?? extent to which it's cheating / helpful / provable to define "good embedding of 1d object into 2d ..." (for example ... ??? ...) in terms of reps of invertible 2-by-2 triangular matrixes ?? .... .... ??? ... ???? ...
??? _does_ p^1 have nice "partial function classifier" property ??? ... ?? toric case ... ??? constraint involving openly / densely defined .... ??? ....
?? comm monoid hom into .... p+ ... ??? .... n+ ..... ???? ..... ??? .... point .... good path .... ???? ..... ?????
?? homomorphisms from commutative monoid into z ... ??? .... ?? lattice where fan lives .... ???? .... "toric rational functions" .... ????? .... duality (?? to say the least .... ??? ....) confusion here .... ???? .....
?? good paths ... ??? ....
?? commutative monoid homs from z into real^1 and / or complex^1 .... ??? .... some special one ?? ... (?? "evaluation of good path at real and / or complex value k" ??? ... any particular good value of k to use ???? ....k = i ???? .... ?? or other root of unity ??? ... ???? ..... ?? k = -1 ??? .... ??? ....) ... to use to interpret lattice point in fan as ... ?? point of real spectrum ?? ... ?? well, or to relate interpretation as real (or complex ... ?? ...) point to interpretation as good path ... ??? ....
Tuesday, February 28, 2012
?? point/path ambiguity in interpretation of fan ... ?? already worked some of this out, but ... ??? should be some very nice clear picture(s ?? ...) .... ???? ....
?? s^1 spectrum of toric variety ???? .... ??? to what extent _did_ we (explicitly or implicitly ...) work this out ?? ...
?? arbitrary sym mon cocontinuous fr between accidental toposes ... ???? "localize and reassemble" ... ???? try to prove that / check whether this shows arbitrary such fr is "good" ... ??? ...
?? try to exploit further idea of ... ??? developing concepts like "good embedding of dualizable objects" in terms of walking examples .... ??? ....
?? tqs(p^1) .... as sym mon cocomplete cat ... ??? epi from 1+1 to invertible object x with mere _property_ ??? ... ??? .... confusion ... ??? filtered vs graded .... ???? dual object as extra stuff .... ???? .... ?? i guess that part of the point is supposed to be that you don't _need_ any extra stuff but that conjecturally it's convenient to use dual object as such stuff anyway ... then bind it tighter (somehow ... ??? ...) so the extraness goes away .... ??? ...
?? s^1 spectrum of toric variety ???? .... ??? to what extent _did_ we (explicitly or implicitly ...) work this out ?? ...
?? arbitrary sym mon cocontinuous fr between accidental toposes ... ???? "localize and reassemble" ... ???? try to prove that / check whether this shows arbitrary such fr is "good" ... ??? ...
?? try to exploit further idea of ... ??? developing concepts like "good embedding of dualizable objects" in terms of walking examples .... ??? ....
?? tqs(p^1) .... as sym mon cocomplete cat ... ??? epi from 1+1 to invertible object x with mere _property_ ??? ... ??? .... confusion ... ??? filtered vs graded .... ???? dual object as extra stuff .... ???? .... ?? i guess that part of the point is supposed to be that you don't _need_ any extra stuff but that conjecturally it's convenient to use dual object as such stuff anyway ... then bind it tighter (somehow ... ??? ...) so the extraness goes away .... ??? ...
Monday, February 27, 2012
?? _cartesian cocomplete smc_ -> _cocomplete smc_ ... ?? just forget cartesianness property ....
?? _cocomplete smc_ -> _cocomplete sm k-algebroid_ .... ?? take k-module objects ... ?? ....
? right adjoints of the above .... ??? cocommutative comonoids in underlying cocomplete smc of cocomplete sm k-algebroid ??? ....
?? ....
?? _cocomplete smc_ -> _cocomplete sm k-algebroid_ .... ?? take k-module objects ... ?? ....
? right adjoints of the above .... ??? cocommutative comonoids in underlying cocomplete smc of cocomplete sm k-algebroid ??? ....
?? ....
?? trying to [express universal property of qs(x) wrt [ag morphism with right adjoint strong monoidal wrt toric convolution]] for toric variety x as [model of tqs(x) as smc cocomplete cat in smc cocomplete cat of toric-convolution stable comonoids in qs(x) .... ???? ..... ???? .... ??? straightforward adjunction ???? .... ???? ... ??? _is_ this something like the way that it works ??? ....
