Sunday, November 6, 2011
?? relationship between tag-geometric and topos-geometric morphism .... ????? 2 contrasting ways of turning comm monoid hom into bi-action ??? ..... ?? and accompanying ways of trying to recover original hom from bi-action .... (??? issue of kinds / degrees of forgetfulness of turning-into processes here .... ??? ...... ?? also somewhat generalized here .... ?? = there ??? .... (?? no ?? ...) .... stackier situations .... ?? ... ?? ... ?? monoidalness structure vs cartesian monoidalness property ... ??? .... ?? ....) .... ?? and for one of the two ways, the turning-into process can be extended to include "localizations" .... ???? .....
?? so what are the alleged two ways ???
1 ?? [f(x),y] .... ???? .....
2 ?? ..... ??? ....
?? different bi-action, vs different way of getting bi-action to act on action .... ?? .... vs ... ???? ....
??? taking half-speed dynamical system of a dynamical system, vs taking double-speed ..... ?? ... ?? double-speed as tensoring with certain bi-action, but is half-speed also ??? ..... ??? maybe converse bi-action ??? ...... ????? .....
(?? funny the way we do seem to have managed to get those two confused ... ??? .... maybe "resolution" instead of "speed" here ??? .... ???? ....)
?? double-speed composed with half-speed .... ???? .... both ways ... ???? .....
?? hmmmm .... original-flavor hecke operator stuff going on here ??? ...... ?????? ...... ????? ......
??? vague heisenberg/hecke/murphy commutation flavor here ... ???? ....
?? hmmmm ... but what about monoid vs group here ??? .... ??? is there already something interesting going on in group case here ????? ...... ?????? ......
?? mono/polynomial vs laurent mono/polynomial here .... ??? .......
?? so what are the alleged two ways ???
1 ?? [f(x),y] .... ???? .....
2 ?? ..... ??? ....
?? different bi-action, vs different way of getting bi-action to act on action .... ?? .... vs ... ???? ....
??? taking half-speed dynamical system of a dynamical system, vs taking double-speed ..... ?? ... ?? double-speed as tensoring with certain bi-action, but is half-speed also ??? ..... ??? maybe converse bi-action ??? ...... ????? .....
(?? funny the way we do seem to have managed to get those two confused ... ??? .... maybe "resolution" instead of "speed" here ??? .... ???? ....)
?? double-speed composed with half-speed .... ???? .... both ways ... ???? .....
?? hmmmm .... original-flavor hecke operator stuff going on here ??? ...... ?????? ...... ????? ......
??? vague heisenberg/hecke/murphy commutation flavor here ... ???? ....
?? hmmmm ... but what about monoid vs group here ??? .... ??? is there already something interesting going on in group case here ????? ...... ?????? ......
?? mono/polynomial vs laurent mono/polynomial here .... ??? .......
?? algorithm as categorified proof, vs ... ???? idea of "correctness proof" (??? ..... ?? and / or "termination proof" .... ???? ....) for algorithm .... ??? ....
?? idea that ... ?? interesting algorithms tend to be hard to be sure that they won't crash ... ??? ....
(?? not really sure whether i agree with that and / or whether it's true .... ??? ....)
?? loss of absolute convincing power ... ???
???? ....... ??? .....
?? idea that ... ?? interesting algorithms tend to be hard to be sure that they won't crash ... ??? ....
(?? not really sure whether i agree with that and / or whether it's true .... ??? ....)
?? loss of absolute convincing power ... ???
???? ....... ??? .....
?? non-toric analog of ... ??? looking at both tag-geometric morphism and topos-geometric morphism as special case of "toric geometric correspondence" .... ??? ....
?? "toric geometric correspondence" as maybe "hecke-flavored" ?? ... ??? ....
?? toric geometric correspondence corresponding (?? ...) to essential topos-geometric morphism between toric varieties as .... ??? ... ?? converse (?? ... ??? .... ??? sesqui- issues here ??? ...) of toric geometric correspondence corresponding (?? ...) to corresponding (??? ...) tag-geometric morphism ??? .... ??? .....
?? "graph of morphism" ... ??? construed as geometric correspondence .... ??? .....
?? test some of this (?? ...) on co- / diagonal example ... ??? .....
?? definitely having some "adjoint vs converse" confusion here .... ??? ......
?? skyscraper sheaf over map graph ... ?? ....
?? "serre duality" .... ????? .....
