notebook 360

2012-02-09T14:58:00.002-08:00

?? universal property of "toric topos" ... ?? how much embodied (?? ...) by "category of classical models" ?? ... ?? or by models in other toric toposes ... which we think we know something about .... ??? .... ???? ...

2012-02-09T12:51:00.003-08:00

?? "concept of cartesian abstract category as making sense in cartesian 2-category because [abstract category in cartesian 2-category is automatically stably comonoidal, so if also stably monoidal then bistably bimonoidal, and then ....] ..." .... ???? ....

??? level slip ??? .... ????? ..... ???? ......

?? microcosm ... ??? ....

2012-02-08T22:17:00.001-08:00

?? ask todd about etymology of "accessible" ?? ....

2012-02-07T10:30:00.000-08:00

?? pixellated vs pixillated ... ??

?? golem vs gollum ... ??

?? idempotent monad and "localization" ... ??? ..... ?? right-exact idempotent monad ... right-exactness vs idmepotence here .... ????? .....

?? frame rate .... james cameron ....

Peter Jackson is shooting “The Hobbit” at 48 ...

douglas trumbull ...

article about trumbull here ...

?? "electron gun" ... ?? ....

?? record maximum frame rate ?? read about scientific applications somewhere recently ??? ....

?? collapsible cup ... telescope ... ??? ....

?? stock market ...

2012-02-05T09:22:00.000-08:00

?? grand unified theory of mathematical politics ...

3 kinds of mathematicians :

1 lack social intelligence

2 normal social intelligence but separate from mathematical intelligence

3 incorporate normal social intelligence into mathematical intelligence

kind 2 use social intelligence for career politics and dominate the field career-wise. kind 2 doesn't know about kind 3, often mistaking it for kind 1, which is the worst possible outcome.

2012-02-04T14:24:00.000-08:00

?? bracket of observable x with hamiltonian ..... ???? ....

?? heisenberg vs schroedinger picture here .... ???? ....

???? .....

??? vanilla base hamiltonian vs "actual" perturbation .... ??? ....

???? .....

??? ....

??? also "interaction picture" ..... ???? ..... ????? .......

???? .....

?? "momentum" .... ??? symplectic .... ??? ....

2012-02-03T19:28:00.000-08:00

?? commutative monoids in a symmetric monoidal poset ... as forming ... ???? ... ??? ... ?? "bi-idempotence" ??? ... ???? ....

?? "walking idempotent monoid" monoidal cat =?= "walking monoid" monoidal poset ??? .... ??? ....

?? right adjoint to forgetful 2-fr from _cartesian sym mon poset_ to _sym mon poset_ .... ??? ....

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

2012-02-03T13:32:00.000-08:00

?? growth, change, _movement_, variation ... ??? ....

?? important to remember here (...) .... be very careful around farm machinery .... ???

?? how many of you have ever found yourself doing something like this ??? .... (row of differences ...) ....

2012-02-03T01:38:00.000-08:00

?? "constant of integration" as "differential galois ambiguity" ?? .... ??? ...

?? "solveableness" ... ??? ....

?? "product integral" ... ????? ......

?? analog for [equation with multiple solutions] of idea that possibility of having to solve [equation with no solutions] is in some sense (and to some extent ...) "protected against" by ... ?? fact that "if it never happens then no one will ever try to reverse it" .... ???? ..... ??? analog as maybe that it's _anti_-protected against ?? ... ???? ....

?? was going to mention "maurer-cartan" here but maybe what i was really trying to get at is simply diff eq for translations on given lie group ... ?? ....

?? analog (wrt discrete : continuous analogy ...) of "lagrange extrapolation" here ?? ....

2012-02-02T13:23:00.000-08:00

?? naive idea of "reducing" integration to taking area under curve .... ??? explaining what's wrong with it ... ?? relatively bizzaro formal properties of taking area vs very simple formal properties of integration ... ?? .....

?? relaitonship to "projection-valued measure" idea ?? ..... ????? .... ??? ...

2012-02-02T12:17:00.000-08:00

?? microcosm principle as tirivial for multicategories, less so for .... ??? ....

"1-sided vs 2-sided" .... ??? ....

2012-02-01T15:43:00.000-08:00

?? given a "cocontinuous monad" (?? ...), and a simplicial set, get .... ???? ....

?? kock-zoeberlein analog ??? .... ??? ....

?? "derived" analog ??? ... ???? .... ???? ....

2012-01-26T10:37:00.000-08:00

?? below permutation threshold ??? .... ???? ....

set,span as walking object [?] with something extra ...

?? 4 ??

set,map as walking object [?] with something extra ...

?? sym mon, co-cartesian ???

set,comap ....

?? sym mon, cartesian ??

set,permutation ...

?? sym mon ??

?? sym mon, ... ???????? .....

??? confusion ???? ..... sym mon .... ????? vs co-sym co-mon .... ???? .... ??? ...

??? 1-sided vs 2-sided version of microcosm principle ???? ....

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

???? everything as the walking object something ??? .... ???? .... ???? ....

2012-01-26T01:25:00.000-08:00

euclidean algorithm endless limit ... fibonacci scaling limit ... ?? pentagram picture ??? ....

?? ask baez about economic equilibrium maximizing "consumption rate" ?? ... ??? ...

set, span, span ... ?? macrocosm /microcosms diagram ... ??? ....

2012-01-23T22:17:00.000-08:00

?? single-particle approximation to gas in piston .... ??? ...."maxwell's demon" .... ?????? ......

2012-01-20T18:32:00.000-08:00

set

set, iso

set, morphism

set, co-morphism

set, span, iso

set, span, morphism

set, span, co-morphism

?? ....

?? as stable n-cat, vs as .... ???? ....

set, iso, iso, iso, ... ??? ....

???? ......

?? as symetric monoidal categories:

set,iso ?? object

set,map ?? cocomm comonoid

set,comap ?? comm monoid

set,(iso class of) span ?? bicomm bimonoid

????? .....

2012-01-20T18:22:00.000-08:00

2 3 5 7 11 13 17 19

-

1

0 2

1 0

0 1

1 2

0 0 1

1 1 2

0 2 3

1 0 4

0 1 0

1 2 1

0 0 2

1 1 3

0 2 4

2012-01-18T16:55:00.000-08:00

?? idea that cartesianness of [m,_set_] (????? ....) seems to be coming mainly from co-domeain ...... ???? .... ???? .... ???? ...

2012-01-18T03:57:00.000-08:00

?? kock-zoeberlein monoid vs ... ??? cartesian bimonoidal abstract category .... ??? .... ??? ...

2012-01-17T22:11:00.000-08:00

??? span map ...

01 -> 11 as "co-projection" ... ??

2 -> 1 as "co-diagonal" ... ??

?? co-domain terminal in both cases ??? .... ???? ....

2012-01-17T14:55:00.000-08:00

?? cartesianness ...... ????? ....

?? as property vs as structure .... ???

?? as something that can be (??2-)predicated of a monoidal abstract category in a cartesian monoidal 2-category only because a monoidal abstract category in a cartesian monoidal 2-category is automatically a co-stable bimonoidal abstract category in the underlying stable monoidal 2-category .... ???? ....

?? contractible ..... ???? .... bit about ... "no interesting span-maps between map-spans" (??? ... ??? ....) .... ??? as maybe related to ... certain paradox / confusion about "contractableness" ... "killing homotopy groups" ... ?? "why isn't everything always boring?" .... ???? hurewicz(??...)'s mistake ...

?? crackpot .... ???? .....

?? matrix .... ???

?? relationships among various variants on "matrix" ....

?? matrix with coefficients in given rig / ring ...

?? operation in prop for certain kind(s ...) of bimonoid .... ?? how did that bit go ??? spans with extra total order structure on certain fibers .... ???? ..... idea that didn't quite seem to work .... ????? .....

?? braid ... ??? ...

?? microcosm .... ??? .... ??? hopf .... ???? .... cartesian .... ???? hopf monoid vs mere bimonoid ... ???? .....

?? kennen vs wissen and ... ??somewhere in moore-postnikoff hierarchy ... ??? ....

??? left-universal property of n-cat given by set,span,span, ... ,span,span,map ?? ..... ????? ...... ???? ....

?? some stuff jeff mentioned ... ?? various "reductions" of span concept ... level slip .... ???? ... ??? .....

2012-01-17T05:16:00.001-08:00

?? "taking cat x and replacing each morphism by an adjunction" ... ???? ....

?? walking adjunction as not (?? quite ??? ...) co-category object ... ??? ... ?? whereas if it were then ... ???? ....

??? some (attempted ... ?? ...) variation on "geometric realization" here ... ???? ....

?? star-cat ?? ... ??? gpd ?? .... ???? .... ??? ...

?? cartesian bicat ?? ... ??? ....

?? adjoint adjunctions ... ?? adjoint strings ... ???

?? "riemann-roch algebra" ... ??? ...

?? "hyperdoctrine" ... ??? .... ??? "adjoint hyperdoctrine" ??? .... ???? .... .... ???? "frobenius reciprocity" .... ??? .... ???? ..... ??? ..... ???? .....

2012-01-17T04:14:00.000-08:00

?? "cartesian bimonoid(al abstract cat)" vs "cartesian bicat" ??? .... ???? ....

2012-01-16T17:52:00.000-08:00

?? ok, some sort of conjecture here about "cartesian stable categorified bialgebra" ... ??? possibly (??? particularly if correct ?? ...) tying in with "dna version of n-cat" ??? .... ???? ....

??? set, span, bijection .... ??? stable categorified bimonoid ??

??? set, span, map .... ??? cartesian stable categorified bimonoid ??

??? set, span, span .... ??? bicartesian stable categorified bimonoid ??

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

?? cartesian not parsing for just plain stable monoidal abstract category; need (... ??? ... egger ... microcosm ... ?? ... "just what you need" ... ???? "generic" ???? .... ???? ... ???? ...) rather bistable bimonoidal ??? .... ???? ....

again, just a stupid conjecture at this point ... strongly doubted it at some points, but back to not really having any idea one way or the other at the moment ... ?? ...

?? "can't see span-maps if you only look at map-spans" .... ?????? ...... ????

?? [map-span]-maps as boring .... ???? ...

?? _can_ see span-spans if you only look at map-spans ??? ... ???? ... ???? [map-span]-spans as not "completely boring" ??? .... ???? though not as interesting as they can really get .... ????? ..... ????? ...... ?????? ....

2012-01-15T21:52:00.000-08:00

?? functorial operation on (?? ...) m-sets given by bipartite graph .... ??? ...

2012-01-15T17:43:00.000-08:00

?? "glueing" ... ?? "global vs local" ... ?? "cohomology" ... ?? "clutching" ... "double coset" / "orientation" ... segal and pressley / brown ... ?? ...

2012-01-15T15:10:00.000-08:00

?? operation in prop for bistable bimonoid .... ??? getting adjoint pair of left adjoints from it, by .... ???? ..... ?? function and measure functoriality combined ?? .... ??? ..... ??? topos ... ??? .....

?? "frobenius ..." ??? ....

?? smallness conditions ... ??? .... associated formal duality structures ??? .... ??? ...

2012-01-15T02:21:00.000-08:00

?? strong stable functoriality of categorified "functions" and / or "measures ?? ....

2012-01-14T23:14:00.000-08:00

?? functoriality that we've been trying to exploit (?? to extract cartesian bi-stable categorified bialgebra from comm monoid ... ?? ...) as not quite right one ?? ...

?? instead we should probably try to use either contravariant ("functions" ... ?? ...) or covariant ("measures" ... ?? ...) version of ... ?? set-valued functors on "k(m,1)" ..... ???? .... ???? .....

