? localization, co-/face, accidental topos .... ???? ....

?? "x^a * (x^b -1) = 0" ....

??? a = 1 or 2 ... b = 1 ... or 2 .... ????? ....

?????? loss of interesting z-gradings ???????? ...... ??? z-graded comm monoid st any ungraded localization satisfies x^m=x^n ... ???? ..... ??? ..... ???hmmmm, but what about z/p-gradings ??????? .... ????? .....

??? loops and tails .... constrained in certain ways .... ???? .... ??? laws for monoids vs for their actions ... ??? .... ???? .....

???? localization of semilattice ??? .... ??? is localization really flat (?? ...) in this generality??? how _does_ proof go?? ??? inverting _monic_ ?? ... ???and / or epic ???? inverting dominance relation in poset ... ???? .... ???? ..... ??? some monoid/action confusion here ?????? .... ???? filtered colimits .... ?????? ........ ??? "torsor" of [walking idempotent] monoid .... ?? colimit of infinite sequence of non-trivial endomorphisms of cayley action .... ?????

??? hmmm, but really just splitting idempotent here ??????? ..... ????? ....
???? some semi-recent discussion with todd ..... ???? splitting idempotents in semi-lattice-as-monoid ....??? .... ???? _was_ that in sort-of this context ???? ..... ???? the two (???? ....) cocone categories ... ???? .... ??? .....

??? TAG structure for actions of semilattice ... ??? .... ?? shortage of gradings by ab gp ... ??? shortage of non-affine possibilities from glueing .... ?????

???? etale groupoid .... ???????? ......

??? endo-ness as involved in relating "localization" (that is, inversion ...) to filtered colimits ???? .... ???? .....

???ab gp objects in (?? generalized ?? ...) accidental toposes here .... ??? ....

??? segal-morse category ... ????? .... ???? arnold / bott /appolonius .... ????

???toric geometry "based over z/n" .... ??? ..... ?? "galois-theoretic" aspect ???? .... ??? spectrum over commutative monoid under Z/n ; for example n=2 and (R,*) ... ???? ....

?? filtered colimit as generalization of retract .... ???? morita .... cauchy .... stable ..... ??????? .....

??? kaehler structures from projective embeddings .... ???? ... baez / minhyong kim ... ??? .....

?? so ... according to current guesses ... ??? .... subdividing a (highest-dim, say for now ....) cone does in fact always "blow-up" the same point ... namely the one corresponding to the unique coset of the linear span of that cone ... but the "blow-up" might not be the canonical blow-up; rather it might be the generalized blow-up corresponding to the ideal power filtration of some "non-radical" (??? or whatever the sensible terminology is here ... ?? ....) ideal whose radical (??? ....) is the maximal ideal corresponding to the special point ... ???? ..... ?? but then there should be just _one_ choice of subdivision (??? ....) which _does_ give the canonical blow-up .... ??? but seems likely that this choice is "lattice-dependent" ??? .... ???? ....

?? "choice of subdivision" vs "choice of subdividing ray" ... ????? ..... ????? .... possibility of filtration more general than ideal power filtration here ???? ..... ???? ......

?? defining "generalized day convolution" (??? ...) as sub-shriek for essential geometric tensor product ... ???? .....

?? compatibility relation between combined structures ... ??? .... ?? doctrinalness ?? ....

?? toric geometry ...

?? string theory ... ??? mirror symmetry ?? ....

?? symplectic geometry ... ??? ....

?? polytopes ??? ....

??? kapranov, gelfand ... ??? ... ??? zelevinsky ??? .....

?? real topologies of 2d affine toric varieties ... ???

??? bruhat cells for sl(2) .... generic as giving plane, coincide as giving cone ..... ???? ...... ??? real vs complex.... ??? z/2 vs z ... ??? .... ??? langlands / tate / fourier duality ???? ...... ??? stern-brocot ... ???? .....

??? ... map from cones of n-dim fan to congruences of (Z/2)^n ... ??? or in complex case, of (R/1)^n ... ???? some sort of nice functor from fan cone poset to "toruses" ..... ????? ..... ???? relationship to "cocone category" ?????? ..... ????.....