?? homomorphism of nice comm monoids .... giving essential (topos-)geometric morphism .... ??? with its extra left adjoint being algebraic manifestation of tag-geometric morphism .... ???? ..... hmmmm .... ??? relationship to ideas about "generalized (or even not so generalized ... ?? ...) day convolution" ?? .... ???? ..... ???? ..... co- / diagonal ... ??? ... ..... ???? special case of localization homomorphism ????? ...... ????? ......
?? "toric geometric correspondence" as maybe "hecke-flavored" ?? ... ??? ....
?? toric geometric correspondence corresponding (?? ...) to essential topos-geometric morphism between toric varieties as .... ??? ... ?? converse (?? ... ??? .... ??? sesqui- issues here ??? ...) of toric geometric correspondence corresponding (?? ...) to corresponding (??? ...) tag-geometric morphism ??? .... ??? .....
?? "graph of morphism" ... ??? construed as geometric correspondence .... ??? .....
?? test some of this (?? ...) on co- / diagonal example ... ??? .....
?? definitely having some "adjoint vs converse" confusion here .... ??? ......
?? skyscraper sheaf over map graph ... ?? ....
?? "serre duality" .... ????? .....
?? homomorphism of nice comm monoids .... giving essential (topos-)geometric morphism .... ??? with its extra left adjoint being algebraic manifestation of tag-geometric morphism .... ???? ..... hmmmm .... ??? relationship to ideas about "generalized (or even not so generalized ... ?? ...) day convolution" ?? .... ???? ..... ???? ..... co- / diagonal ... ??? ... ..... ???? special case of localization homomorphism ????? ...... ????? ......
Saturday, November 5, 2011
?? general "toric correspondence" between toric varieties ... ??? in both "toric spectrum" and "topos spectrum" pictures? .... ?? with both tag-geometric morphisms and topos-geometric morphisms as special cases .... ?? what these special cases look like in the "wrong" pictures ??? ....
?? also possible "torus-equivariant geometric correspondence" concept ... ??? .... relationship .... ?? ....
??? torus-equivariant quasicoherent sheaf in torus case ??? .... ???? ....
?? "correspondence vs skew-correspondence" ??? ..... ???? ..... commutativity tricks .... ??? ....
?? fourier duality tricks ... ??? ....
?? "sesquicoherence" issues somewhere here ??? .... ??? ....
?? also possible "torus-equivariant geometric correspondence" concept ... ??? .... relationship .... ?? ....
??? torus-equivariant quasicoherent sheaf in torus case ??? .... ???? ....
?? "correspondence vs skew-correspondence" ??? ..... ???? ..... commutativity tricks .... ??? ....
?? fourier duality tricks ... ??? ....
?? "sesquicoherence" issues somewhere here ??? .... ??? ....
doctrine .... level slip .... "predicate theory vs propositional theory" .... ???.... "intended environment" doctrine morphism .... ??? ....
?? propositional formula interpreted as degenerate predicate formula .... ??? ..... underlying (?? ...) propositional model of predicate model .... hmmm, level slip _here_ ... theory vs model ... ??? vs ... ??? formula vs model ??? .... ???? .....
tannakian twist here ... coalg vs its comodule cat ... ??? ..... ???? ..... doctrine ... ??? .... toric .... ??? ....
?? i think that i'd been going to ask here something about ... ?? how certain level slip stuff here (?? ...) gets along with .... morphisms of some kind ... but at the moment my memory of what i meant by that seems hazy ... ?? ....
?? propositional formula interpreted as degenerate predicate formula .... ??? ..... underlying (?? ...) propositional model of predicate model .... hmmm, level slip _here_ ... theory vs model ... ??? vs ... ??? formula vs model ??? .... ???? .....
tannakian twist here ... coalg vs its comodule cat ... ??? ..... ???? ..... doctrine ... ??? .... toric .... ??? ....
?? i think that i'd been going to ask here something about ... ?? how certain level slip stuff here (?? ...) gets along with .... morphisms of some kind ... but at the moment my memory of what i meant by that seems hazy ... ?? ....
tensor powers ... of bialg .... prop .... as toric convolution powers of unit quasicoherent sheaf over toric variety ... ... .... ???? props involved in non-affine case ???? ...... ..... fan .... ???? ..... ???? ... ??? "re-using fan" ??? ..... ???? see "flat toric monic" below ... ??? .....
?? also projective case and graded bialgebra ... ??? ....