(?? ... ??? not just "base change" but this extra level shift .... ???? .... ???? .... ?? recently wondered about things like this ... ?? ... ??? hint about unstable case ???? ..... ?? maybe instead of saying "base change" here i should say "choice of coefficients" ... ??? ... ?? realizing now that the level shift here seems strongly related to "hopf / bialgebra duality" ....?? slip vs flip ???? ... z vs z/2 ... ??? ... ?? bott periodicity ??? .... fourier duality ... hopf / bialgebra duality .... ??? .... 2 ... 8 ... 4 ... 2 ... ??? .....)

??? !! finish copying stuff from paper here .... !! ??? ...

2012-01-14T23:10:00.000-08:00

?? various variants of "right adjoint given by taking cocommutative comonoids" ... ??

?? with cocompleteness ?? ... ?? ...

?? stable categorified bialgebra case ??? .... ?? ...

2012-01-14T22:52:00.000-08:00

?? poset case of cartesian bi-stable categorified bialgebra ... ???

2012-01-14T17:17:00.000-08:00

affine vs non-affine in toric case vs in non-toric case .... ??? .... ?? "total distributivity" vs ... ??? hmm, i was going to say "coherent cohomology" ... ?? but ... ??? coherent vs quasicoherent here ??? .... ??? .... not that i know anything relevant here ... ??? ....

2012-01-13T01:35:00.000-08:00

?? field : scheme : stack :: property : structure : stuff ???? ..... ???? .....

?? ?? stack / field ?? ... ??? .... "champs" ... ?? ....

?? field / "local comm ring" .... ??? "localization" / "anti-conservative" .... ??? local vs localization ... ??? ... object vs morphism .... "local localization" ... ??? .... ??? ... moore-postnikov factorization ..... ???? .... ???? ....

2012-01-13T01:28:00.001-08:00

?? bistable "categorified bialgebra" where multiplication is cartesian ??? ....

?? or anything in between "distributive" (?? ...) and "grothendieck topos" ?? ... ????

?? "cocomm comonoid right adjoint to inclusion" game here ?? ... ??? ...

?? "toric category" .... ??? "torus category" if topos is boolean ... ??? ....

?? universal property of particular otric variety ... nice simple / semi-famous examples ... ??? ....

2012-01-11T20:24:00.000-08:00

?? so .... ?? contravariant functoriality of homming ringoid (?? ...) into ringoid of all ab gps ... ???? ... ?? also preservation of tensor product of ringoids ???? ... ???? .... ?? allowing [x,_ab gp_] to become model of p^op (?? ...) where x is a model of prop p ... ??? .....

?? but maybe also some covariant functoriality ?? ... ?? and looking at the codomain slot as variable as well .... sesquivariance .... ???? .... ?? p^op # q ?? ... ??? ....

?? idea maybe we're suggesting module cat of comm ring is more saliently co-monoidal than monoidal ??? ..... ??? weird from "doctrine" viewpoint ?? ... ???? .... ????? ....

[note added several days later : hmm, i was going to add a note here remarking how screwed up this was ... ?? but maybe it's only about as screwed up as we already realized it was at the time ... ?? ...]

2012-01-11T19:59:00.000-08:00

?? "restoration of exactness" / "reconciliation between limits and colimits" .... ??? .... ?? "original sin" .... ??? .... ??? adjoint vs converse .... ????? ...... ???? .....

2012-01-10T21:16:00.001-08:00

?? "black hole" .... ??? "vacuum" ..... ???? ...... "pair production" .... ??? .....

2012-01-10T14:34:00.000-08:00

?? so ... can we actually get nice "categorified bialgebra" from toric variety by taking either ordinary or toric quasicoherent sheaves and dualizing (?? ...) one of {ordinary tensor product, toric convolution} ?? ... ??? does one of these choices work better and if so then which one ?? ...

?? add addendum to thesis sketch about this ?? ...

?? so ... ??? consider accidental topos t of toric variety ... but just as cocomplete category ??? ..... with also "comultiplication" given by ... ?? diagonal functor t -> tXt ... ?? hmmm, maybe not quite make sense yet ??? ...... ?? want to go from t to t#t, not to tXt .... ????? .... hmmmmm ..... ??? ....

?? but now doesn't it seem like ... we're heading in a direction that would work just as well in the non-toric case ??? .... ??? which we're pretty sure that we don't want to be ..... ??? ..... ???? ....

?? or .... ???? ordinary tensor product and this "duplication" operation as having "frobenius" relationship to each other ??? .... ???? whereas .... maybe this "duplication" (??? ...) operation and toric convolution really do have a "bimonoid" relationship ... ??? ..... ???? ..... ?? and maybe it really is non-vacuous for that to happen ..... ????? ...... ????? .....

?? some peculiar confusing things going on here, but ..... ????? ...... duality flips ...... ???? .....

???? "toric duplication" ???? ..... ????? .....

?? stable comonoidal category ..... ???? ....modules of a commutative ring .... ???? more generally, quasicoherent sheaves over a nice (?? ...) scheme .... ???? ..... ??? ....

?? look concretely at alleged "duplication" operation in case of accidental topos of projective line ??? .... ??? ....

?? let's try "the other choice" here ... ?? for example in the affine case ... ?? tensor product of reps of comm monoid by just thinking of them as modules of comm monoid ring .... ?? "duplication" by .... ???????? .... ????? ....

??? adjoint vs converse here ??? .... ????? ....... ????

?? level/duality slip/flip .... ???? ......... ?????? ......

?? comm monoid m .... rep r of mXm .... ???? ..... ?? "first duplicate then tensor, vs vice versa" .... ??? .....

?? duplicate using multiplication, tensor using comultiplication .... ???? slip/flip .... ???? ...

?? accidental topos .... ???? cartesian product and "duplication" ???? ....

??? seem to be transporting bialgebra structure _contra_variantly along "module category" process ??? ..... ????? ..... ??? ....

?? "coefficients" idea here ?? ...

?? algebra / bialgebra duality .... ??? ..... "glueing" ... ?? ... ??? ...

2012-01-09T12:29:00.000-08:00

?? underlying toric quasicoherent sheaf of unit quasicoherent sheaf (?? wrt "ordinary" tensor product ...) over 1d torus ...

?? hmmm .... unit toric quasicoherent sheaf wrt "tensor product" .... corresponding (??? ....) quasicoherent sheaf ....... ?????? ....

?? unit quasicoherent sheaf wrt ordinary tensor product as comonoidal wrt toric convolution ??? ..... ???? ... ?? maybe contrary to something that i wrote the other day ?? ... ... ?? ...

?? still confusion here .... ??? ....

?? hmm, does that (?? namely, idea of ordinary unit being toric convolution comonoid ... ??? ...) conflict with "divergence" idea that we had ?? .... of hypothetical (?? ..) comultiplication .... ???? ....

?? laurent polynomials in x1, x2 as module over laurent polynomials in x ..... ?? ....

???? comonoidal reps of abelian lie alg ...... ?????? ..... ??? then specially nice such ??? ...... as forming accidental topos of correponding torus ..... ????? ....

?? did we accidentally perform some fourier duality flip here ????? .... ????? .... .... acting vs grading .... acting by lattice and grading by torus vs vice versa ...... ???? .....

???? hmmm, acting by lattice and grading by lattice do both enter here (?? ... ??? graded commutative monoid and sections of toric line bundles ... ??? ...) , right ???? .....

??? hmmm, lie algebra idea here as seeming to apply directly only to toruses and not to more general toric varieties ....... ????? ......

????? .....

?? ok, yes we made a silly duality flip here .... quasicoherent sheaf over torus as rep of lattice, not of torus ...

?? so in affine case, simply cocomm coalg acted on by lattice ,,, ??

?? more generally, simply cocomm coalg in accidental topos ... ?? ....

?? "spectrum" interpretation of toric quasicoherent sheaf here ... again ... ?? ... ??? case of ordinary unit ... ?? ....

?? "comultiplication divergence" idea here ??? .... ???? ....

?? these cocomm comonoid cats ... ?? as enriched over plain (...) cocomm comonoids .... ???? ....

2012-01-09T12:26:00.000-08:00

?? "marked moduli stack" (and associated "quantification operations" .... ???? ....) and ... ?? "alg geom" as sort of "substitute" for "logic", vs as "enriched environment" (??? ...) in which for it to take place ... ??? ....

2012-01-08T11:36:00.000-08:00

?? idea of ... ?? ... i was going to say, ... in thinking in recent days about for example trying to interpret various quasi-["algebraic monoid" ... ??? ...]s as giving categorified (?? symmetric but not necessarily co-symmetric ?? ...) bialgebras whose objects are the quasicoherent sheaves over the underlying scheme (?? ...) of the ["alg monoid" ...] ... ?? that it felt as though we were simply dealing with some sort of "base change(s ?? ...)" between a comm ring of "quantities" and a categorified comm ring of "structures" .... ?? and that we should try to figure out what such relevant base change(s ?? ...) might be, although it also feels like there's something a bit wrong with the idea, in that it's hard to imagine what such base changes might be like .... ?? though it might help if we settled for some sort of zig-zag of base changes instead of a single one ... ??? but then i thought, ... but surely we know in principle "what's really going on" in situations of the sort that we're dealing with here, and should be able to make it much more explicit ... ?? namely, in dealing with the ag doctrine, we know how to interpret regular (?? ...) functions and quasicoherent sheaves side-by-side as structure and stuff resepctively ... ?? and we know a bit about how to relate the pure structure world of schemes to the stuff-y world of stacks ..... ????? .... ?? so then how does all this latter stuff relate to the base change / zig-zag idea ??? ..... ???? ....

??? "cheapo categorification" .... ???? fourier duality .... ???? z/2 (?? and / or z/4 ???? .... ?????) vs z .... ???? ...... ???? ...

(?? "stuff as structures" ... ?? "structure as properties" ????? .... ???? .... ??? known parsing glitches here ??? .... ??? ....)

?? polynomial functions form a comm ring, rational functions form a sheaf of local comm rings ..... ???? ...... ?? field ?? .... ?? division ?? .... ..... ???? .... ?? "being able to divide anywhere the value isn't zero" .... ????? ... ???? .... .... lawvere theory, vs .... ???? ..... affine line vs projective line ... (cockett .... ??? .... did i ever remember to write down idea about .... ?? projective line as classifier not for arbitrary partial functions but for something like "openly-defined" such ??? ....)

?? ask yetter about possibility of some sort of informal seminar ... ???? .... ?? hmmm, warn some people (non-[category-theorists], roughly ?? ...) about "mackey effect" here ??? ....

2012-01-07T18:30:00.000-08:00

?? stable categorified hopf algebra associated to abelian variety ?? ... ??? ....

?? hmm, _is_ there really a convolution operation here ?? ... ??? ....

2012-01-07T11:52:00.000-08:00

?? categorified bialgebra associated to toric variety .... ??? ... taking it (the idea ... ?? ...) seriously ...

(?? some confusion here between .... tensor product of comm rings, vs tensor product associated to a comm ring .... ???? ....)

?? module of comm ring (?? also comm monoid case ??? ... ???? ...) giving bi-module in hopefully obvious way .... ???generalization of this to other stable tensor cats ??? ..... ??? is this a significant extra thing to ask for, or more automatic, or what ??? .... ??? .... ?? traditional quasicoherent sheaf case ... ??? ....

?? "tannaka-krein" ideas here ... ??? .... ?? lots of level slipping here ??? .....