?? niceness of functor (...) here .... ??? ??lattice homomorphism ?? .... "level-preserving" ??? .... ???? ....

??? "geometric realization" .... ??? "kan extension" .... ???? .... ??? maybe sort of straightforward .... ??? .....

???? "weighted colimit" .... ???? ... ??? homotopy colimit ??? .... ????

?? real : complex :: 3! : gl(2,Z) ??? 3! = gl(2,z/2) ??? ... ??? ....

?? consider ... inclusion of checkerboard quadrant into chess quadrant ... ?? .... comparison of real glueing patterns ... ??? ....

??? "complex glueing pattern" .... ????????? .......

?? alleged relevance of toric varieties to string theory ??? .... ???? ....

?? "mirror symmetry" ?? ...

?? for given "fan pattern" (omitting lattice information ...), seems to be only finite number of "real glueing pattern"s ?? .... ??? ... ?? all of which actually occur ??? .... ??? .... ??? but how do change-of-lattice morphisms work here ??? .... ??? .....

?? well also ... the ("real" ...) gross topological structure must not be encoding the whole toric variety structure, it seems ... ??? .....

??? "reflection" ... ????? .....

?? so consider a 2d affine toric variety ... ??? .... and consider the (non-toric ... ???....) subvariety given by .... "omitting the torus" .... ???? so this is ... ?? modding out by the "non-face" ideal .... ??? and is independent of the original variety... in certain senses ... ??? as TAG theory?? ... ??? ...

??? still built out of toruses, sort of .... ???? .....

??? so is there some weird tensor product of commutative monoids ... ?? something like "pointed union of toric varieties" ..... ?????? ..... ??? also non-toric ... ????? ..... ??? ....

??? trying to "attach ray to origin and then attach angle to ray, without attaching other ray forming the angle" ??? ... ??? ... ??? ....

?? specification of quotient torus, vs of "attachment information" ... ??? for example, 2d fan with single ray .... ?? the line of the ray, vs its direction ... ??? hmmm, should we then just reduce higher co-dimension attachment to co-dimension 1, on the grounds that the higher case is mediated by co-dimension 1 steps ... ??? vs ... ??? thinking about, for example, ways of attaching 1d torus to 3d ... ??? 3d fan formed from single "angle" ... ???

?? attachment ... ??? artin-wraith... and/or other stuff ... ???? ...

?? if intersection of cones isn't a mutual face then does that give "non-separation" ??? ... ??? ...

?? "newton polygon" .... "tropical geometry" ... relationship to toric geometry ... ??? .....

?? "supporting hyperplane ... " .... ??? ....



??? ?? "toric variety as built out of toruses" ... ??? then "non-linear fan" .... ??? ....

??? nerve of cover ... ???? simplicial complex as element of free distributive lattice ..... ????? free boolean algbera and (P,+)-spectrum vs (P,*) (??? ...) of toric variety ... ????? ....... ??? ....

?? "blowing up corner of polytope to extra face" ?? .... ??? ... poincare dual ... ??? ....

?? quasicoherent sheaves over toric variety as ab gp objects in an ab cat, but tensor product arising that way not the "right" one ... ??? ... ?? ringed topos with external Z as the ring ... ??? quasicoherence here ??? ...

?? so (P,*)-spectrum can (?? reflecting further on the "ray cone as direction in which you can go to infinity ..." idea ... ???? ....) be seen as some sort of "poincare dual" of fan ??? ...?? only "schematically" so far, but ... ?? .... then (R,*)-spectrum as ... ??? some copies of this glued together in way involving lattice .... ??? ... ?? change of lattice ??? ... ??? ....

??? weylotope as example here ... ?? vs as example of that other idea ... toric projective embedding ... ??? multi-projective ??? .... ?? weylotope degeneration vs scaling ... ??? .... each root independently ... ???? .... ???? ...

?? polytope edge as blow-up of ... ??? .... ???

??? blow-up ... operad .... ??? opetope ... ??? "flat" ... ???? ....

?? at first thinking that "monomial-flavored" (?? ...) ideals in comm monoid rings were (generally ... ?? ... ?? infinitesimal neighborhoods ... ??? ....) "non-toric" in having corresponding affine sub-variety being non-toric ... ??? but then realizing (?? ...) that what's actually "non-toric" here is mainly the embedding of the subvariety, rather than the subvariety itself .... ?? ....