?? hmmm ... ?? this idea above of tensor powers of bistable bialgebra as toric convolution powers of unit ordinary quasicoherent sheaf over toric variety (?? and then maybe doing some stuff with this that i didn't spell out too clearly yet ...) ... ?? combining this with relationship between toric convolution of ordinry quasicoherent sheaves and cartesian product of toric quasicoherent sheaves ... ??? .... hmmmm ..... ??? relationship to ideas about .... ??? ... revisionist idea of defining "group" as "not neccessarily cocommutative hopf object" and recovering original (?? ...) case as special case where symmetric monoidal environment of model has property (...) of being cartesian ... ?? ..... ???? .... ??? .... topos ... ??? .....
what we're supposed to be working on ... ?? tensor product as "generalized day convolution" ... various sort sof geometric interpretations ... trying to straighten out .... for paper .... ???? ....
flat toric monic ... ?? ... ??? does monic requirement for flatness in toric case make good conceptual sense ?? .... ????? .....
?? also projective case and graded bialgebra ... ??? ....
?? hmmm ... ?? this idea above of tensor powers of bistable bialgebra as toric convolution powers of unit ordinary quasicoherent sheaf over toric variety (?? and then maybe doing some stuff with this that i didn't spell out too clearly yet ...) ... ?? combining this with relationship between toric convolution of ordinry quasicoherent sheaves and cartesian product of toric quasicoherent sheaves ... ??? .... hmmmm ..... ??? relationship to ideas about .... ??? ... revisionist idea of defining "group" as "not neccessarily cocommutative hopf object" and recovering original (?? ...) case as special case where symmetric monoidal environment of model has property (...) of being cartesian ... ?? ..... ???? .... ??? .... topos ... ??? .....
what we're supposed to be working on ... ?? tensor product as "generalized day convolution" ... various sort sof geometric interpretations ... trying to straighten out .... for paper .... ???? ....
flat toric monic ... ?? ... ??? does monic requirement for flatness in toric case make good conceptual sense ?? .... ????? .....
Friday, November 4, 2011
?? prototypical "false tannaka(-krein) theorem" as basis of alleged "tannakian philosophy" ??? .... ?? in particular more prototypical than example we first thought of ... toric case ... ?? but ... ?? perhaps not too surprising / perverse ... ?? maybe somewhat familiar pattern .... ??? nilpotent ideals as at first bug rather than feature in "nullstellensatz correspondence" .... ??? ......
?? so ... ?? consider prototypical toric variety x = "the torus" ... ?? and consider all tag morphisms and all topos-geometric morphisms between x and x^2 .... ???? more particularly from x^2 to x ??? ..... well, i guess we should look fairly broadly ... from x to x^2 as well, and considering everything in each adjoint string ... ?? and see whether / where "tensor product of toric quasicoherent sheaves" is showing up here ... ??? ....
?? also ... ??? trying to straighten out lack of / correlation between "flatness" and affineness and so forth here .... ??? ....
?? tag morphisms and topos-geometric morphisms between affine toric varieties / commutative monoids as almost tthe same thing, except "backwards" to each other ??? ..... ?? with the "localizations" as the only reconciliation ... ????? ....
?? topos-geometric morphism from lattice^2 to lattice, simultaneously with tag-geometric morphism
from torus to torus^2 ??? ...... ???? ..... ???? fourier duality ??? ...... ????? ...... ????????? ...... ???? .....
?? also ... ??? trying to straighten out lack of / correlation between "flatness" and affineness and so forth here .... ??? ....
?? tag morphisms and topos-geometric morphisms between affine toric varieties / commutative monoids as almost tthe same thing, except "backwards" to each other ??? ..... ?? with the "localizations" as the only reconciliation ... ????? ....
?? topos-geometric morphism from lattice^2 to lattice, simultaneously with tag-geometric morphism
from torus to torus^2 ??? ...... ???? ..... ???? fourier duality ??? ...... ????? ...... ????????? ...... ???? .....
Thursday, November 3, 2011
?? doctrine of bialgebras .... ???? level slips .... ???? ..... ?????? .......
?? "commutative ring equipped with extra toric convolution operation on its modules" ... ??? ??? toric convolution powers of the cayley-yoneda module ... ???? .... ????? .....
?? "graded commutative ring equipped with extra toric convolution operation on its graded modules" .... ???? .... graded bialgebra ..... ???? ..... ???? .....
?? trying to work out whole bit about ... "toric geometric morphism interpretation of tensor product of toric quasicoherent sheaves" .... ????? .....
?? "commutative ring equipped with extra toric convolution operation on its modules" ... ??? ??? toric convolution powers of the cayley-yoneda module ... ???? .... ????? .....