?? relationship between ["mayhem vs doctrine" ... ??? ...] and ["operations on models" and / or "multi-model stuff ..." ... ??? .... ??? ...] ?? ..... ??? unclear ?? .... ??? ... to me at the moment, at least .... ??? ....

?? relationship between [relationship between toric quasicoherent sheaf (?? ...) and "toric convolution cocomm comonoid" ... ?? ...] and [relationship between toric line bundles (?? ...) and "graded commutative monoid" ... ?? ...] ?? ... ??? "grading" ... "convolution" .... ???? ..... ??? fourier duality ... grading / acting .... ?? ... ???? ....

?? some days you get the woozle and some days the woozle gets you .... ??? ...

2012-01-06T11:10:00.000-08:00

?? bed of variable height nails and 2-variable calculus .... ???? ....

?? intro and extro ??

?? convolution comonoid on abelian variety associated with finite subgroup ... ??? any others ???? ..... ?? was worried (?? ...) about infinite subgroups, but ... ??? attempted comultiplication seems "divergent" ??? .... ??? maybe that's good ??? ... ??? ... ??? ...

?? maybe get nice topos from abelian variety convolution nice (?? ...) comonoids, but seeing only abstract group structure ???? ..... ???? what _would_ such alleged topos be like, and might this suggest anything interesting about accidental topos of toric variety ?? ....

?? convolution comonoids on discrete (?? fe finite ... ??) comm monoid .... ???? .... "fourier duality" here ... ???? ..... ?? grading vs acting .... ???? ....

?? comvolution comonoids where fiber over unit point is counting measures on a discrete set .... ??? whether such always (??? ...) form a topos ... ???? ..... ???? ....

?? convolution unit as ordinary monoidal ... ?? ordinary-comonoidal too ??? ... ?? or _is_ it ??? .... ?? ordinary unit as convolution-monoidal but not comonoidal ??? .... ???? ..... ?????? .....

?? dual / converse wrt convolution ??? .... ????? ..... ?? toric case where inverses don't always exist .... ???? ....

?? "agfct 0" .... ??? main goal as to get category-theorists interested in alg geometry ... ??? first, by perhaps teaching them some .... toric case as toy case ... good / bad aspects of that .... ?? danger of getting too interested in "special" / "peculiar" aspects (though of course you could try to show that these are actually interesting .... ??? ....) ... ?? second, by revealing (?? to some extent) certain way of relating category theory to alg geom .... ??? though to some extent postpone this till [agfct 1, ...] ... ?? with emphasis on "moduli stack" / "doctrine theory" idea .... ???

?? agfct 1 as about dimensopnal analysis bit ??? ... ?? some chicken / egg expository problems here ??? .... ??? "moduli stack" idea here ... how relates to 0 vs 1 vs .... ???? .... sections of agfct, that is ... ???? ....

?? issue of how [doctrine vs meta-theory of more complicated (?? ... ?? "mayhem ...") meta-doctrine ...] issue relates to [?? ... trying to relate ordinary ag "moduli stack" ideas to toric case as worked out in somewhat hypothetical agfct 0 ... ?? ... ?? "operations on models" idea .... ??? .... as maybe intrinsically mayhem-ish ... ??? ... ?? "(co- ???? ...) walking model" .... ???? ..... ??? categorified bialgebra ??? .... ??? ask yetter about such ???? .....] ... ???? .....

?? try checking up on teaching / employment at ksu math dept website ??? ... ?? applicaiton process ?? ... ?? try comparing listed instructors (??? ...) to listed faculty (?? ...) ....

?? print out ksu math dept grad student application at ucr ....

2012-01-01T14:19:00.000-08:00

?? tag morphisms (?? ...) between accidental toposes and toposes of actions of non-discrete comm monoids and / or semigroups ... ??? .... ?? also glueing together latter sort of toposes .... ??? .... ??? ....

?? "commutative semi-monoidal topos" ... ??? ..... ?? ...

?? hmm, idea that discrete mapping to continuous is in a way more interesting than the other way around, and similarly for lower eilenberg-maclane space mapping into higher ... (?? analogy here ??? ...) ... ?? and that here (?? ...) these two trends sort of conflict ... ??? .... ?? maybe leading to non-interestingness ... ?? ...

?? but ... ?? maybe interesting to at least temporarily forget about "continuous" aspect here and focus on other aspect .... ?? [tag topos (??) given by _set_^n with tensor product coming from commutative monoid structure on n ... ?? ...] mapping (?? ...) into accidental topos of toric variety ..... ???? ..... ??? non-toric analog also ??? .... ??? .... ??? ag theory given by _ab gp_-valued functors on small cat .... ??? .... ??? ...

2012-01-01T14:14:00.000-08:00

?? (?? "small-ly presented" ??? ...) "filteredly cocomplete" (infinity,1)-cat .... ??? .... ?? and (infinity,1)-topos coming from it .... ???? .....

2011-12-31T23:05:00.000-08:00

?? double negation topology and torus of toric variety, vs and forcing ... ??? ....

?? "generic" .... ??? "toric birational geometry" ... ??? "generic point" .... ??? .... ?? zariski locale of comm ring .... ???? ..... ???? .....

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

2011-12-31T13:43:00.000-08:00

?? alleged analogies between 3-manifolds and number fields ... ??? ... ??? maybe ask yetter about this ?? .... ???? other stuff to ask yetter about ?? ...

2011-12-30T11:31:00.000-08:00

?? 2-cat of small (?? ...) cats as sort of zeroth approximation to 2-cat of toposes ... ?? 2-cat of filteredly cocomplete small (?? ...) cats as sort of 1th approximation ??? ....

(?? 0,1 here as intended very informally, though ... ?? ...)

?? toric varieties as nice examples of stuff outside the zeroth approximation because to significant extent (?? though not completely ... ???) stay inside the first approximation .... ??? ....

?? object vs morphism ... ??? ...

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

?? affine ... ???? .... infinity-topos ??? .....

?? graded comm monoid ... toric line bundles .... ??? ....

?? "flat" .... ???? "something going the other way" ... ??? .... ?? string of adjunctions ... ??? ..... ?? geometric vs algebraic-geometric ?? ... ?? .... "fiber as function of codomain point" .... ???? ...

?? ...

?? non-toric analog for (?? way ??) above ... ??? ... ??? zeroth and first approximations to 2-cat of ab cats .... ??? ....

2011-12-29T19:48:00.000-08:00

[toric quasicoherent sheaf topos as nice simple example of non-[totally distributive] topos] as related to [toric quasicoherent sheaf topos as example of topos nicely associated to filteredly cocomplete category] .... ??? ...

2011-12-24T04:54:00.000-08:00

?? accidental infinity-topos of toric p^1 ... ?? ....

?? given "topological dynamical system", consider ... ??? space of "weak z-histories" ... ??? ....

?? vague "quantum measurement" feeling here ?? ...

?? idea of derived category (and / or "weak"-er version ... ?? ...) as about "unification of limits and colimits" .... ??? problematicness of "unstable analog" here ??? .... ??? ....

?? infinity-geometric morphisms between accidental infinity-toposes ... ??? ....

?? "pair of topological dynamical systems ew z-anti-equivalence between their weak z-history spaces" ... ??? ...

2011-12-19T05:41:00.000-08:00

?? rose is a rose is a rose .... rose by any other name .... ??? bijection-invariance ... ??? .....

2011-12-19T02:20:00.000-08:00

?? "structuralism" as means of evading (?? ...) certain issues that lawvere focuses on .... ??? "abstract vs concrete" / "specific vs general" ... ??? in lawvere's sense ... ??? .... ?? ....

tannakian correspondence for toric varieties (sketch for a doctoral thesis)

2011-12-11T15:39:00.000-08:00

in this paper we present some answers to the question: what happens to the tannakian philosophy of algebraic geometry (roughly, "to know a variety is to know the tensor category of quasicoherent sheaves over it") when the varieties that you're studying are toric varieties?

one answer is: an extra "toric convolution" tensor product of quasicoherent sheaves appears. (for example in the case of an affine toric variety, its coordinate algebra is a monoid algebra and thus a bialgebra; toric convolution is then the tensor product associated to the bialgebra comultiplication.)

(an extra "convolution" tensor product of quasicoherent sheaves also appears in the case of for example an abelian variety, though with somewhat different formal properties.)

another answer, more oriented towards the study of toric varieties as objects in themselves than as ordinary varieties with extra structure, is: instead of studying quasicoherent sheaves of modules over a structure sheaf of commutative rings, we study quasicoherent sheaves of actions of a structure sheaf of commutative monoids ("toric quasicoherent sheaves"), and the "tensor category" of these has an underlying grothendieck topos instead of an underlying abelian category.

the most direct relationship between these two answers is that the ordinary k-based quasicoherent sheaves over a toric variety x appear as the k-module objects in the topos t(x) of toric quasicoherent sheaves over x. these k-module objects can be "tensored" in two different ways: either making use of the tensor product on t(x), or in the way that k-module objects in any topos can be tensored. the former amounts to the usual tensor product of quasicoherent sheaves, while the latter is the extra "toric convolution" product.

the topos t(x), when x is non-affine, is among the simplest sort of example of a grothendieck topos that is not a presheaf topos and not "totally distributive". it is glued together from toposes of presheaves over single-object symmetric monoidal categories (aka "toposes of actions of commutative monoids"), but the glueing is along non-essential "localization" inclusions, which results in the lack of total distributivity. this may make t(x) interesting even from the standpoint of pure topos theory.

more generally, although the relationship between algebraic geometry and topos theory discussed here is related to more oft-mentioned such relationships in a somewhat peculiar way, this work does seek to revive a particular form of interaction between algebraic geometry and category theory that has not been developed to its full potential, the idea that the spaces (or more generally "stacks") that algebraic geometers study should be seen as "classifying stacks" for "models" of some kind of "theories" in lawvere's sense (that a "theory" is a category equipped with some sort of extra algebraic structure).

the specific example of this philosophy explored in this paper, focusing on toric varieties, is intended as a toy example, in the same way that toric varieties give toy examples of many phenomena.

(in order to get a fresh viewpoint in this preliminary version of the paper we have to some extent avoided consulting the literature on toric varieties; thus some later rewriting to account for this is to be expected.)

1 construction of the topos t(x) from the fan of a toric variety x

the cones of the fan of a toric variety x form a finite poset under inclusion, or equivalently a finite t0-space, with the open subsets being the downward-closed ones. this finite t0-space is a toric analog of the zariski t0-space of a scheme. being finite it is covered by the minimal neighborhoods of its points, which correspond to the affine open toric subvarieties which glue together to give x.

in this context, the concept of "quasicoherent sheaf of actions of the structure sheaf of commutative monoids of the toric variety x" goes through straightforwardly, in imitation of the usual concept of "quasicoherent sheaf of modules of the structure sheaf of commutative rings of x" for a scheme x. the main difference is that instead of forming a tensored abelian category they form a tensored grothendieck topos.

definition: the "toric quasicoherent sheaves" over x are the tensored grothendieck topos t(x) described above.

alternatively, t(x) can be defined as the filteredly cocontinuous set-valued functors on the cocone category x# of x, which is the category where an object is the cocone dual to a cone in the fan of x and a morphism is a translation map. x# can be recovered as the category of models of the topos t(x).

theorem 1.1: the two alternative definitions of the topos t(x) are equivalent.