?? "quasicoherent artin-wraith glueing" ... ?? maybe using some sort pf "local ring (?? ...) of closed subvariety" ????

?? interpretation of highest-dim cone as single point .... (vs points in that cone as also showing up in the torus ... ?? ...) .... ??? "generic point" ???? ... ???? nonlinear fan ???? ... ???? ....

notes for next discussion with alex

?? day convolution ... adapted to filteredly cocontinuous case .... ?? geometric morphism ... ??? ... ??? non-toric analog ??? ... ???? ....

?? further concrete understanding of toric spectrums ... "built out of toruses" ... "attachment" ... "travel along coset in ray direction" .... ??? .... ?? "real glueing pattern" ... ??? .... examples ... ?? change-of-lattice morphism ... ??? ....


???set-valued functors preserving some class of colimits .... ???? .... ?? well, don't bother alex with this particular stuff too much yet ... until/unless it gets interesting enough to seem relevant ... ??? ...


??? get form/s ??? .... ?? focus on toric stuff ... ??? ....

??? toric varieties in string theory ???? ....

?? in (P,*)-spectrum, coset of ray completion as corresponding to ideal point reachable by traveling in the ray direction along the coset .... ??? ...

?? then ... ?? ideal point associated to coset of angle joining two rays ?? ?? how to reach it ??? .... ?? maybe sort of clear ... ?? ...

?? then ... ?? (R,*)- and (C,*)-spectrums as bundle over (P,*)- ... ??? ... ?? combinatorial treatment ... ??? ....

??? projective plane example .... ?? ....

?? blow-up ... ??? ....

?? watching linear functionals on cone degenerate to linear functionals on face ... "infinite slope" .... ??? ... "lagrangian grassmanian" ??? .... ??? "infinite slope" ... "crashing wave" ... ???? .....

?? something backwards here ?? ... ??? .....

?? understanding torus attachment from many viewpoints .... cocone, cone ... ??? ....

?? "attachment" as ... ??suggesting "natural boundary" to be mapped to actual boundary ... ??? .... ???? .....

??? again, "quasicoherent artin-wraith glueing" as seeming problematic .... ??? doesn't seem to be enough "stuff" lying around ?? .... ??? ....

?? polytope associated to toric projective embedding ???? .....

?? kaleidoscope case ??? ??? weylotope ??? .... ?????? single projective embedding ??? .... ?? segre embedding ?? ... ??? weylotope degenerations ... ??? ....

?? non-/singularness ?? ... ???? .... ????? simplicial number ... ???? .....

dominant weights and positive co-roots as dual ??? ....????? .....

dual of weight lattice as _co_-root lattice ?? .... ??? way in which this complicates attempt at linking galois-schubert correspondence with kaleidoscopic toric variety ... ???? ....

?? so ... ?? toric variety as built out of toruses, obviously ....

repercussions for examples ... kaleidoscopic, generalized (?? including stacky ...??) ... ??? ....

?? kaleidoscopic .... ?? bruhat cell, vs ... ?? ....

?? quotient torus / sub-lattice .... section / retraction thereof ... ??? ?? extra information in "fringe attachment" data ?? ..... ????? .....

?? other real forms of complex toric variety ??? ... ?? .... ??? x^2+y^2=1 ??? ....

(C,*) -> (P,*) "magnitude" ... ??? ....

??? so ... non-fullness of progression from toric to non-toric stuff ... especially, comm monoid alg hom coming from partial monoid hom with domain a "fringe" (?? ...) submonoid ...

toric vs ordinary quasicoherent sheaf over toric variety ... functor both ways ... adjunction ... ??? ....

??? universal property of category of filteredly cocontinuous set-valued functors on x ??? ..... ??? where x _has_ filtered colimits ???

?? analogy : discretely cocontinuous set-valued functors on x where x has discrete colimits ??? ....

??? tensor product of toric quasicoherent sheaves in filteredly cocontinuous picture, and geometric morphism involving accidental topos ... "structure constants" ... "generalized day convolution" ....