?? "graded commutative ring equipped with extra toric convolution operation on its graded modules" .... ???? .... graded bialgebra ..... ???? ..... ???? .....
?? trying to work out whole bit about ... "toric geometric morphism interpretation of tensor product of toric quasicoherent sheaves" .... ????? .....
?? so ... for an affine toric variety, the tensor product of toric quasicoherent sheaves is clearly just day convolution wrt the symmetric monoidal structure on the corresponding commutative monoid (as one object category) .... being somewhat careless about opposite categories here ... and this day convolution can be thought of as the extra left adjoint of an essential geometric morphism ..... ???? ..... ?? meaning essentially just a functor, namely that tensor product functor on the one-object category ... ??? ....
?? then non-affine case .... ???? ....
?? also .... ?? geometric morphism here as _flat toric geometric morphism_ .... ????? ..... ??? .... ?? fan picture ?? ... flatness ... ??? contrast between binary and nullary cases ... ???? .....
?? snake eating own tail here ... ??? ....
?? hmmm .... ?? isn't x -> 1 just as flat (?? ...) as x -> x^2 ?? .... ??? no wait, it's flatter .... ?????? ...... ???? .... ???? confusion .... ????? ..... ??? affine vs non-affine here ??? .....
?? linear / bilinear confusion here ??? .... ??? .....
?? so ... ?? possibility what we've got here is the multiplication operations as "toric geometric morphisms" ??? .... ???? 1 -> x as pretty non-flat ... ?? x^2 -> as pretty flat .... ???? .... ??? hmmmm .... ???? torus case vs more general case ???? ...... ???? .....
?? "non-toric analog" .... ???? ..... ??? of ... ??? ..... ??? ... ?? problematic ?? .... ???? ....
?? toric geometric interpretation of essentialness here ?? ..... ??? ...... hmmmm ....
?? weird interrelationship between ordinary tensor product and "toric convolution" here ??? .... ???? ...... ????? .......
?? then non-affine case .... ???? ....
?? also .... ?? geometric morphism here as _flat toric geometric morphism_ .... ????? ..... ??? .... ?? fan picture ?? ... flatness ... ??? contrast between binary and nullary cases ... ???? .....
?? snake eating own tail here ... ??? ....
?? hmmm .... ?? isn't x -> 1 just as flat (?? ...) as x -> x^2 ?? .... ??? no wait, it's flatter .... ?????? ...... ???? .... ???? confusion .... ????? ..... ??? affine vs non-affine here ??? .....
?? linear / bilinear confusion here ??? .... ??? .....
?? so ... ?? possibility what we've got here is the multiplication operations as "toric geometric morphisms" ??? .... ???? 1 -> x as pretty non-flat ... ?? x^2 -> as pretty flat .... ???? .... ??? hmmmm .... ???? torus case vs more general case ???? ...... ???? .....
?? "non-toric analog" .... ???? ..... ??? of ... ??? ..... ??? ... ?? problematic ?? .... ???? ....
?? toric geometric interpretation of essentialness here ?? ..... ??? ...... hmmmm ....
?? weird interrelationship between ordinary tensor product and "toric convolution" here ??? .... ???? ...... ????? .......
Wednesday, November 2, 2011
?? toric convolution on category of vector spaces over a field .... ??? as just ordinary tensor product ?? ... ?? lax interchange structure as standard .... ??? .... ???? theory of object invertible wrt both tensor products ???? ...... ?????? ....... ..... ??? confusion ..... ???? .....
??? "2 kinds of toric spectrum .... preserving toric convolution vs not ..." ....... ?????? ....... ???? .... ??? flat / toric analogy again ??? ...... ???? ...... ???? .....
_propositional toric th_ -> _dim toric th_ -> _toric th_
_prop ag th_ -> _dim th_ -> _ag th_
???? .....
?? hmmm ... ??? .... beck-chevalley ??? .... ???? ... ??? failure thereof ?? ... ??? ...
??? toric convolution for at dimensional and/or propositional levels ??????? ...... hmmmm .... ???? .... ????????? ...... ????? comultiplication ????? ...... ????? ..... ????? ...... ??? non-doctrinalness here ?????? ..... ???? .....
?? doctrine of "intended environments" .... ???? .... ???? .... ??? what was that fairly recent bit about .... ???? .... object vs stack ... ???? ...... ???? ..... ???? .....
??? "2 kinds of toric spectrum .... preserving toric convolution vs not ..." ....... ?????? ....... ???? .... ??? flat / toric analogy again ??? ...... ???? ...... ???? .....