2 geometric morphisms from t(x) to t(y)

a geometric morphism from a grothendieck topos t1 to another such t2 is a left-exact left adjoint functor from t2 to t1. as in the ordinary non-toric case, an arbitrary left adjoint functor between categories of toric quasicoherent sheaves has a geometric interpretation as a sort of "correspondence", so a geometric morphism from t(x) to t(y) will have a geometric interpretation as a particular kind of toric correspondence. even if there's no obvious fundamental geometric significance to this particular kind of correspondence, it will be useful to know what it is.

theorem 2.1: consider the (weak) 2-category where an object is a toric variety, a morphism from x to y is a geometric morphism from t(x) to t(y), and a 2-morphism from f:x->y to g:x->y is a natural transformation from the left-exact left adjoint of g to that of f. this 2-category can equivalently be described in the following two ways:

1) a morphism from x to y is a filteredly cocontinuous functor from x# to y#, and a 2-morphism from f:x->y to g:x->y is a natural transformation from f to g.

2) a morphism from x to y is a toric map m from a dense toric open subvariety o of y to x, such that inverse image under f preserves affine toric varieties, equipped with a toric line bundle i over o.

(a "toric line bundle" over toric variety x is an object in t(x) invertible under tensor product. after "toric convolution" is defined, "toric line bundle" can alternatively but equivalently be defined as an ordinary line bundle with a cocommutative comonoid structure under toric convolution.)

a 2-morphism from (o,m,i):x->y to (o',m',i'):x->y requires o contain o' and m' = m restricted to o', and consists then of a morphism from i' to the pullback of i to o'.

composition of 1-morphisms and 2-morphisms is fairly straightforward; thus the composite of (o,m,i):x->y and (o',m',i'):y->z is defined on the intersection of o' and m'^*(o), and the line bundles are tensored after both being pulled back to this intersection.

for many of the ideas in this paper we can ask whether an analog is known or exists in the case of ordinary non-toric varieties; for theorem 2.1 perhaps the right analog would give the geometric interpretation of left-exact left adjoints between categories of ordinary quasicoherent sheaves.

the subtopos of t(y) given as the "image" of a geometric morphism (o,m,i) from t(x) to t(y) is again of the form t(z) for some toric variety z; in fact z=o. (thus (o,m,i) is surjective in the topos-theory sense iff m is totally defined from y to x in the toric variety sense; that is, iff o=y.) in fact, every nonempty subtopos of t(y) is of the form t(z) for some dense open toric subvariety z of y. a prominent example is the "double negation" subtopos of t(y), in which case the corresponding dense open toric subvariety is the dense torus in y; thus the double negation topology is responsible for the "toric" nature of toric varieties.

3 construction of the tensor product on t(x)

in the affine case, the tensor product of toric quasicoherent sheaves is simply the tensor product of actions of the corresponding commutative monoid (that is, day convolution for the commutative monoid viewed as single-object symmetric monoidal category). in the non-affine case, the usual construction of the global tensor product by glueing together the local tensor products goes through straightforwardly.

alternatively, we can understand the tensor product with the help of theorem 2.1.

given a toric variety x, consider the following four partial toric maps:

1) "binary multiplication" xXx -> x. this is total when x is affine but in general only partial; inverse image preserves affineness.

2) "nullary multiplication" 1 -> x. total, and inverse image preserves affineness.

3) "co-binary diagonal" x -> xXx. total, and inverse image preserves affineness.

4) "co-nullary diagonal" x -> 1. total, but inverse image preserves affineness only when x is affine.

applying theorem 2.1 to these four partial maps, we get geometric morphisms from the first three, but from the last one as well only when x is affine.

the partialness of 1) corresponds under theorem 2.1 to the non-essentialness of the co-binary diagonal operation of t(x); this shows that t(x) is totally distributive iff x is affine.

the totalness of 3) gives by 2.1 an essential geometric morphism, making t(x) into a symmetric semi-monoidal topos (which by the failure of 4 to give a geometric morphism is not fully monoidal). the extra left adjoint due to essentialness gives the tensor product of toric quasicoherent sheaves.

the filteredly cocontinuous functor corresponding to this geometric morphism can be thought of as the tensor product of categories enriched over the discrete symmetric closed monoidal category given by the dual lattice of the torus of the toric variety x, interpreting the objects of x# as such enriched categories. this makes x# into a symmetric semi-monoidal filteredly cocomplete category, as the unit object for this tensor product exists as an enriched category but not as an object of x#.

3 relationship of toric quasicoherent sheaves to ordinary quasicoherent sheaves on a toric variety

theorem 3.1: for a commutative ring k, the k-module objects in the topos t(x) are essentially the k-based ordinary quasicoherent sheaves over x.

thus since the ordinary quasicoherent sheaves over x are the k-module objects in a grothendieck topos, they can be tensored together like k-module objects in any such topos. this is not the ordinary tensor product of quasicoherent sheaves, however.

definition: "toric convolution" of k-based ordinary quasicoherent sheaves on a toric variety x is the tensor product arising here.

alternatively, toric convolution can be defined by patching together the "tensor product" functors corresponding to the bialgebra comultiplications on each affine toric open.

taking the free k-module object on a toric quasicoherent and then applying the equivalence of theorem 3.1 gives a quasicoherent sheaf which is cocommutative comonoidal wrt toric convolution. this is a full embedding from the topos of toric quasicoherent sheaves to the cartesian closed category of toric convolution cocommutative comonoids. it would be nice to have a good characterization of the image of this embedding so as to recover the topos t(x) from the k-module-enriched category of k-based ordinary quasicoherent sheaves over x equipped with its two tensor products (ordinary tensor product and toric convolution).

4 toric "proj" construction

let m be a commutative monoid "graded by an abelian group g"; in other words with a homomorphism h to g. this is equivalent to a strict symmetric monoidal category c(h) where all of the objects are strictly invertible and all of the self-braiding morphisms are identity morphisms. (the objects are the elements of g and the hom-set [g1,g2] is the fiber of h over g2-g1.) let t(h) be the symmetric monoidal object in the 2-category of toposes obtained as the presheaf category over c(h).

specialize to the case where g = the integers and grade 1 is finite and generates m, and consider the grothendieck pretopology on c(h) containing for each object n the cover of it by the morphisms from n-1. the sheaf topos for this topology is the "toric proj" construction proj(h). when h is the grading of the homogeneous coordinate monoid of a projective toric variety x, proj(h) is naturally equivalent to t(x) as a symmetric semi-monoidal topos.

addendum 2012-1-11: there should be a lot of material here from the "categorified bialgebra" viewpoint. i neglected this viewpoint originally because it didn't fit with a certain big picture that i was trying to develop, but after further consideration this viewpoint seems too central to the geometry of toric varieties to neglect.

2011-12-05T23:01:00.000-08:00

?? "[n^2]-torsor ew z-frame" ... ?? must have thought about this before ... ??? maybe even sort of noticed the anomalous multiplicity of "irrelevant models" ??? ... ?? and thought it somewhat perverse ?? ... ??? ....

2011-12-05T19:58:00.000-08:00

?? show that nice (??) topos corresponds to nice (??) filteredly cocomplete category, and that .... in sufficiently nice case subtopos corresponds to sub-[filteredly cocomplete category], and that ... ??? in "toric" case it's hopefully more or less obvious that subcategory being closed under filtered colimits implies "compatibility with (?? binary ?? ...) tensor product" .... ??? ... ??? ....

2011-12-05T17:30:00.001-08:00

?? was trying to write up "toric proj" stuff "from filteredly cocomplete picture" when noticed complications .... ??? ....

?? relationship to accidental geometric morphisms associated with toric line bundles ... ?? ...

?? formula / model confusion here associated with "mayhem" ?? ... ??? ....

?? but even more ... ??? bit about "inverse image preserving toric affineness" ... ??? .... ??

?? slice topos approach here ??? ....

?? maybe loss of ... ?? being in nice simple case where "filteredly cocomplete picture" sees everything ... ??? ...

?? double negation topology on graded action topos ??? .... ???? .....

?? so ... ?? slice topos of _set_^[_n_^2] .... ??? over _z_ ??? ....

?? so ... can we make some sort of table of what we think the hom-cats are like .... and the composition functors ... ??? ... ?? hmmm, maybe lots of ... identification of model objects taking place ... under various "localizations" .... ?? try straightening it out ... ??? ...

?? hmm, so maybe ... ?? ... the models of the graded action topos are .... well, "z+1+1+1" ... if you know what i mean ... ??? ....

2011-12-05T07:15:00.001-08:00

? was thinking (for paper ...) about idea that "saying what accidental topos classifies" ("describing arbitrary geometric morphism with accidental geometric co-domain") should subsume "giving toric-geometric interpretation of accidental geometric morphism" ... ?? but probably not really ?? .... ?? or at least not in way that works nicely for paper ... ?? ...

2011-12-04T18:17:00.000-08:00

?? concrete picture of endofunctor corresponding to toric line bundle ... ??? ....

?? p^1 ...

n- 0 0
z z z
0 0 n+

?? identity homomorphism on each diagonal entry .... ??? .... but something non-trivial on [n-,z] and/or [n+,z] ??? ...

2011-12-03T23:27:00.000-08:00

?? so is cockett taking projective line as some sort of "partial function classifier" ?? ...

2011-12-03T07:57:00.000-08:00

?? "toric localness" (?? vs mere localness ?? .... ?? toric localness as stronger ?? ....) .... ?? cockett ??? .... accidental geometric morphism ... ??

??? toric localness _of_ toric convolution, vs _as defined in terms of_ it ... ??? ...

?? various apparent near-paradoxes / confusions here ...

??? ordinary tensor product vs toric convolution ... ?? as associated to geometric morphisms in different ways, so seems problematic for both to acquire their toric localness that way ... ???

?? if toric localness is to do with some sort of compatibility with tortic convolution, then why ordinary tensor product has only lax exchange property with it ?? ... ... ??? more mayhem ... ??? ....

?? way existence of toric convolution is seeming to me now (in terms of toric lcoalness ...) vs ... ?? way it seemed to me originally ?? ... i mean without relying on topos trick ... ?? seemed weird because of partialness of toric multiplication ... ?? but having trouble at the moment seeing how that owuld entenr into any really rigorous (?? ...) treatment .... ??? ....

2011-12-02T14:56:00.001-08:00

?? can "defective modules" of a bialgebra be tensored ??? .... (?? bialgebra structure on v given one on tensor and/or symmetric algebra over v ??? ... ??? ...)

x # b -> x

y # b -> y

x # y # b -> x # y # b # b = x # b # y # b -> x # y ....

?? defectiveness doesn't interfere here ???? ?? geometric interpretation of defective modules ??? ?? non-defective as defective with what sort of extra ... ???? .... mere property, but ... ??? ....

?? egger ... tensoring of modules as related to tensoring of vector spaces .... ??? .....

?? bialgebra homomorphism preserves multiplication and comultiplication ... ?? indiced functor on modules preserves "ordinary tensor product" ... while some adjoint of it preserves "convolution tensor product" ??? ... ??? ... ?? ... ?? try to straighten this out .... ???? .......

2011-12-02T13:09:00.000-08:00

?? to what extent can module category of bialgebra be thought of as "categorification" of the bialgebra ??? ....

?? some very basic presumably old questions here that i don't have much idea how to answer ... ??? ...

?? whatever it is that bialgebra structure on commutative algebra lets you do to modules, to what extent does toric structure on variety let you do it to quasicoherent sheaves ?? ... ??? maybe a lot ??? ... ?? feeling that there's something missing as ... ?? slippery phantom ?? ... ??? ?? co-binary operation adjoint to toric convolution ??? ... ??? ....

?? co-modules as what's missing ?? ... ??? .... ??? .... ??? ....

?? functorial operations preserved by bimodule homs ... and/or "toric maps" ... ??? ....

bimonoid vs bimonoid .... ???? ...hopf ... frobenius ... "algebraic" manifestation of "haar measure" .... compactness ... serre duality ... verdier duality .... ??...