?? lax interchange .... ??? plus extra mere property ??? ... .... ??? ....
?? lax interchange .... ??? plus extra mere property ??? ... .... ??? ....
Sunday, February 26, 2012
?? case where the two tensor products are the same ... ???? ... ?? what various proposed sortd of morphisms and / or compatibility conditions specialize to in that case ... ??? ...
?? possibility that ... ?? formal properties of morphisms between "toric categories" might improve in some way in "projective" case ??? .... ???? .... ?? or maybe even of the categories themselves ... ?? ...
?? possibility that ... ?? formal properties of morphisms between "toric categories" might improve in some way in "projective" case ??? .... ???? .... ?? or maybe even of the categories themselves ... ?? ...
?? arbitrary sym mon cocomplete cat ...
?? or maybe with underlying cocomplete cat good enough so that ab gp objects can be tensored ..... ??????? .....
??? taking cocomplete algebroid of ab gp objects in it .... ??? ....
?? "toric convolution" live here ???? .... ???? ....
?? ag morphism with right adjoint strong monoidal wrt toric convolution ... ??? .... ?? as arising from arbitrary sym mon coocntinuous fr here ??? ... ??? .... ???? .... ???? .....
?? usefulness of actually having good description of left universal property of sym mon cocomplete cat tqs(p^1) (for example ...) for ... ??? trying to figure out what the "good" morphisms are here .... ????? .....
?? sym promonoidal category .... ??? extra structure making it sym mon ... ??? ... ?? "day convolution" wrt actual tensor product vs wrt pro-tensor .... ??? ... "generalized day convolution" .... ???? ....
?? or maybe with underlying cocomplete cat good enough so that ab gp objects can be tensored ..... ??????? .....
??? taking cocomplete algebroid of ab gp objects in it .... ??? ....
?? "toric convolution" live here ???? .... ???? ....
?? ag morphism with right adjoint strong monoidal wrt toric convolution ... ??? .... ?? as arising from arbitrary sym mon coocntinuous fr here ??? ... ??? .... ???? .... ???? .....
?? usefulness of actually having good description of left universal property of sym mon cocomplete cat tqs(p^1) (for example ...) for ... ??? trying to figure out what the "good" morphisms are here .... ????? .....
?? sym promonoidal category .... ??? extra structure making it sym mon ... ??? ... ?? "day convolution" wrt actual tensor product vs wrt pro-tensor .... ??? ... "generalized day convolution" .... ???? ....
?? so consider ... ??? ag morphism from projective line to "1" ... ?? ....
_set_ -> tqs(p^1) .... ???? "constant" ... ??? ....
[x X n_-] -> [x X z] <- [x X n_+]
?? doesn't preserve terminal object .... ???? .....
?? sense in which process "s |-> sXm" from _set_ to _set_^m (m single-object smc ...) induces process "v |-> v#k[m]" from _k-module_ to [k[m],_k-module_] .... ???? .....
?? some sort of 2-fr f : _cat_ -> _k-algebroid_ .... ??? .... ??? k-module objects in category c .... ???? niceness requirement on cat and / or k-algebroid here ..... ???? .........
?? applying hypothetical f to 1-cell given by "s |-> sXm" ..... ???? ....
?? maybe try case where 1-cell here is a bi-presheaf ..... ????? ..... ?? f simply takes free k-bimodule on bi-presheaf .... ???? ...... ??? but can we adapt this to the case where bimodules are replaced by general cocontinuous functors ?? .... ???? .....
?? trying to use eilenberg-moore category ??? .... ??? canonical presentation ?? ... ??? ....
?? how tensor product ("#") gets along with presentations .... ??? ....
???? simply .... ???? considering how the _right_ adjoint acts on k-module objects, and then taking the left adjoint of that .... ?? ... ?? perhaps also "more explicitly" (?? ...) describable along lines suggested above .... ??? ....
?? global sections ..... ??? ....
?? how automatic _is_ it that there's a good tensor product of k-module objects coming from the tensor product of plain objects ?? ... ??? ...
?? consider for example categorified study multiplication here .... ??? .... hmmmm .... ??? k-module objects maybe work nicely in this case ??? ....