_propositional toric th_ -> _dim toric th_ -> _toric th_
_prop ag th_ -> _dim th_ -> _ag th_
???? .....
?? hmmm ... ??? .... beck-chevalley ??? .... ???? ... ??? failure thereof ?? ... ??? ...
??? toric convolution for at dimensional and/or propositional levels ??????? ...... hmmmm .... ???? .... ????????? ...... ????? comultiplication ????? ...... ????? ..... ????? ...... ??? non-doctrinalness here ?????? ..... ???? .....
?? doctrine of "intended environments" .... ???? .... ???? .... ??? what was that fairly recent bit about .... ???? .... object vs stack ... ???? ...... ???? ..... ???? .....
Tuesday, November 1, 2011
?? alleged semi-monoidal structure on co-fan category .... ??? is it really there ?? .... ???? what's it supposed to be like ???? .....
?? "categorified inf" .... ???? nullary inf as top .... ??? clearly problematic for basic opens of a toric variety .... ?? but hmmm, perhaps the categorified binary inf really does exist here .... ????? .... binary intersection of basic opens .... ???? .....
??? relationship between lax interchange transformation and "generalized day convolution as extra adjoint of essentiaql geometric morphism" .... ???? .....
?? "categorified inf" .... ???? nullary inf as top .... ??? clearly problematic for basic opens of a toric variety .... ?? but hmmm, perhaps the categorified binary inf really does exist here .... ????? .... binary intersection of basic opens .... ???? .....
??? relationship between lax interchange transformation and "generalized day convolution as extra adjoint of essentiaql geometric morphism" .... ???? .....
?? "toric environment" where a model is a pair of models ... ??? or where a model is .... ?? ... "solution to some categorified system of equations for a tuple of models" ... ?? ...
?? "torically algebraically complete" toric environment ... ??? ....
"galois" ideas in toric geometry ... ???
?? actual absolute galois group action ... ???? relationship to stuff like "weil conjectures" and "motive" ... ??? .... toric case ... ??? ...
?? other ideas ... interactions .... kummer's chemistry analogy ... complex multiplication ... abelian langlands .... artin reciprocity ... ??? .... lattice ... braiding ... ???? ....
??? 2-topos .... ??? where semantic 2-cat is "something like" _ag th_, vs where syntactic 2-cat is ... ??? ....
?? "torically algebraically complete" toric environment ... ??? ....
"galois" ideas in toric geometry ... ???
?? actual absolute galois group action ... ???? relationship to stuff like "weil conjectures" and "motive" ... ??? .... toric case ... ??? ...
?? other ideas ... interactions .... kummer's chemistry analogy ... complex multiplication ... abelian langlands .... artin reciprocity ... ??? .... lattice ... braiding ... ???? ....
??? 2-topos .... ??? where semantic 2-cat is "something like" _ag th_, vs where syntactic 2-cat is ... ??? ....
[date uncertain ...]
i often find the n-category cafe somewhat annoying... perhaps for more than one reason, but overwhelmingly for just one reason... the same reason that i tend to find any other participatory writing forum annoying, namely that it's unfairly stacked against me because the medium of communication is writing rather than talking...
anyway, for this reason i sometimes go for long periods without reading it, and sometimes miss interesting discussions that way... and i seem to have just found one that i missed that looks like it might be interesting... on the langlands program... so i want to make a pretty serious effort to go back and read it and try to understand it pretty soon...
there's also "math overflow"... one of my problems with that... basically just a special case of the already mentioned problem... is that by the time i manage to write up a question in presentable form, it usually requires/causes me to already pretty much figure out the answer...
in principle it ought to be possible to overcome that problem to some extent...
i often find the n-category cafe somewhat annoying... perhaps for more than one reason, but overwhelmingly for just one reason... the same reason that i tend to find any other participatory writing forum annoying, namely that it's unfairly stacked against me because the medium of communication is writing rather than talking...
anyway, for this reason i sometimes go for long periods without reading it, and sometimes miss interesting discussions that way... and i seem to have just found one that i missed that looks like it might be interesting... on the langlands program... so i want to make a pretty serious effort to go back and read it and try to understand it pretty soon...
there's also "math overflow"... one of my problems with that... basically just a special case of the already mentioned problem... is that by the time i manage to write up a question in presentable form, it usually requires/causes me to already pretty much figure out the answer...
in principle it ought to be possible to overcome that problem to some extent...
Subscribe to:
Posts (Atom)