?? toric convolution on category of vector spaces over a field .... ??? as just ordinary tensor product ?? ... ?? lax interchange structure as standard .... ??? .... ???? theory of object invertible wrt both tensor products ???? ...... ?????? ....... ..... ??? confusion ..... ???? .....

??? "2 kinds of toric spectrum .... preserving toric convolution vs not ..." ....... ?????? ....... ???? .... ??? flat / toric analogy again ??? ...... ???? ...... ???? .....

_propositional toric th_ -> _dim toric th_ -> _toric th_

_prop ag th_ -> _dim th_ -> _ag th_


???? .....

?? hmmm ... ??? .... beck-chevalley ??? .... ???? ... ??? failure thereof ?? ... ??? ...

??? toric convolution for at dimensional and/or propositional levels ??????? ...... hmmmm .... ???? .... ????????? ...... ????? comultiplication ????? ...... ????? ..... ????? ...... ??? non-doctrinalness here ?????? ..... ???? .....

?? doctrine of "intended environments" .... ???? .... ???? .... ??? what was that fairly recent bit about .... ???? .... object vs stack ... ???? ...... ???? ..... ???? .....

?? alleged semi-monoidal structure on co-fan category .... ??? is it really there ?? .... ???? what's it supposed to be like ???? .....

?? "categorified inf" .... ???? nullary inf as top .... ??? clearly problematic for basic opens of a toric variety .... ?? but hmmm, perhaps the categorified binary inf really does exist here .... ????? .... binary intersection of basic opens .... ???? .....

??? relationship between lax interchange transformation and "generalized day convolution as extra adjoint of essentiaql geometric morphism" .... ???? .....

?? "toric environment" where a model is a pair of models ... ??? or where a model is .... ?? ... "solution to some categorified system of equations for a tuple of models" ... ?? ...

?? "torically algebraically complete" toric environment ... ??? ....

"galois" ideas in toric geometry ... ???

?? actual absolute galois group action ... ???? relationship to stuff like "weil conjectures" and "motive" ... ??? .... toric case ... ??? ...

?? other ideas ... interactions .... kummer's chemistry analogy ... complex multiplication ... abelian langlands .... artin reciprocity ... ??? .... lattice ... braiding ... ???? ....

??? 2-topos .... ??? where semantic 2-cat is "something like" _ag th_, vs where syntactic 2-cat is ... ??? ....

[date uncertain ...]

i often find the n-category cafe somewhat annoying... perhaps for more than one reason, but overwhelmingly for just one reason... the same reason that i tend to find any other participatory writing forum annoying, namely that it's unfairly stacked against me because the medium of communication is writing rather than talking...

anyway, for this reason i sometimes go for long periods without reading it, and sometimes miss interesting discussions that way... and i seem to have just found one that i missed that looks like it might be interesting... on the langlands program... so i want to make a pretty serious effort to go back and read it and try to understand it pretty soon...

there's also "math overflow"... one of my problems with that... basically just a special case of the already mentioned problem... is that by the time i manage to write up a question in presentable form, it usually requires/causes me to already pretty much figure out the answer...

in principle it ought to be possible to overcome that problem to some extent...

?? idea that existence of non-affine toric variety can be interpreted as failure of certain naive generalization of certain "tannaka-krein" theorem from case of bicommutative hopf algebras to case of bicommutative bialgebras, and that such failure might be expected to persist beyond bicommutative case, giving examples of some sort of "non-affine generalized toric variety" ... ??? ....

?? was idly wondering whether this might relate to some concept of "generalized toric variety" (?? ...) that i think alex mentioned ben webster as having worked on ... maybe still wondering that, but may have changed my mind about relaxation of which half of bicommutativity might be involved ... ???

?? try glueing together oppositely "oriented" copies of m(2) along common gl(2) ?? .... ???? .....

?? particularly possibility of _"projective"_ (??? .... ???? ....) generalized toric variety here .... ???? .....

?? in trying to clarify various alleged analogies between various "combined doctrine" situations ... ?? idea that "cartesian product" recurrently shows up as "the extra product" ??? ... ?? in particular in disguised / mutated form in at least one case ??? .... ?? in 4-product case of "ag theory with extra toric convolution product", the 4 products are "tensor product", "toric convolution", "cartesian product", and "sum" ... ??? and not only is cartesian product "extra" here, but the other "extra", namely toric convolution, is itself a sort of disguised / mutated version of cartesian product ... ?? via relationship to accidental topos .... ??? ....

?? "extraness" and "flatness" here, vs "extraness" and "toricness" ??? ....