_set_ -> tqs(p^1) .... ???? "constant" ... ??? ....
[x X n_-] -> [x X z] <- [x X n_+]
?? doesn't preserve terminal object .... ???? .....
?? sense in which process "s |-> sXm" from _set_ to _set_^m (m single-object smc ...) induces process "v |-> v#k[m]" from _k-module_ to [k[m],_k-module_] .... ???? .....
?? some sort of 2-fr f : _cat_ -> _k-algebroid_ .... ??? .... ??? k-module objects in category c .... ???? niceness requirement on cat and / or k-algebroid here ..... ???? .........
?? applying hypothetical f to 1-cell given by "s |-> sXm" ..... ???? ....
?? maybe try case where 1-cell here is a bi-presheaf ..... ????? ..... ?? f simply takes free k-bimodule on bi-presheaf .... ???? ...... ??? but can we adapt this to the case where bimodules are replaced by general cocontinuous functors ?? .... ???? .....
?? trying to use eilenberg-moore category ??? .... ??? canonical presentation ?? ... ??? ....
?? how tensor product ("#") gets along with presentations .... ??? ....
???? simply .... ???? considering how the _right_ adjoint acts on k-module objects, and then taking the left adjoint of that .... ?? ... ?? perhaps also "more explicitly" (?? ...) describable along lines suggested above .... ??? ....
?? global sections ..... ??? ....
?? how automatic _is_ it that there's a good tensor product of k-module objects coming from the tensor product of plain objects ?? ... ??? ...
?? consider for example categorified study multiplication here .... ??? .... hmmmm .... ??? k-module objects maybe work nicely in this case ??? ....
[transcribed from paper ... semi-recent ... ?? couple of weeks ago ?? ...]
[m,_set_]__cat_ ?? co-[comm monoid] ?? ... m -> mXm -> _set_X_set_ -> _set_ .... ?? cartesian ...
[comonoid,monoid] ... ???
?? functoriality we've been trying to exploit as not quite right one(s ?? ...) ... ??? (?? and we _are_ jumping into idea about "stably co-(??? note added later : ?? thinko/typo here ??? ... ???) again ... ?? ... geometric morphsm being homomorphism here ... ??? ...) ... ???? "functions" vs "measures" ..... ???? adjunction vs monad ??? .... z vs z/2 ??? .... ??? but just -1,0,1 as reliable ?? ... ??? ... ?? "co-arity" operations as very unavoidably involving "X going to #" ?? ... ?? ...
?? "haar measure" ... ??? "serre duality" ....
?? "kronecker delta" (?? extent of functoriality thereof ... ?? ...) ... "frobenius reciprocity" ... ??? ....
?? specialness of omultiplication |-> specialness of multiplication ... ???? .... ?? diagonal .... cartesian product .... ???? ?? funny level slips ??? ... ??? riding ?? ...
?? "projection formula" ??? ... ???
?? prop for [[?? indecipherable ??] plus bialg] ... ??? ...
?? sets and relations inducing vsps and operators ... ?? and / or categorified ... ?? .... ?? hecke ??? ... ??? categorified bialgebra ... ???? cartesian bicat ... ??? ....
[m,_set_]__cat_ ?? co-[comm monoid] ?? ... m -> mXm -> _set_X_set_ -> _set_ .... ?? cartesian ...
[comonoid,monoid] ... ???
?? functoriality we've been trying to exploit as not quite right one(s ?? ...) ... ??? (?? and we _are_ jumping into idea about "stably co-(??? note added later : ?? thinko/typo here ??? ... ???) again ... ?? ... geometric morphsm being homomorphism here ... ??? ...) ... ???? "functions" vs "measures" ..... ???? adjunction vs monad ??? .... z vs z/2 ??? .... ??? but just -1,0,1 as reliable ?? ... ??? ... ?? "co-arity" operations as very unavoidably involving "X going to #" ?? ... ?? ...
?? "haar measure" ... ??? "serre duality" ....
?? "kronecker delta" (?? extent of functoriality thereof ... ?? ...) ... "frobenius reciprocity" ... ??? ....
?? specialness of omultiplication |-> specialness of multiplication ... ???? .... ?? diagonal .... cartesian product .... ???? ?? funny level slips ??? ... ??? riding ?? ...
?? "projection formula" ??? ... ???
?? prop for [[?? indecipherable ??] plus bialg] ... ??? ...