2011-12-02T10:21:00.000-08:00

?? bialgebra module categories ... (left-adjoint) functors preserving toric convolution ....

??"fourier duality" here ?? .... ??? ... ??? some confusion ... ??? co-modules ... ???? .... ??? "dual bialgebra" .... ???? ....

2011-12-02T07:46:00.000-08:00

?? accidental geometric morphism ... ?? both left and right adjoint parts as preserving products, so k-module objects get preserved ... ???? so then does this give an adjunction at the k-module level ??? .... ??? and does the left adjoint preserve toric convolution ??? ... ??? .... ?????... .... ???? ....

?? look at example of affine line vs projective line ... map to terminal toric variety ... ??? ... ?? issue of preservation of affineness by imverse image ... ??? ...

?? tensoring by "toric" line bundle as preserving toric convolution ... ?? is that precisely equivalent to being cocommutative monoid ??? .... ??? .... ???

2011-12-01T11:51:00.000-08:00

?? additional item for section 3 ... ??? modified version of theorem 2.1 ... ?? for symmetric semi-monoidal 2-morphism ... ?? ... ??? but ... ??? confusion about .... ??? whether that's mainly just supposed toget rid of the line bundle, or does it get rid of the toric map as well ??? ..... ??????? ........

?? concrete stuff about toric convolution and functors preserving it ... ?? affine line and projective line cases ??? ....

?? preserving toric convolution alone ... preserving it and also ordinary tensor product .... ?? "mayhem" games ?? ...

?? "mayhem" and "quantum double" ?? ... ??? modules and co-modules together .... ??? .....

?? idea of extra clause in theorem 2.1 concerning functors preserving toric convolution .... ???? ....

2011-11-30T18:32:00.000-08:00

?? so do we really have a reasonable expectation of being able to nicely describe toric map in terms of toric convolution somehow ??? ..... ???

2011-11-30T16:44:00.000-08:00

?? so is it true for abelian varieties that a functor preserves both ordinary tensor product and "convolution" iff it corresponds to "inclusion of a zariski-open subgroup" ??? .... ??? ...

?? first of all, this might be a bad example ... ??? shortage of zariski-open subgroups here ?? .... ???? .... ?? maybe should try something like "algebraic monoid" instead .... ??? .... ??? ....

?? second ... ??? whether the idea makes any sense might depend on ... ??? whether the (?? ...) idea made any sense in the toric case .... ???? ...... ?? "combined doctrine" .... ???? ....

??? confusion about .... "toric map" .... ??? and how such relates to toric convolution .... ??? hmm, taking "toric mayhem" idea somewhat seriously ??? ..... ?? index-raising/-lowering games ... ??? .... ?? relationship between functor f preserving some operation and some adjoint of f preserving some "raising/lowering" of it .... ??? ....

??? "lax interchange law" as maybe hinting that one of the two operations involved has an adjoint (?? ...) that strongly interchanges with the other one .... ??? .....

?? cartesian product as adjoint to diagonal ..... ???? ..... level slip games here ??? .... ??? some sort of "microcosm principle" ??? .... ???? ..... and / or antithesis thereof ... ???? ..... ?? microcosm principle as maybe prototypical level slip ??? ... ??? ....

(reminds me of idea from paper notes that i still haven't copied here yet ... "s |-> _set_^s" as a sort of categorification (2-)functor .... ??? .....)

?? adjoint to toric convolution .... ????? ..... ????? ....

?? well, so what sort of adjoint string are we getting toric convolution and/or cartesian product to belong to, from theorem about accidental geometric morphisms ?? .... ???? ....

?? so ... partialness of toric multiplication in non-affine case as giving non-essential geometric morphism which is "geometric diagonal" of topos .... ???? .... so there's topos-wise f^* and f_* here .... ?? topos-wise f^* for geometric diagonal of a presheaf topos, for example, is ... ?? ... "diagonal presheaf of a double presheaf" ????? ..... ???? hmmm, so _is_ this what cartesian product of presheaves looks like when you "construe it as a single-variable functor" in the way that we've been practicing ?? ....(?? using universal property of tensor product of cocomplete categories ?? .... ??? ....) ... ??? bit hard to tell when everything seems so tautological .... ???? ... ??? more hints about "cartesian microcosm principle" ???? ... ???? ..... ?? let's try assuming so for a moment .... ???? .... ?? then .... ??? f_* as right adjoint to that .... ???? ..... ?? right kan extension ??? .... ?? taking a single presheaf x to the double presheaf x# with x#(y,z) = x(y) X x(z) ??? ..... that was just a stupid guess ... ?? .... ??? diagonal presheaf of representable (by object pair ... ?? ...) double presheaf as ..... ?? hmm, might take a bit of time to straighten this out ... ??? ....

??? "diagonal action of bi-action" .... ?? "diagonal module of bi-module" .... ??? "using comultiplication to tensor modules together" .... ???? ..... ???? ..... ?? then to define cocommutatie comonoids .... ????? .... ??? ... ????

??? "toric hom" ... ??? "toric deconvolution" ??? ..... ???? ...

2011-11-30T14:14:00.000-08:00

??? so to what extent have we been clear up to now that .... ?? whereas "ordinary tensor product" is given by extra left adjoint ("left-left") of essential geometric morphism, toric convolution is given by non-extra left adjoint of (in general ... in particular in non-affine case ... ?? ...) non-essential geometric morphism .... ???? ....

?? of course seems pretty obvious in various ways, but that doesn't guarantee we actually noticed it ... ??? ....

?? relationship to ... "index-raising/-lowering" issues here, and ... ??? associated relationship among various "compatibiltiy conditions" ... "lax interchange" .... ??? .....

?? can we get away here with .... talking about "toric convolution" as though it lived in topos ??? ..... ?? which maybe it sort of does, and is essentially just cartesian product ??? ..... ???? ...

2011-11-30T12:12:00.000-08:00

?? another piece of propaganda for idea that semi-monoidalness of accidental topos isn't that fundamental .... ??? that the _essentialness_ of the "binary multiplication" geometric morphism is crucial in getting tensor product of toric quasicoherent sheaves to exist ... ??? .....

?? we sort of did implicitly almost notice this before ... ??? topos whose diagonal is non-essential, but with essential co-diagonal (though peculiarly not in co-nullary case ... "co-nullary co-diagonal" doesn't exist as geometric morphism ... ???? ....) .... ???? .....

???? so _does_ "second right adjoint of tensor product" generally exist in algebraic geometry ?? ... construing tensor product as working on bimodules .... ?? so i guess that i'm really asking about first right adjoint of pullback along diagonal ...... ????

?? some confusion here ... ???

??? toric case ... length 3 adjoint string ... middle (?? in affine case ... ??? ....) = "pull back along mult hom mXm->m to turn m-set into [mXm]-set .... ??? .....
left-left adjoint as .... ??? ....

??? same triple reverse confusion we keep running into ??????? ......

?? topos-wise we have "tensoring of torsors" as an essential geometric morphism ....
?? the left-left adjoint is "tensor product of filteredly-cocontionuous set-valued functors .... the left-right adjoint is ... "pull back along tensor product of torsors to turn fccsvf of one variable into fccsvf of two variables ... ???

?? toric-wise we have .... left-left adjoint and left-right adjoint together form toric geometric map ....

topos-wise we have f_!, f^*, f_* ??? ....

toric-wise we have f^*, f_*, f^! ??? .....

??? how do general "toric maps" (?? ...) get along with toric convolution, and to what extent does this explain (?? or make even more confusing .... ???? ....) relationship / overlap between toric maps and topos-theoretic geometric morphisms ??? .....

???? functors that get along (?? ...) with toric convolution but _not_ especially with ordinary tensor product .... ?? relationship to topos-theoretic geometric morphisms that .... ?? well, that stretch the relationbship to "toric maps" .... ??? ....

??? hmmm .... ??? so "topos-wise f^*, f_* lining up with toric-wise f^*, f_*" as "combined doctrine" idea ... ?? comparatively rare but "nice" ... ??? ... ?? and fits with (or maybe _is_ ... ???? .... coincides with ... ??? ....) "single functor preserving both ordinary tensor product and toric convolution" .... ????? ....

while, "topos-wise f_!, f^*, f_* lining up with toric-wise f^*, f_*, f^!" is more like "combined mayhem" ... ?? more frequent and "normal", but .... ?? often confusing for perhaps obvious reasons .... ???? ...

?? and then there's the idea that maybe, at least in the toric case, "everything factors into combined mayhem followed by combined doctrine" ...... ????? ..... ??? understanding better why this happens, or at least whether it really does .... ??? .... ?? also understanding overlap / residue given by "line bundles" or whatever ... ??? ....

(?? relationship to general topos pattern of surjection-injection image factorization ???? .... ?? ... adjunction ... co-/monad .... ??? ....)

??? then ... ??? questions about possible non-toric analogs of everything (?? ...) here .... ????? .....

?? in particular, some question that i may have been trying to get at in the first place here .... ?? as to geometric interpretation of "f^!" in general algebraic geometry .... ???? any relationship to "inverse image preserves affineness" ??? ..... ???? ..... and extent to which tendency for ag geometric diagonal to have f^! holds .... ??? and relationship to stuff that simon what's-their-name may have tried to explain to us .... ??? "grothendieck's six operations" ?? ... ??? ... ??? .....

2011-11-30T10:01:00.000-08:00

?? passage from toric variety to accidental topos as .... ??? getting along with "cartesian product" _how_ ??? .... ??? ?? somewhat confusingly ?? ... ??? maybe important to straighten out to some extent for purposes of ... semi-monoidal structure on accidental topoa .... ??? ....

?? hmm, maybe sort of ... but the thing that we're imagining trying to explain here seems so tied up with the thing that we were imagining trying to explain it with that ... ?? well, it's more like there both the same thing to be explained .... ??? ...

?? sort-of paradoxical sort-of contravriant functor sort-of preserving cartesian products .... ??? ....

2011-11-30T07:03:00.000-08:00

?? idea that [realization pairing between accidental topos formulas and models] as equivalent to [pairing "global sections of tensor product" between toric quasicoherent sheaves and the basic localizations among them] as pretty obvious from certain viewpoint ... ?? that of course (?? ...) the formula-model pairing gives the "basic local sections", and of course (? again, certain viewpoint ... ?? ?? hmm, so maybe obvious from certain simultaneous pair of viewpoints ... ??? ...) the local sections are given by "global sections of tensor product" pairing between quasicoherent sheaves and the special ones given by the basic localizations .... ?? but then there's the issue of what structure on those basic localizations you remember or forget .... ??? and it seems that there's some sort of analogy here .... [more rigid structure on basic localization ...]:[less rigid structure on basic localization]::[accidental geometric morphism ... ?? where "line bundle" phenomenon shows up ... ?? ...]:[semi-monoidal accidental geometric morphism ... where "line bundle" degree of freedom (?? ...) gets rigidified ...] .... ??? .... ??? level slip here between model as object and as morphism (?? "single-variable correspondence" ... ?? ...) ... line bundle ... ?? extra nonrigidity seeping into transition functions .... ??? ..... (?? "stackification" .... ??? .....)

localization of comm monoid (?? or ring ?? ... ???? non-troic analog issue everywhere (?? ...) here ...) .... as .... comonoid (?? and / or "frobenius (??bi-??)monoid" ??? ... ???? ...) in module category ... forgetting vs not forgetting that (?? ...) extra structure ..... ??

??? each object as semi-monoid in the semi-monoidal filteredly cocomplete category .... ??? equivalence between quasicoherent sheaves of actions and filteredly cocontinuous functors here ... ???? .....