?? "integeral toric quasicoherent sheaves" .... ???? ?? (Z,*)-spectrum of toric variety ?? .... ??relationship to (R,*)-spectrum ???? ..... ???? .... ???? ... ??? any levle slip ??? .... ???? ...... ????? .... ???? ....

?? so ... ?? seem to be getting ... ?? complex point of toric orbit of toric variety corresponding to cone c in fan f as .... given by "magnitude coordinate data" (coset of linear span of c in vsp(f)) and "angle coordinate data" (coset of linear span of [c and lattice(f)] in vsp(f)).... .... then such a complex point as real if angle coordinate data is "half of zero" ???? ......

??? trying to apply this prescription out of bounds ... "fan" that's not quite a fan .... ????? ...... ??? any interesting (?? "TAGgish" ?? ...) generalization of toric variety here ??? .....

??? in turn this (...) seems to say .... ??? co-dimension j cone in fan corresponds to toric orbit shaped like j-dim torus ... ??? maybe somewhat obvious ??? in retrospect ??? at least to extent to which it's true ... ???.... ???true over what commutative monoids ????? ....

??j-dim torus in question arising from ... ??? dual lattice of lattice obtained by ... ?? fan lattice modulo span of part in cone ... ???.....

??? any artin-wraith glueing going on here ??? ..... localization as AG/TAG analog ??? ... ??? "ringed (or monoided) artin-wraith glueing" ... ????? ??? "fringe functor" ???? ... .... ???? .... "attahcing map" ... "mapping cone" ... ???? ....

"attaching map" ... ??? "cw-complex" ... ????? ......

?? line bundles and other bundles over toruses ...?? ....

?? tropical geometry and toric geometry ... (P,+) .... also (N,+) .... ??? ....

?? idea of trying to describe (P,*)-spectrum of toric variety as .... ?? point consists of coset of linear span of some cone in fan ... ??? ... ??? but ... ??? not clear how dependence on lattice enters here ??? .... ?? must affect glueing / convergence ... ?????.....

?? mapping cone ... ??? ....

?? "unitary spectrum" ??? ....????....

fan spectrum ....

flag variety ... subvariety ....

fan cone as toric orbit ,,,, ???? ....

?? "ghost point of cone" ... ??? .... (R,+)-spectrum vs (P,+)-spectrum .... ??? todd's subtlety and blow-up ????? ....

??? "available non-invertibles take over the role of zero" .... ???? .....

??? combinatorial ("schematic") aspect of fan as "2"-spectrum ... ??? comm monoid hom (P,+) -> 2 corresponding to mapping from "real" to "combinatorial" aspect of fan ... ???? other interesting homomorphisms here ??? .... ?? or non-homomorphisms or missing homomorphisms or something, to explain some of the weird stuff ??? .... ??? non-preservation of non-invertibility .... (?? somewhat dramatically so ??? .... "localization" .... ???? .....) ..... free (?? ... or ... ??? ....) boolean algebra .... ???? .... ??? also non-commutative diagram of homomorphisms ... ??? ....

?? does blow up map literally induce bijection on P-spectrum ???? .... ???? ....


???? hmmm .... ???? climbing from fan spectrum to whole real spectrum by ... inclusion (P,+) into (R,+), understood in terms of "ghost points" ... hopefully incorporating blow-up in a nice way .... .... ??? then .... (R,*) almost onto (R,+) ..... ????? ..... ??? .... ????? .....

?? fan as spectrum and blow-up ???? ..... ?? .....

??? canonical representative point of ,,, ?? toric orbit ??? ... ??? but .... ??? that point as typically not appearing in fan spectrum ??? ... ??? ....

?? so let's try to figure out whether "conical wrapping-up of toric variety" makes any sense ??? ....

?? set equipped with commuting operators x,y ... ??? and isomorphism between .... ??? "forcing x to become invertible while action of y survives, then x and y trade places" and "forcing y to become invertible while action of x survives" ??? .... ???? ?? good tensor product here ?? ... ??? ....

?? fan spectrum as not completely faithful .... ??? .... ?? todd's subtlety ... ??? ..... ???? .....

?? "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 ... ??? ....