?? sets and relations inducing vsps and operators ... ?? and / or categorified ... ?? .... ?? hecke ??? ... ??? categorified bialgebra ... ???? cartesian bicat ... ??? ....
Saturday, February 25, 2012
?? adjunction (?? ...) between modules of bialgebra and co-modules of dual bialgebra ??? .... ??? .... ?? hmmm, or vice versa ... ??? not clear how close to being the same .... ?? ...
?? fourier duality .... level slips ....
??? family resemblance between ... ?? geometric interpretation of toric quasicoherent sheaves, and ... ?? geometric rekationship between modules and co-modules .... ???? ...
?? confusion about continuous dual of discrete vs discrete dualk of discrete ....... ??? ....
??? "spin" manifesting as "dimension" ??? ...... ?????? ..... roots of unity .... ??? ....
?? fourier duality .... level slips ....
??? family resemblance between ... ?? geometric interpretation of toric quasicoherent sheaves, and ... ?? geometric rekationship between modules and co-modules .... ???? ...
?? confusion about continuous dual of discrete vs discrete dualk of discrete ....... ??? ....
??? "spin" manifesting as "dimension" ??? ...... ?????? ..... roots of unity .... ??? ....
between accidental toposes ... ?? especially non-affine ?? ...
?? "symmetric semi-monoidal geometric morphism" ... ?? geometricness requirement as not allowing for "cohomological" / "(relatively ...) non-affine" case ... ??? ... ?? absolute vs relative case here ??? .... ??? in absolute case topos manages to keep toposness, whereas in relative case geometric morphism loses its geometricness ??? .... ??? confuision ... level slip ... ?? ....
?? "symmetric semi-monoidal essential geometric morphism" ... ??
?? "essential symmetric semi-monoidal geometric morphism" ... ??
?? "symmetric semi-monoidal essential morphism" ... ?? maybe "correct" ?? ...
?? ambiguities / glitches in parsing some of these terms ?? ... (?? "nullary ambiguity" ?? ...)
?? further complications connected with "semi-" here ?? ...
?? "symmetric semi-monoidal geometric morphism" ... ?? geometricness requirement as not allowing for "cohomological" / "(relatively ...) non-affine" case ... ??? ... ?? absolute vs relative case here ??? .... ??? in absolute case topos manages to keep toposness, whereas in relative case geometric morphism loses its geometricness ??? .... ??? confuision ... level slip ... ?? ....
?? "symmetric semi-monoidal essential geometric morphism" ... ??
?? "essential symmetric semi-monoidal geometric morphism" ... ??
?? "symmetric semi-monoidal essential morphism" ... ?? maybe "correct" ?? ...
?? ambiguities / glitches in parsing some of these terms ?? ... (?? "nullary ambiguity" ?? ...)
?? further complications connected with "semi-" here ?? ...
Friday, February 24, 2012
?? so .... ?? topos picture equivalent of "ag morphism whose right adjoint is strong monoidal wrt toric convolution" as .... ??? strong monoidal left adjoint between accidental toposes, whose right adjoint ... ??? well, maybe two possibilities here, and maybe i think that i know which one is correct ... not the one that i was originally guessing .... vacuous property (my original guess ... "strong monoidalness of arbitrary right adjoint wrt cartesian product as automatic" ... ??? .... but cartesian product vs its cocontinuous extension (again ...) ... ??? this sort of thing as complicating things here again .... ???? ....
some of the ideas sloppily stated here ... as usual ... uncertainties all around ... ??? ....
?? so when does "homming from a bi-presheaf" (in one variable ...) get along nicely (?? ...) with cocontinuous extension of cartesian product ?? .... ?? just that diagonal / diagonal pun ???? .... ??? .... hmm, how messed up _is_ that pun ?? ....
?? vacillating over vacuousness here ... ?? possibility that some tameness factor is making something vacuous here even though the vacuousness arguments that we've been trying to give don't quite make sense ??? ... ??? .... ??? ....
?? toric convolution vs cocontinuous extension thereof ... ??? ....
?? cocontinuous extension of cartesian product (?? along universal bicocontinuous ?? ...) in "discrete" case as "taking diagonal of matrix of sets" .... ????? ....
?? "discrete" case as somewhat antithetical to case we might be interested in ??? ...