?? more (explicit ...) in paper about "ulterior ..." ... ?? ...

?? "single-variable correspondence" ... ??? ....

??? "frobenius" ... "bi-lax" ... ???? .... ??? comonoid vs monoid ... ??? ...some confusion or at least unclarity / ignorance .... ??? ....

??? trying to connect up toric line bundle / accidental geometric morphism connection with "toric proj" / "toric serre's theorem" ??? ....

?? "accidental infinity-topos" idea ... ?? maybe all toric opens get promoted to basic ?? ... ??? ...

2011-11-28T17:24:00.000-08:00

?? are we really claiming that geometric morphisms between accidental toposes are surjective precisely in case they're essential ??? .... ?? and if so then why didn't we notice it before ... if we didn't .... ??? ....

?? maybe even for geometric morphisms merely into accidental toposes ??? ...

?? filteredly cocomplete picture here ??? ....

?? well, for merely into .... ??? couldn't you have a geometric morphism defined on a discrete sum, with one component taking care of the surjectiveness, and the other taking care of the non-essentialness ??? ..... hmmmmmm ...... ?????

?? well, what about merely out of ???? .... ?? for merely out of, essentialness clearly doesn't imply surjective ??? ... ?? from looking at affine case ???

?? while for merely into, surjective pretty clearly doesn't imply essential .... ????

?? for merely into, "essential implies surjective" seems plausible at the moment .... ???? ....

?? for merely out of, how plausible is "surjective implies essential" ???? ?? not terribly, at the moment ??? .....

?? surjective as "anti-injective" by factorization theorem ??? ....

?? hmmm, so how does "essential is equivalent to surjective" stand up in case where domain and co-domain are both possibly non-affine accidental toposes ?? ... ???? ...

??? consider inclusion of "plane minus axis" into punctured plane .... ??? ....

?? hmmm ... point included into line .... torus included into line .... ??? both surjective ??? .... ???? .... ???? .....

n to 1 .... "restricting" along this as "inclusion of trivial actions" .... left adjoint to that as .... ?? taking orbits ??? ....

?? coalgebra for comonad here .... ??? right adjoint then left .... ????

2011-11-28T15:04:00.000-08:00

?? so ... ?? "eckmann-hilton" "property vs structure" for accidental topos as more delicate now ??? ..... ?? because of line bundle aspect ... ?? ....

2011-11-28T06:56:00.000-08:00

?? issue of ... pushout of underlying toposes of symmetric monoidal toposes ... ?? circumstances under which this maybe has some reasonable chance of coming close to being symmetric monoidal again ... "glueing of toric varieties" .... ??? non-affine .... ???? why / under what circumstances semi-monoidal but not full monoidal structure (?? ...) survives .... ??? ...

2011-11-28T05:59:00.000-08:00

?? just how annoyingly far would you have to relax the requirements in order to get toric varieties to qualify as monoidal instead of semi-monoidal wrt geometric morphisms ?? ....

?? hmmm, derived / homotopy level here ??? ... ?? "infinity-topos" ?? (?? maybe avoid being pinned down here ... ?? ...) .... ?? maybe mention in paper ... ?? ... ?? "secret motive" squared ???...

?? also mention ... ?? non-toric analog of "toric-geometric interpretation of accidental-topos-geometric morphism" ... ???? ... ?? arguably, left-exact left adjoint ... ?? ...

?? also ... arbitrary grothendieck topology being compatible with "toric structure" ... ?? non-vacuousness of that ??? ... ?? special role of double negation topology ... = "the torus" ... ??

2011-11-28T05:58:00.000-08:00

... i had to use paper for writing notes more than usual in the last couple of days, partially because of one day when my computer wasn't working ... haven't managed to transfer them to here yet, but i should try pretty hard to do that soon ...

2011-11-27T21:03:00.000-08:00

?? if tensor product and toric convolution correspond respectively to "diagonal" and "multiplication", then ... ?? should / do they act just like those act, namely in some sort of "(categorified ... ?? ...) bialgebra" way ???? .... ???? .... ??? index raising/lowering games here ???? ..... ???? ..... ?? "lax interchange" .... ???? .....

?? egger ??? ... ??? micricosm principle ?? .... ??? ...

??? "symmetric semi-monoidal topos" ... ??? semi-_co-_monoidal cocomplete category ??? .... ??? ..... ??? "semi-bi-monoidal ..." ??? .... ???? ..... ???? ....

?? ... adjoints .... ??? .... theory (?? of some doctrine, o maybeof something more general .... ??? ....) formed by some operations together with adjoints of them .... ???? .... ???? ....

2011-11-27T11:58:00.000-08:00

?? some peculiar stuff going on here ... ?? inclusion order structure on subtoposes ... ?? in particular, on (certain ...) 1-morphisms with varying domain but fixed co-domain ... ?? somehow becoming (?? ...) order on morphisms in single hom-space, associated with 2-morphisms .... ??? maybe factorization system where some part is systematically endo ? ... ??? well, maybe already have something ilke that, with line bundles (?? ...), but then also with localizations ?? ... ??? ...

?? "relatively invertible" ... ?? relative inverse ?? ... ?? ...

?? anyway, take another stab here at giving nice explicit description of geometric morphisms and 2-morphisms between accidental toposes .... before trying to test/prove guess ... ?? ...

?? so let x and y be toric varieties ... ?? then we want to describe category of geometric morphisms from accidental topos of x to that of y .... ?? ...

?? so ... object as .... ??? .... ?? dense toric open o of y, together with invertible tqcs i over o, together with "inverse-affine-preserving" toric map m from o to x ....

??? morphism from (o,i,m) to (o',i',m') as ... ?? inclusion of o in o', with m = m' restricted to o, and a morphism f from i to the pullback of i' to o ...

?? maybe hopeless to straigthen out whether f should go that way or the other, on the grounds that usinf the inverse line bundles would switch the convention .... ??? ....

2011-11-27T04:29:00.000-08:00

???? .... accidental geometric morphism .... factorization ... distributivity .... stuff .... flag geometry .... extension .... ??????????????????????? ....

?? confusion about first vs second vs extremes against middle here ... ??? ....

?? trying to understand geometric morphisms _and 2-morphisms between them_ exploiting lowbrow or highbrow treatment of "factorization system" here ...

?? how 2-morphisms get along with factorization of 1-morphisms .... ??? possible funny interaction between variance choice at different levels ?? ...

?? hmmm .... canonical / natural / functorial / 2-functorial (... ?? ...) vs unique (??up to ... ??? ...) factorization ..... ???? .....

?? "strictly canonical factorization" ... ??? including .................... strict equality of intermedite object .... ??? for example canonical surjections and injections, or injections and canonical surjections ..... ???? ....

"borrowing/[acquiring by contamination] [canonical-but-not-unique]ness (... ??? perhaps somewhat different-from-usual (?? perhaps more literal ... ??? explicit act of canonization without which the choice wouldn't stick out ... ?? issue of morphisms being required to preserve the previously (?? ...) arbitrary choice ... ??? ...) meaning of "canonical" here, though might be able to shoehorn it in ... ???) from that of coequalizer of er and/or equalizer of co-er ..." ... ??? .... ?? actually that as funny way to try to put it because of uniqueness for er in turn / in the first place ... ??? ....

?? nonce braiding in monoidal category of endo-modules of discrete category vs of non-discrete .... ???? ...

?? anyway ... ??? (2- ??)category with all morphisms endo ... objects as toric varieties, morphisms as "toric" line bundles ... ?? use in "factorization system" ... ??? as "substrate" and / or as one of the composants ???? .... ?? vague resemblance to .... ?? certain semi-famous examples of group and/or monoid extensions / semi-direct products ... ?? automoprhisms acting on translations ... ??? ..... ??? substrate vs superstrate here ?????? ...... ???? ..... "jordan-holder" and nonce-braiding ..... ???? ..... ?? vaguely reminding me of question coming up in flag geometry ..... ??? "parabolic induction" ??? ..... "what is the extra stuff that [blank] has?" ... ?? having trouble remembering exactly how to formulate the question at the moment, though pretty sure that i could reconstruct it ... ??? "residual geometry" .... ?? in "residual geometry" situation there's some obvious "extra stuff", but then it becomes clear that that's not _all_ the extra stuff ... thus provoking that question .... ??? ....

??? anyway, could certainly try to take very general (?? "highbrow" ?? ...) approach here, but should really mostly sticking with lowbrow for now i think ...

(?? general ... ?? ...)

(?? "educational experiment" ... ?? ...)

?? first try to outline how 2-morphisms between geometric morphisms fit in with surjection/injection factorization in general topos context, then try specializing/adapting (?? ...) to case of accidental topos ... ??? ...

?? so somewhere pretty near beginning of paper (?? new section on geometric morphisms between accidental toposes ?? ...), maybe should introduce "toric" line bundle in bootstrappy way ... ??? as comonoidal wrt toric convolution ??? .... ??? hmmmm, maybe a bit _too_ bootstrappy (in the bad sense ...) ... ?? well, the idea of quasicoherent sheaf of actions instead of of modules is accessible enough, i guess .... ???? ... ?? hmm, "invertible sheaf" terminology as maybe helpful here ?? ...

left exact comonad on geometric domain topos ....

idepotent left exact monad on geometric codomain topos ... .... ?? ...

????? maybe something nice about ... idea of "first tensor by localization of unit object, then tensor by "(relatively- ... ??? ...)invertible" object" ... ?? i meant "nice" just in ... having two vaguely allied things ("localization of unit object" and "invertible object" .... ??? ...) somewhat lining up next to (?? ...) each other in "factorization system" ... ??? .... but ... ?? now also .... ??? ... ?? idea of ... ?? understanding maybe nice interaction here ... ??? this idea of "relative invertibility" .... ?? i was going to suggest "torsor" approach here (?? maybe guided to some extent by "cohomological" (?? ...) spirit ?? ...), but ... ??? maybe (?? categorified ?? ...) addition/multiplicaiton confusion here ??? ....

(?? so ... ?? _are_ "bundles of 1d affine spaces" interesting here (?? ...) ??? ..... ???? and how much did we ever really know about them ?? ... ?? certainly there was _something_ that we thought that we understood about ... short exact sequence approach to "affine vector space" ... and some sort of "cohmology" idea connected with it ... ??? ...)

(?? cockett .... ???? glueing ... ??? trying to get general ag case to work somewhat nicley ... ?? like tag case ??? .... ??? and trying to interpret this as secret behind ... ?? certain stuff ... ?? topos theory in ag ... ?? .... glueing in ag ... ??? lots of things here seem to be having trouble trying get to fit .... ???? ....)

2011-11-26T20:43:00.000-08:00

?? unstackiness and "property vs structure" aspect of eckmann-hilton ?? ....

?? adding new elements and/or new equations to filteredly cocomplete category .... ??? ..... ?? and / or to symmetric semi-monoidal such .... ??? ....

2011-11-26T19:05:00.000-08:00

?? so ... given invertible accidental geometric morphism, consider effect on skyscraper sheaves ... ?? also interaction with double negation topology torus ... ??? ....

2011-11-26T14:02:00.000-08:00

?? so does tensoring with a non-trivial line bundle give an endo-[geometric morphism] of the accidental topos of the projective line, for exsmple ??? .... ??? or auto, maybe ... ?? ...

?? hmmm .... ??? how could it possibly not ?? ... (?? also ... consider non-toric analog .... ??? ....) ... ?? so then ... ?? how complicated is this going to make it giving a nice explicit description (in paper ...) of geometric morphisms in general here ???? ..... ???? .....