?? property/structure ambiguity (kock-zoeberlein ...) of cartesian products here (?? ...) and ... ?? "flatness" .... geometric interpretation .... ??? .....

?? hypothetical adjunction ... ?? between "_lax interchange bi-tensor category_" and "_bi-cartesian bi-tensor category_" ... ??? .... ????? ..... ???? ..... relationship of unit and/or co-unit for such to "_the_ analogy" here ??? .... "toric convolution"and cartesian product in accidental topos .... ???? .....

?? vague memory of stuff baez talked / asked about once ... ?? embellishment on right adjoint property of "cocomm comonoid(_)" ?? .... ???? bi-stable bi-monoid .... ??? ...

?? was thinking a bit about "preserving extras" in 4-product situation ... ??? but maybe not as much leeway as might guess at first .... ?? "abelian" overlap between product and sum .... ??? also using binary products as stand-in for limits with more general-shaped diagram schemes ... ??? .... ??? ....

?? various weirdnesses connected with various "tannakian" ideas .... ??? some / all contributing to weirdness of using name "tannakian" for certain things .... ???

?? old (...) idea of "hierarchy of tannaka-krein theorems" ... ?? i thought that it was weird that some items seem to want to appear in more than one place in the hierarchy ... ?? like ... "commutative bi-algebra" ... once in connection with "symmetric monoidal category" (???) but then again in connection with "category with two stable tensor products and lax interchange map between them" (??? ....) ..... ?? then halfway decided that this was connected with ... ?? multiple hierarchies ... (?? ...) ?? module hierarchy and co-module hierarchy, for example .... ??? .... ???? but then also ... ?? maybe different flavors of tannaka-krein theorems, some where you (?? try very hard to ??) get precise equivalence between algebra of certain sort and category of certain sort, but then others where algebra of certain sort gives only some special (?? "affine" ??? .... ??? ...) examples of category of certain sort ..... ???? ...

?? again, funny way certain "tannakian" ideas "fold over" on each other / themselves here ... ??? ....

?? idea of ... smc as "generalized affine toric variety", then gluing such together along "open inclusions" corresponding to "localizations" ?? ....

(?? non-toric analog as well ?? ...)

?? to what extent do we understand why one-object smcs are unstacky here ??? .... ??? hmm, maybe we _do_ understand it pretty well ... ???? unit object as not really extra stuff ??? ... hmmm .... i'd been going to ask whether this was part of some "stabilization slope" that we should understand ... and i guess that that question still applies, but "unit object as not really extra stuff" seems pretty close to right track ... ??? ....

[?? disjunction] : [?? existential quantification] :: [?? pushouts and/or discrete sums] : [?? higher-dim colimits ... ??? coequalizers and/or co-invariants ... ??? ....] ??? ....... ??? existential quantification as sort of "decategorification" .... ??? .... ?? unstacky vs stacky (?? ...) version of "cover" / "covering" .... ??? ..... ???? .... ??? "no decent theory has existential quantification at outermost level" as less defensible than "no decent theory has disjunction at outermost level" ??? .... ??? ....

?? non-toric analog of conjecture that grothendieck topologies on commutative monoid correspond to open sub-[toric variety]s of affine toric variety ??? ...

?? bit about "blow up, then blow up resulting singular points" ... ??? "geometric mackay correspondence" ... ?? nakijima ?? .... ?? maybe related to some more systematic (?? ...) resolution process ... ??? .... ?? toric case ??? ....... ???? .....

?? (?? bi-stable) bi-monoids for lax interchange map of toric variety ... ?? (toric) geometric interpretation ... ?? ...

?? formal properties of co-/slice cat of cat of comm monoids ... ?? ?? hmm, maybe co- makes big difference here ?? .... ???? .... ??? paradox here ??? ... (?? obviously based on some stupid mistake i'm making ....) ?? .... "affine ..." ... hmmmm .... comm monoids under given such, vs comm rings under given such .... ???? vs ab gps under given such vs over given such .... ???? ..... ???? .... ??? .... "formal properties" of categories here .... ??? ...

??situation where taking vector space objects in x is right adjoint functor of x, vs situation where it's left adjoint ... and / or both .... ??? .....

??? doctrine morphism here ?? .... ???? .....


?? getting torus-equivariance involved here .... ?? ...

?? lax interchange law .... adjunction ... ?? ... comonoids wrt one tensor product vs monoids wrt other ... ??? ..... ?? geometric interpretation in toric context ???? ...... hmmmmm ..... ??? ....

??? case of finite ab gp, or maybe locally compact ... ???? ....