?? arbitrary strong symmetric monoidal functor between affine accidental toposes as coming from comm monoid hom ??? ... ?? because of left universal property of accidental topos .... categorified monoid alg of single object smc .... .... ??? ....
?? "fourier duality" argument for why "toric ag morphism" should have right adjoint strongly monoidal wrt toric convolution ?? ... ??? ....
some of the ideas sloppily stated here ... as usual ... uncertainties all around ... ??? ....
?? so when does "homming from a bi-presheaf" (in one variable ...) get along nicely (?? ...) with cocontinuous extension of cartesian product ?? .... ?? just that diagonal / diagonal pun ???? .... ??? .... hmm, how messed up _is_ that pun ?? ....
?? vacillating over vacuousness here ... ?? possibility that some tameness factor is making something vacuous here even though the vacuousness arguments that we've been trying to give don't quite make sense ??? ... ??? .... ??? ....
?? toric convolution vs cocontinuous extension thereof ... ??? ....
?? cocontinuous extension of cartesian product (?? along universal bicocontinuous ?? ...) in "discrete" case as "taking diagonal of matrix of sets" .... ????? ....
?? "discrete" case as somewhat antithetical to case we might be interested in ??? ...
?? arbitrary strong symmetric monoidal functor between affine accidental toposes as coming from comm monoid hom ??? ... ?? because of left universal property of accidental topos .... categorified monoid alg of single object smc .... .... ??? ....
?? "fourier duality" argument for why "toric ag morphism" should have right adjoint strongly monoidal wrt toric convolution ?? ... ??? ....
Thursday, February 23, 2012
?? universal property ... of structure type category wrt proposed morphisms ... ???
?? duality ... z-elt vs z-torsor ... ??? module vs co-module ... ??? .....
?? terry bisson .... ???? vague memories ... categorified fourier duality ... ???? cartesian vs otherwise .... ??? ....
?? equivalent .... in topos picture of proposed morphism in "ag+t" picture ... ??
?? duality ... z-elt vs z-torsor ... ??? module vs co-module ... ??? .....
?? terry bisson .... ???? vague memories ... categorified fourier duality ... ???? cartesian vs otherwise .... ??? ....
?? equivalent .... in topos picture of proposed morphism in "ag+t" picture ... ??
?? ag morphism whose right adjoint is strong monoidal wrt toric convolution ??? .... ?? is this a/the correct idea ??? ..... ??? ....
?? idea of "trying to squeak by with just the middle morphisms" as maybe ruined / complicated by ... ??? ... possible problematicness of "pairing" here ??? ....
?? "operations" sometimes partial ... in one (?? ...) case "cohomological" ... ?? morphisms allowed to be cohomological in general .... ???? .....
?? possible clue to "stuff needed for theory" (?? ...) in this "cohomological" bit ??? .... ???? ....
?? how "semi-monoidal topos" (...) bit compares / relates to other approaches to developing structure of toric category that we've been pursuing ... ?? ...
?? idea of "trying to squeak by with just the middle morphisms" as maybe ruined / complicated by ... ??? ... possible problematicness of "pairing" here ??? ....
?? "operations" sometimes partial ... in one (?? ...) case "cohomological" ... ?? morphisms allowed to be cohomological in general .... ???? .....
?? possible clue to "stuff needed for theory" (?? ...) in this "cohomological" bit ??? .... ???? ....
?? how "semi-monoidal topos" (...) bit compares / relates to other approaches to developing structure of toric category that we've been pursuing ... ?? ...
Wednesday, February 22, 2012
?? toric categorified bialgebra at chain complex level .... ??? .... hmmm ....
?? "middle morphism" morphisms between toric categories ?? ... ??? ... "co-logic" ?? ... ??? ...
compatibility conditions ... ??? ....
?? "inverse image of affine is affine" ... ?? using to assemble single object .... ??? ....
?? "middle morphism" morphisms between toric categories ?? ... ??? ... "co-logic" ?? ... ??? ...
compatibility conditions ... ??? ....
?? "inverse image of affine is affine" ... ?? using to assemble single object .... ??? ....
Tuesday, February 21, 2012
?? so is it a convincing argument that the diagonal of the accidental topos of the projective line (for example) is non-essential because "toric multiplication" is only partially defined on the projective line ?? ....