?? was going to suggest idea of using "categorified moore-penrose inverse" (somehow ...) here, but .... ?? confusion ??? ...

?? by the way, to what extent have we thought about how toric automorphism group of projective n-space fits in its ordinary (??) automorphism group ?? ... ???? ....

?? kock-zoeberlein .... ???? ....

?? endomorphism vs morphism here ... ???? .....

?? op(?? or sesqui???)-/lax vs strong monoidal structure here .... ???? ...... ... ??? ....

?? idea of getting toric map from surjective (?? ...) geometric morphism here .... ?? how screwed up is that now ??? ...... ??? maybe tensoring with any line bundle gives the identity ???? ..... ??? hmmm, and maybe treating injective and surjective separately makes it easier to keep various sorts of complications (non-trivial line bundle vs sub(?? in _some_ sense .... ??? localization ... ??? ...)-bundle of trivial line bundle .... ??? ...) from interacting with each other too horribly .... ??? but where are we suggesting that the twisting line bundle should live ??? .... i was going to suggest 3 possibilities domain, image, co-domain .... ?? but maybe there's some motivated way to choose one of these as the right one by thinking in terms of pullback ..... ??? so, very naively, seems to me like that would suggest taking line bundle to live on domain of toric map ... ?? on principle that if it lives on one of the other two it can be pulled back to the domain ..... ????? .....

?? hmm, maybe key example is .... geometric morphism from geometrically terminal topos (?? as accidental topos of terminal toric variety ... ??? ...) to accidental topos .... ???? ....

2011-11-26T04:46:00.000-08:00

(?? categorified ...) presentations of filteredly cocomplete categories ...

?? seems like easy to do lots of calculations with them .... ?? already seems to help to (??near-)confirm conjecture about toric map where inverse image preserves affineness ... ??? .....

?? factorization systems here ...

?? theory (?? in various sufficiently rich doctrines ??) of filteredly cocontinuous functor from filteredly cocomplete category x .... ?? duality games here ??? ... ??? ... ??? .....

2011-11-26T04:36:00.000-08:00

?? issue of ... ?? more 2-morphisms between accidental geometric morphisms than i was expecting ... consider affine case for example ... ?? things to think about here, but main idea seems to be that ... ?? am mainly interested in "symmetric semi-monoidal geometric morphisms" here .... ??? ....

?? annoying how this complication bumps into the complication of mere semi-monoidalness vs actual monoidalness ... ?? ...

??? "translation vector space" functor of affine space ... ?? relevant here ??? .... ?? as retraction of ... ???? .....

?? theme of "modules of commutative rings vs representations of groups" here (...) ??? .... ???? ..... (?? ... contrasting flavors of tensor products ... ?? ...)

??? "unstackiness" condition on symmetric semi-monoidal grothendieck topos ... ???

?? hmmm, this "more 2-morphisms between accidental geometric morphisms than i was expecting" issue .... ?? hoping now that it's really more or less just same as issue of what kind of structure correspondences have ... co(???sesqui ???)-/monoidal wrt "ordinary" tensor product, vs ... ???? ... ??? .... ... ??? getting toric convolution involved here ?? ... and / or abelian variety cousin of it ....?? ...

?? idea that has bugged me sometimes .... ?? situation where suddenly revealed cryptomorphism links previously inscrutable concept x to .... about equally inscrutable concept y ... so if you were hoping to reduce complex to simple in most strsightforward way then that's not what's happening here ... ?? but then, arguably, the cryptomorphism can still be highly valuable ... ?? ...

?? hero in disguise/hiding vs in disarray/disgrace ... ?? ambiguity between these ... ??? ....

destry .... ??? hamlet ... superman ... zorro ... pimpernel ... hermes/zeus ... ??? ... kenobi ...

?? ray ... ?? reappearance of hero ?? .... ?? particular example on tip of my tongue .... ??? .... resurrection / regrowth .... ??? ...

2011-11-25T19:50:00.000-08:00

?? "inverse image of affine is affine" ... ??? geometric / algebraic / conceptual aspects .... ??? .... ??? toric vs non-toric case .... ??? ....

?? ... accidental injective geometric morphism ...

?? ... accidental surjective geometric morphism ... ?? lex comonad ... coalgebras .... ?? ...

?? non-affine quasi-affine ... ??? is it true that this gives injective geometric morphism for "domain-narrowing", but not surjective geometric morphism as toric map ??

?? trying to straighten out blame-apportionment ... lex vs left adjoint ... ?? also injective and surjective .... ??? .... ?? toric and non-toric ... ??? ... ??? ... ?? being a (partially-defined) map vs some more general sort of "correspondence" ... ?? ....

?? forcing pullback condition to hold, vs inclusion of those actions satisfying the condition ... ???? ...

2011-11-25T12:33:00.000-08:00

?? so given geometric morphism between accidental toposes ... ?? we think that we can factor this into .... ???? well, thinking in toric picture ... ??? ... "first", narrow domain to dense toric open ... ??? meaning that we've got a geometric morphism that's also a tag morphism .... with the "algebraic left adjoints" coinciding .... ??

??? surjection-injection factorization of geometric morphism between accidental toposes .... ??? ...

?? injecction part as corresponding to narrowing domain to dense toric open .... ??

?? surjectionpart as .... ???? .....

?? tag morphism where algebraic left adjoint has left adjoint ... ?? vs ag morphism where that happens ... ??? vs g morphism where it happens .... ???? ..... ??? geometric interpretations ??? .... ???? .....

?? "enough global functions" ???? ..... ????? .....

?? ag morphism ... geometrically from p^1 to 1 ... ?? algebraically taking vector space to tensor by it of unit quasioherent sheaf .... ?? right adjoint given by ???.... global sections .... ????

??? non-projectiveness of unit object ..... ????? ..... ??? homming from it as not preserving cokernels ????? ?? epi defined as having zero cokernel ??? ... ?? not preserving epis ... ????.....

??? homming from unit object as always (??) right adjoint ... ?? quotient of whether it in turn has right adjoint ????? ....

??? right adjoint to "underlying vector space of module" .... ???? "histories" flavor ??? ....

?? bi-module formulation here ???? .... ??? adjoint bi-module .... ???? ..... ???? .....

?? "topos-geometric morphism fitting together with tag-geometric morphism in way that happens with arbitrary totally-defined toric map between affine toric varieties" ...

?? vs "... in way that happens with arbitrary narrowing of domain to dense toric open" .... ?? ...

??? adjunction / monad /co-monad associated with injective / surjective topos-geometric morphism .... ???? .....

?? somewhere here seemed like i was anticipating ag / tag morphism with "algebraic left adjoint" having extra left adjoint, but then instead seemed to run into its right adjoint having extra right adjoint ..... ???? .... ?????? ......

2011-11-25T04:33:00.000-08:00

?? so we're still trying to straighten out the correspondence between "toric" operations on a toric variety and (so-called ...) "geometric" operations on its accidental topos, with the totally defined among the former corresponding to the essential among the latter ... ?? ...

?? so the toric variety has its binary multiplication operation, totally defined (?? precisely ??) in the affine case ... ?? and also its nullary multiplication operation, always totally defined ... ??? and it has its co-binary diagonal operation, and its co-nullary "total projection" operation ... ?? except that the latter is perhaps not actually toric in the non-affine case ???? .... ?? not quite sure yet ... ?? might depend on haggling over definition of "toric" here ?? ... ?? relationship to issue of "preservation of basicness of toric open subsets by inverse image of toric map" ??? ... anyway, the latter two seem to be totally defined when they exist torically ... ??? ...

anyway, on the other side of the correspondence, we seem (?? with the help of some somewhat wild guesses ...) to have ... ??? the co-binary diagonal operation on the accidental topos, essential precisely in the affine case ... ??? which it would probably be nice if this is strongly linked with the total distributivity of the topos ... ??? ...

?? and we also have the co-nullary total projection operation on the topos ... ?? always essential ???? ...

?? and the binary "multiplication" (?? ...) operation on the topos ...

?? and what about nullary "multiplication" ???? ..... ???? apparently missing in filteredly cocomplete picture ... ??? ...

??? still lots of confusion here .... ???? ....

?? testing above guesses in affine case should be somewhat good consistency check here ?? ...

2011-11-24T03:11:00.000-08:00

?? "correspondence" ... "spectrum" .... ????? .....

?? various ideas about "correspondence" .... ?? one being concept of simply "adjunction" ????? ...... ???? ..... (??? this as similar / related to idea of "sesquiherent sheaf" as correspondence ... ??? ...) ?? but ... ??? well for one thing, consider trying to apply that interpretation of "correspondence" to the case of "pure stuff" ... ag (?? ...) theory of g-torsors for some g some nice "algebraic group" .... ???? ..... ?? say work over field of complex numbers or something like that, maybe .... then underlying cocomplete algebroid of the theory probably becomes something like just a "2-vector space" in (roughly ...) kapranov-vovoedsky sense ... ?? which would make the "correspondences" here rather "bland" ... though this sort of blandness can it seems still be the setting for interesting stuff ... thinking here of vaughan jones / kapranov-vovoedsky / "lowbrow mackay correspondence" style "categorified matrix algebra" stuff ... ?? but ... ?? two extremes here, each annoying in their own way ... ?? ... correspondence category mysteriously knows too much, vs knows too little ..... ???? ....

??? "spectrum" ... ??? "mysterious spectroscopic data" .... (?? "reductionism" .... ?? ...) ... vs "moduli stack" ... "world of possibilities" ... ??? .....

?? idea that "correspondence" should be sensible "logical/geometric" object .... ??? span (?? of groupoids vs of ... ??? ...) idea vs "sesquicoherent sheaf" idea vs "sesquiherent cosheaf" idea .... ???? which of these have good "logical" / "geometric" / "conceptual" status .... ??? maybe first two but not last one ???? ..... ???? .... ??? ...

??? derived level reconciliation between sesquicoherent sheaf and sesquiherent sheaf ??? ....

??? "sensible logical / geometric / conceptual status of (?? specific version of .... ??) correspondence concept" vs (?? ....) .... ?? "doctrine" interpretation of correspondence ??? .... ??? correspondence as simply theory interpretation wrt poorer doctrine .... ??? how this relates to logical / geometric idea .... ?? "hecke operator" .... ??? "orientation" .... ??? .....

??? paper ... ?? arranging certain part of section 1 (?? ...) in something like "dictionary correspondence" form ... ??? so get to mention stuff like ... ??? well, that alleged bit about "affine corresponding to totally distributive" ... ?? in connection with grothendieck topologies, or maybe with toposes ... ??? .... ??? ... also, double negation topology as corresponding to (?? the ?? ...) torus .... ??? ....

?? hmmm .... ?? stating (?? "dictionary correspondence" style ?? ...) stuff in terms of grothendieck topologies vs in terms of geometric morphisms .... latter viewpoint as not completely more general (?? without a bit of going out of way ... ??? ...) in that ... ?? interesting if _all_ grothendieck topologies on toposes with toric interpretation participate in that interpretation ... ?? closedness of sub-2-category here under taking arbitrary sheaf subtoposes ???? .... ??? raising issue of other closedness properties of the (?? ...) subcategory ... ?? ... ?? also (?? thus ?? ...) raising issue of "reflectiveness" of subcategory ... ??? .... and / or of similar issues ... "birkhoff theorem" - style theorems ... ?? closure of semantic category (??) under certain processes / relationships as giving information about nature of corresponding (?? ...) syntactic category ... ??? ... ??? hmmm, extent to which we've already encountered this idea of relating this "birkhoff"-style stuff to "doctrine" and / or "generalized gabriel-ulmer duality" (?? ... ??? level slip ?? ... ???? ..... ?? anyway "daulity" of some kind ... perhaps various kinds .... ??? ...) ideas .... ???? makkai-??? .... .... ???? ..... ???? ....