?? bit about toric geometry and generalizing fourier duality to monoids instead of just groups .... ??? .....

?? toric case ... ?? as toy case of tannakian philosophy, in keeping with status as toy part of ag ... ?? ...

?? other toy cases of tannakian philsophy .... ??? ....

?? lax interchange map from cartesian product outermost to convolution outermost corresponding to "adding solutions to x+y = k1 and x+y = k2 to get solution to x+y = k1+k2" ???? ..... ??? does this make any sense ?? ...

?? hmmm ... some fourier-duality confusion here ... pointwise product vs convolution ... which of these cartesian product qualifies as ... ??? ..... ?? maybe ok though ??? ....

?? cartesian product as "pullback along diagonal" ... ???? ..... ??? convolution as "pushforward along codiagonal" .... ???? ..... ???? ....

?? pointwise tensor product of graded vector spaces, vs convolution tensor product ... ???? .....

(a toric b) ordinary (c toric d) -> (a ordinary c) toric (b ordinary d) ... ??? ....

(a cartesian b) convolve (c cartesian d) -> (a convolve c) cartesian (b convolve d) ... ??? ....

hmmm .... ??? ..... ?? "components given by functoriality of convolution, applied to projections" ??? ....

(a convolve b) cartesian (c convolve d) -> (a cartesian c) convolve (b cartesian d) ... ?? ...

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


(.... , a_-1 * b_1 , a_0 * b_0 , a_1 * b_-1 , ...) ...... ???? .....


??? hmmm, so maybe arrow really is backwards from expected here, due to duality flip .... ???? .....

?? skyscraper sheaves here .... ??? and how they get along with duality ... ?? ....

?? this (?? ...) lax interchange property as vacuous on topos side, but not on ag side ??? ... ??? .... ?? adjointness here ???? .....

??? cartesian smc with extra tensor product ... ?? vs smc with extra tensor product and lax exchange map .... ???? ....

?? other arity lax exchange properties here ?? .....

?? idea of trying to turn [relationship between [?? "accidental topos" with its extra "tensor product" ...] and [ag theory of toric variety, with its extra "toric convolution" ...] .... with latter as k-module objects in former and former as special cocommutative comonoids in latter ... ??? ] into some sort of nice [adjunction and/or interpretation of doctrines ... ??? ....] ?? ....

?? comonoid process as wanting to be right adjoint here and k-module process as wanting to be left adjoint ???? .....

?? maybe implicit compatibiity conditions here ??? .... ?? between toric convolution and "tensor product" (... ??? ...) so that "tensor product" survives to comonoids .... ?? and / or between ??? and ??? so that ??? survives to ????? ...... ????? ....

?? "between cartesian product and "tensor product" so that "tensor product" survives to k-module objects" ???

??? concept of "k-module object" as involving cartesian product .... ?????? ......

?? "generalized day convolution" .... ???? .....

??? distributivities here ????? ...... ??? decategorified case ??? ... ??? pointwise product and convolution .... ??? .....

?? graded action x of g-graded commutative monoid m ... ?? m-equivariant map from x to g ??? ...

?? case x invertible ... ??? ..... cartesian power of x ... ??? ....

m -> g .... ???? .... displacement .... ???? .....

m^n -> g .... ??? ....

?? re-expressing toric convolution of given quasicoherent sheaves as particular quasicoherent sheaf .... ??? .... universal property wrt combined doctrine as stronger .... ??? .......

??? "flat" ..... ????? .....

?? projective n-space as toric variety ... ?? "theory (?? of doctrine "ag + toric convolution" .... ??? ...) of line object x equipped with nice embedding into standard [n+1]-space, plus toric convolution cocommutative comonoid structure on x ..." .... ????? .....

?? why did brandenburg mention "locally free" at all ??? .... ... bad sign .... ???? .....

?? grothendieck topologies on n^2 as one-object category ...

?? "if you can recover from any nonzero injury at all, then you can recover from losing just the single tail cell ..." ?? ...

?? if you have any regeneration powers strictly beyond that, then .... ??? .....

?? situation where stability under pullback is vacuous because pullback is monotone increasing ... ??? ....

?? toric topos write-up ...

?? themes ...

?? toric tannakian correspondence ... ???

?? presheaf topos glued together using non-essential inclusions ...

?? missed opportunity ... eisenbud ... ????.... ??? opportunity to try again .... synergy .... ??? ....

?? combine ... recent alex outline ... octoberfest talk .... "lax interchange map" idea .... ??? ....

?? "toric varieties under the tannakian correspondence" ...

?? contrast between topos / ab cat analogy and topos / ag theory analogy ?? ...