?? also, ... ??? possibility we just got it backwards as to which half (?? 1/4 ?? ... ???1/6 ??? ...) is missing ?? .... ??? aesthetic / philosophical / ??? preferences as to which way it should be ??? ... ??? "functions vs measures" ideas ??? .....
?? right adjoint of cocontinuous extension of tensor product acts as comultiplication of categorified bialgebra, with cartesian product as multiplication ?? .... ??? .....
???? hmmm, possibility we're really seeing here that we're _not_ (in any obvious way) getting a categorified bialgebra in the non-affine case .... ?? because sometimes (?? binary toric multiplication ?? ...) the "essentialness" is missing, and sometimes (?? co-nullary toric diagonal ?? ...) the "geometricness" is missing ... ??? .... ??? ....
?? and again this should have been pretty obvious all along ??? .... ???? ...
?? on the other hand can we salvage a compatibility relation here .... ???? ....
?? well, we already have the idea of getting a semi-monoidal topos here ... ?? do we get anything beyond that ?? ... still, might be worth making some things (compatibility conditions ...) explicit here ... ??? ....
?? "logic" .... "model" .... confusion .... ???? .... ?? confusion which i think i described somewhat incorrectly recently ... ?? ...
?? derived level .... ??? .... ??? "cohomology" ..... ?????? ......
?? "middle of the morphism" .... ???? ....
?? "mayhem" .... ???? ....
??? hmmm .... ??? cocontinuous extension of cartesian product, and biaction coming from action .... ?? as "main structure" ... (?? "middle of morphism" ... ?? ...) ?? all bialgebra compatibility as parsing here ??? .... ???? "logic" as not obviously really living here .... ?? backwards ??? .....
?? the missing nullary multiplication ..... ??? as .... ???? maybe existing in some "compensatorily shifted" form ??? ... ??? .... ??? ... derived level .... ???? .... ???? .... ???? .....
?? (?? which ?? .... ??? ....) adjoint to "global sections" in affine case ... ??? .... ???? .....
?? also, ... ??? possibility we just got it backwards as to which half (?? 1/4 ?? ... ???1/6 ??? ...) is missing ?? .... ??? aesthetic / philosophical / ??? preferences as to which way it should be ??? ... ??? "functions vs measures" ideas ??? .....
?? right adjoint of cocontinuous extension of tensor product acts as comultiplication of categorified bialgebra, with cartesian product as multiplication ?? .... ??? .....
???? hmmm, possibility we're really seeing here that we're _not_ (in any obvious way) getting a categorified bialgebra in the non-affine case .... ?? because sometimes (?? binary toric multiplication ?? ...) the "essentialness" is missing, and sometimes (?? co-nullary toric diagonal ?? ...) the "geometricness" is missing ... ??? .... ??? ....
?? and again this should have been pretty obvious all along ??? .... ???? ...
?? on the other hand can we salvage a compatibility relation here .... ???? ....
?? well, we already have the idea of getting a semi-monoidal topos here ... ?? do we get anything beyond that ?? ... still, might be worth making some things (compatibility conditions ...) explicit here ... ??? ....
?? "logic" .... "model" .... confusion .... ???? .... ?? confusion which i think i described somewhat incorrectly recently ... ?? ...
?? derived level .... ??? .... ??? "cohomology" ..... ?????? ......
?? "middle of the morphism" .... ???? ....
?? "mayhem" .... ???? ....
??? hmmm .... ??? cocontinuous extension of cartesian product, and biaction coming from action .... ?? as "main structure" ... (?? "middle of morphism" ... ?? ...) ?? all bialgebra compatibility as parsing here ??? .... ???? "logic" as not obviously really living here .... ?? backwards ??? .....
?? the missing nullary multiplication ..... ??? as .... ???? maybe existing in some "compensatorily shifted" form ??? ... ??? .... ??? ... derived level .... ???? .... ???? .... ???? .....
?? (?? which ?? .... ??? ....) adjoint to "global sections" in affine case ... ??? .... ???? .....
?? actually i'm at least momentarily confused ... cocontinuous extension of tensor product as essentialness of geometric morphism in affine case .... ??? ..... ?? then in non-affine case the "essentialness" is still there but the "geometricness" is missing ??? ... ??? .... ?? but then ... ??? where's alleged non-essential geometric morphism here ?? ... ??? ... ???? ....
