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

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

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

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

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

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

????? .....

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

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

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

??? span map ...

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

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

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

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

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

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

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

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

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

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

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

?? 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 .... !! ??? ...

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

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

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

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

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 ... ??? ....

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

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

?? 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 ... ?? ...]

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

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

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

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

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

?? 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 ??? ....

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

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

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

?? 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 dimensional 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 ??? ... ?? application process ?? ... ?? try comparing listed instructors (??? ...) to listed faculty (?? ...) ....

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

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

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