?? perhaps part of what i meant to be trying to say here was ... ??? ... birkhoffian "closedness" properties of semantic (?? ...) categories as maybe conceptually "dual" (?? or almost so ??? ... ?? some funny aspects ... ?? ... maybe some effortful decategorification ... ?? ...) to abstract syntactic operations (realized (somewhat ... ?? ...) concretely on syntactic categories of theories ...) defining doctrines ... ?? .... .... ??? "concreteness" (??? .... ??? ...) of semantic categories ... ??? anything dual on syntactic side ???? ..... ???? single- versus multi-environment semantics here .... ??? .... single- case and (?? "generalized gabriel-ulmer" ?? ...) "duality" ... ??? ..... ??? .....

??? anyway ... ?? so what about actually trying to take some sort of appropriate "birkhoffian" hull of the toric varieties among the toposes ??? .... ???? ..... ?? if it's not already some such sort of hull, which it likely isn't ... ??? ....

cockett .... ???? ..... partial maps .... ??? .... ?? certain (?? "accidentally" ??? ...) nice things about the toric case .... ?? dense opens and analytic continuation .... ?????? .... ?? connectedness .... ??? (??? new discrete sums ??? ... ????) ... ?? extent to which these might be unavailable in more "adult" context ... ??? ...

"birational toric geometry" as maybe a sort of "homotopy theory" ??? ... ?? model category ?? ... ... ??? .... 2-category .... ??? .... ?? torus as fibrant (?? ...) object or something ???? .....

???? (bad ... ?? ...) idea of regarding general grothendieck topos theory as "noncommutative toric geometry" .... ???? .....

2011-11-23T14:38:00.000-08:00

?? both ordinary tensor product and toric convolution form geometric morphism viewpoint ??? ..... ??? ....

?? reversal games ??? .... ??? .....

2011-11-23T14:17:00.000-08:00

?? so ... ??? toric map from p^1 to 1 ... ???? ..... ?? corresponding geometric morphism ??? ... ?? maybe "global sections" ??? .... ??? as lex left adjoint ??? .... ???? ..... ?? or does this not work ?? ... ???? .... ???? ..... ....

?? trying to straigthen out ... "conservation of basicness" issue .... ??? ..... ... ?? ...

2011-11-23T12:47:00.000-08:00

?? so consider some "anti-conservative" commutative monoid homomorphism ... "does nothing but invert stuff" .... ?? inclusion of n into z should do, i think ... ?? then we want to try to get two distinct geometric morphisms between accidental toposes corresponding to this ... ??

?? so first we get an essential geometric morphism, with lex left adjoint given by pulling back along the homomorphism .... ???

then we cannibalize this a bit ... since it's essential, there's another left adjoint invlved, namely its own left adjoint ... ??? and this is lex just in this special anti-conservative case, i think ... ?? ...

?? so maybe what's going on is something like .... the 2-category of accidental toposes, geometric morphisms, and natural transformations is equvalent to the poset-enriched category where an object is a toric variety, and a morphism is a toric-dense-openly defined toric map, and a 2-morphism is an "extension" relationship ... ?? with maybe some hopefully more or less obvious arrow reversal here ...

?? so this "extension relationship" stuff seems like somewhat of a baby (?? ...) version of idea of "extension of correspondences" ... ??? ....

(?? b(??? ...)-series flag geometry ??? .... ???? ....)

??? and ... we have these adjunctions (?? and asociated co-/monads) in this poset-enriched category .... ??? ..... ?? endo-map of toric variety with domain of definition given by some toric dense open, with the map equal to the identity on its domain of definition .... ??? unit / co-unit here ??? .......

(?? vague memories ... grothendieck and/or lawvere-tierney topology ... ?? analogy to kuratowski closure operator ... anti-kuratowski .... semilattice ... ?? distributive lattice ??? ..... partition of unity ???? ..... ??? qm "observable" ... ??? .... commutative vs noncommutative ... diagonal matrix vs matrix .... ???? ..... projection operator ... 0 / 1 .... ???? .... ???? ....)

?? in other direction get identity morphism ??? .... ??? localization as retract here ???? ..... ????? .....

??? hmmm .... ???? using extension relationship to get quotient (2-)category here ??? .... ??? "decategorificaiton" ??? ... ??? maybe some sort of "half-decategorification" ?? .... ???? model category structure here ????? ......

2011-11-23T09:57:00.000-08:00

?? so ... ??? category of toric varieties and "dense-toric-openly defined toric maps" as full subcategory of (2-)category of toposes ... ???? ....

?? also filteredly cocomplete categories .... ????? ....

2011-11-23T06:15:00.000-08:00

?? at the moment i'm confused about ... ?? toric map from arbitrary toric variety to 0d torus ... whether it's really there ... ??? ... ?? terminalness of 0d torus wrt certain concept of "toric map" ... ??? ... 0d torus .... corresponding commutative monoid as initial, so .... looks like these maps are there in the affine case, but ... ??? non-affine case ??? ..... why _was_ i worrying about this ?? ... ??? something about essentialness ?? ... ??? .... ??? .... ... ... actually i guess because of ... ?? issue of preservation of "basic"ness ... alleged interpretation of topos-geometric morphism between accidental toposes, acting on models, as something like ... ?? "inverse image of basic open under dense-toric-openly defined toric map" ?? .... ???? .... ?? try to straighten out .... ???? ....

2011-11-22T23:48:00.000-08:00

?? so let's try understanding all of the filteredly cocontinuous functors from for example the model category of _set_^z to the model category of _set_^x for some (nice ... toric-geometry-wise ... ?? ...) commutative monoid x, for example ... ?? ... treating exponents here as single-object categories ... ?? ...

this should be in direction that includes ... ?? just plain functors from z to x ..... ??? ..... with toric geometric interpretation as some sort of nice "toric map"s from spec(x) to spec(z) .... ?? so ... ??? we're somewhat expecting / hoping that the general geometric morphisms here are something like "nice partially defined toric maps from spec(x) to spec(z)" .... ???? ....

??? divisor of a "toric meromorphic function" ????? ...... ????? ....

?? " ... fibration ... " ???? .... ???? equivalence between slice categories ... ??? ..... ?? relationship to stuff that cockett may have been hinting at ?? ....

?? so given morphism m from z to groupization of x ... ??? .... ??? consider morphism from z to .... "least groupized version of x for which m exists" .... ??? .... ?? hmmm, but then there's also morphisms from z to "unnecessarily groupized versions of x", is that right ??? ... ?? do these really give distinct geometric morphisms ?? .... ???? .....

?? presumably issue of "do you get to explicitly specify the domain of definition of the partial map ?" .... ????? .....

??? so for example x := n .... ?? zero hom from z to groupization(x) here ?? ... ?? as only example where "unnecessary groupization" is available ??? .... well, in fact only example of hom z -> n, which maybe is more or less saying the same thing ??? ...

??? so .... ?? given n-set s ... ??? construed as filteredly cocontinuous set-valued functor on model category of _set_^n ... ?? .... ??? pull back along two allegedly somewhat different allegedly filteredly cocontinuous functors from model category of _set_^z to model category of _set_^n .... ??? .... ?? to get a z-set, more or less ??? ...

?? "assign to a z-torsor t the set s, and to a z-torsor iso the identity map of s" .... ???? .... ??? vs "assign to a z-torsor t the set "s tensored over n with z", and to a z-torsor iso the identity map of "s tensored over n with z" ..." ... ????? .... ?? really does sound like two inequivalent functors (from ... to ... ?? ...) ... ??? are they really both lex left adjoints ?? ....

?? "assign to an n-set s the z-set "s with trivial z-action"", vs "assign to an n-set s the z-set "[s tensored over n with z] with trivial z-action" ... ???? .... ?? bi-action interpretations here ??? ...

2011-11-22T15:28:00.000-08:00

?? moore-postnikov factorization of filteredly cocontinuous functor between accidental toposes .... ??? corresponding to what factorization of corresponding partial toric map ?? ....

?? might as well throw in moore-postnikov factorization of (?? left adjoint part of ???) topos-geometric morphism here too ??? ....

?? "injective" (topos-)geometric morphism ... ?? ... lawvere-tierney topology .... ?? maybe corresponding to full-and-faithful on model level ?? ... ?? precisely, or more or less ... ??? ... ??? idempotent monadic on formula level ??? .... ???? .....

?? surjective/injective factorization of geometric morphisms as corresponding to

(?? hyperconnected ??? .... ??? ...)

?? hmmm .... you don't get to independently specify domain of definition; it happens automatically ??? .... ??? part of cockett's point ??? ...... ????? ....

???? co-span ???? .... ??? ...

??? map torus of domain to co-domain .... ?? determine domain of definition from that ?? ... ???? ....

?? "toric birational geometry" .... ????....

??? topology on commutative ring as generalized ideal ?????? .... ???? ..... ?? ...

?? topology on commutative monoid ??? .... ???? ..... ??? .... "toric ideal" ... ??? .... ????? .....

?? essentially surjective filteredly cocontinuous functor followed by full-and-faithful filteredly cocontinuous functor .... (?? _is_ "image" here guaranteed to be filteredly cocomplete ??? ....) ??? toric map from toric variety to some (????) completion of another toric variety .... followed by "openly defined isomorphism" ... ???? ..... ???? ..... ??? blow-up/down here somewhere maybe ?? ....

??? (?? weak ?? ...) pullback of co-span here ??? .... ... ?? ...

?? "maximally extended map" .... ??? .... ??? zorn ... cohomology ... ??? .... ?? "obstruction" ... ??? ....

2011-11-22T14:32:00.000-08:00

?? "wearing your goedel number on your sleeve" vs remaining a moving target ... ??? .... ?? arms race ... ??? ....

2011-11-22T00:38:00.000-08:00

?? recently went through episode hallucinating that although tag theories for projective line and other way of glueing together the same pieces (?? ...) are inequivalent, their underlying accidental toposes are equivalent .... ?? so have to be careful about not letting memories of things that i believed during it contaminate my current thoughts .... ??? ....

2011-11-21T02:02:00.000-08:00

?? maybe what we want is something like ... ??? showing that in certain pretty special case, "global sections of tensor product" pairing and "cotqcs/tqcs pairing" coincide ?? ... ??? ...

?? maybe fairly general specialness or more special such, very special to toric variety situation ... ??? ....

?? preservation of filtered colimits by taking global sections ?? ... ?? in certain special case, maybe ?? .... ?? combined with tensoring always (?? ...) preserving colimits ... ?? ... ??? whether filtered colimits of flat things is special case of colimits of general things ... ??? ... ??? ....

?? "flatness" ... ???? .... ?? affineness ... ??? ...

2011-11-20T22:05:00.000-08:00

?? "global sections of tensor product" ... ?? ...

?? try to "use as categorified quadratic form" ... ??? .... problems with non-exactness (before reaching derived level ... ?? ...) ... ?? ... "mixed half-exactness" ....

?? but instead of trying to get "linear" (?? ...) equivalence to own dual (?? ...), trying to get full and faithful embedding .... ??? "parody of yoneda embedding" ... ??? .... ???? ....

???? restricting (?? ...) to "flat" (?? toric ?? ...) quasicoherent sheaves here .... ??? ...

?? how ambiguous is "flat" here ???? ...... ????? .....

tqcs(x) X tqcs(x) -> tqcs(x) -> _set_ ... ??? ....

?????