?? some nefarious fourier duality confusion that i fell for _again_ ?? .... ??? ....
?? hmm, "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." .... ?? certainly _something_ is screwed up here .....
?? some nefarious fourier duality confusion that i fell for _again_ ?? .... ??? ....
?? hmm, "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." .... ?? certainly _something_ is screwed up here .....
Monday, February 20, 2012
?? essentialness of diagonal ... ?? for sober space ....
?? essential inclusion of sober spaces .... ??? essentialness (= extra left adjoint ...) evaluated at total open set x of essential subspace x .... l(x) <= u iff x <= [u intersect x] ..... ???? .... ?? l(t) as smallest open set containing x ??? ....
??? existence of such smallest open set as maybe implying x is itself that open set, given some reasonable "separation property" ???? .... ????? ......
?? maybe converse too ... ???? ...... ???? .... ??? essentialness for inclusion of open subspace as given by .... ??? simply "inclusion", noting that this is totally-defined because relatively open subspace of open suspace is open .... ???? ..... ???? ....
?? maybe explaining to some extent how i've gotten confused about interrelationships among "essentialness" and "openness" and "logicalness" .... ???? .... ??? ... ???? ....
?? open inclusion vs open map .... ???? ....
??? mystical stuff about "open" and "cover" and "grothendieck topology" ...... ????? .....
?? "separation conditions" and compactness .... in topos context ... ?? ....
?? so if (...) essentialness of diagonal is some substantial part of the way towards total distributivity, then what supplementary condition are accidental toposes of non-affine toric varieties missing ?? ... ?? ...
?? essential inclusion of sober spaces .... ??? essentialness (= extra left adjoint ...) evaluated at total open set x of essential subspace x .... l(x) <= u iff x <= [u intersect x] ..... ???? .... ?? l(t) as smallest open set containing x ??? ....
??? existence of such smallest open set as maybe implying x is itself that open set, given some reasonable "separation property" ???? .... ????? ......
?? maybe converse too ... ???? ...... ???? .... ??? essentialness for inclusion of open subspace as given by .... ??? simply "inclusion", noting that this is totally-defined because relatively open subspace of open suspace is open .... ???? ..... ???? ....
?? maybe explaining to some extent how i've gotten confused about interrelationships among "essentialness" and "openness" and "logicalness" .... ???? .... ??? ... ???? ....
?? open inclusion vs open map .... ???? ....
??? mystical stuff about "open" and "cover" and "grothendieck topology" ...... ????? .....
?? "separation conditions" and compactness .... in topos context ... ?? ....
?? so if (...) essentialness of diagonal is some substantial part of the way towards total distributivity, then what supplementary condition are accidental toposes of non-affine toric varieties missing ?? ... ?? ...
?? universal property of [category of structure types] as "toric category" ... ??? ...
??? relationship between operational structure on object wrt operation o (?? microcosmic level shift here ?? ...) and ... ??? co-operational structure wrt (left ?? ...) adjoint operation o' ?? ... ???? .... ???? .... co-monoidal category ... microcosm ... level slips/shifts ... ??? ..... cartesian ..... 2-sided .... ??? .... ??? ...
?? "co-x object" vs "co-object" ... ??? ....
l(x) -> y .... x -> r(y) ....
??? x#x .... ??? ....
f(x) <-> x#x .... ??? ....
x <-> g(x#x) ..... ???? .....
l(a) -> b#c ....
a -> r(b#c)
?? r categorified multiplication, l categorified comultiplication .... ??? .... ??? ....
"operad" .... ?? "co-operad" ... ???? .... ???? ....
?? diagonal and cartesian product here .... ??? .... .... hmmmm ..... ???? ... ??? model of operad in (_set_^op,X) ... and / or in (_set_,+) .... ???? .... ???? .... ??? "component" of operation .... ??? ....
??? relationship between operational structure on object wrt operation o (?? microcosmic level shift here ?? ...) and ... ??? co-operational structure wrt (left ?? ...) adjoint operation o' ?? ... ???? .... ???? .... co-monoidal category ... microcosm ... level slips/shifts ... ??? ..... cartesian ..... 2-sided .... ??? .... ??? ...
?? "co-x object" vs "co-object" ... ??? ....
l(x) -> y .... x -> r(y) ....
??? x#x .... ??? ....
f(x) <-> x#x .... ??? ....
x <-> g(x#x) ..... ???? .....
l(a) -> b#c ....
a -> r(b#c)
?? r categorified multiplication, l categorified comultiplication .... ??? .... ??? ....
"operad" .... ?? "co-operad" ... ???? .... ???? ....
?? diagonal and cartesian product here .... ??? .... .... hmmmm ..... ???? ... ??? model of operad in (_set_^op,X) ... and / or in (_set_,+) .... ???? .... ???? .... ??? "component" of operation .... ??? ....
Sunday, February 19, 2012
?? essentialness of diagonal ..... ????? ....
?? theorem about when classical model category of topos of filteredly cocontinuous set-valued functors on x is equivalent to x as expected to have "at least as many complications" as diaconescu's theorem .... ???? ....
?? diaconescu's theorem for locales ??
?? discussions of essential inclusions between sober spaces in johnstone's or other books maybe ?? ...
?? theorem about when classical model category of topos of filteredly cocontinuous set-valued functors on x is equivalent to x as expected to have "at least as many complications" as diaconescu's theorem .... ???? ....
?? diaconescu's theorem for locales ??
?? discussions of essential inclusions between sober spaces in johnstone's or other books maybe ?? ...
?? 3/4 ... herring (red) .... derived .... claws (hanging on by ...) ......... boolean ....
?? presheaves on single-object symmetric monoidal category x as same as for x^op ... ??? not so in multi-object case ??????? ..... ????? .... ????? ..... i guess : single-object smc as self-opposite; not so in multi-object case ?? ... ???? ... ??? identity fr as "star-structure" ??? ..... ????? ...... ???? ..... ??? "serre duality" and inverse line bundle ??? .... ????? .....
?? so i'm hopeful that a bunch of stuff (about the algebraic structure on the category tqs(x) of "toric quasicoherent sheaves" over a toric variety x, roughly) is fitting together reasonably well now ... i'll try to describe it here ... it seems to be fitting together well enough to give me that feeling of "why didn't i see it before when it seems so clear now?" (and associated feeling of "everyone (or at least someone) probably knows this already") even though i think that i actually understand some of the reasons that i didn't see it all so clearly before ...
it seems useful to organize the ideas here by first considering the ("natural" in some sense ... though there's some trickiness involved here ...) algebraic structure on tqs(x) when x is affine, and then picking out from among that considerable amount of structure the part that survives to the non-affine case ...
in the affine case, tqs(x) is a symmetric monoidal object in the (cartesian symmetric monoidal) 2-category where the objects are the totally distributive toposes and the morphisms are the essential geometric morphisms ...
(more generally, the category of presheaves on any symmetric monoidal small category is such a symmetric monoidal object ... for pretty straightforward reasons ...)
in the non-affine case, tqs(x) is a symmetric monoidal object in the (symmetric monoidal) 2-category where the objects are the small-cocomplete categories and the morphisms are the left-adjoint functors, and it's simultaneously also a grothendieck topos whose diagonal geometric morphism is essential, with the extra left adjoint of the diagonal acting as a categorified comultiplication engaged in categorified bialgebra compatibility with the tensor product acting as categorified multiplication.
it seems reasonable to take more or less the above description as an axiomatic definition of some sort of "generalized toric category" ... more, in that for example we might want for some (not all ...) purposes to impose the further condition that the double negation topos is a categorified hopf algebra ... less, in that for example, we might want to separate out the categorified bialgebra and the topos as separate categories, conceptually "dual" to each other in some sense ... not sure how that idea might play out ...
there's a bit more of the structure in the affine case that survives to the non-affine case, but it seems not so fundamental offhand ... the totally distributive topos with essential symmetric monoidal structure hangs on just barely as a non-totally-distributive topos with non-essential symmetric semi-monoidal structure ... one of the reasons that this seems less fundamental is because of the (contestable ...) philosophy that "semi-" concepts are less fundamental ... still, you could try incorporating this extra structure in the axiomatic definition ...
?? hmmm .... ?? 3/4 vs ?/6 ??? .... ???? .... essential geometric morphism as made of three parts ... ??? ....
?? stuff about cartesianness of categorified bialgebra as red herring ???? ... ???? .... still ..... ???? .... ???? ...... ??? trying to study cateogrified prop of natural operations here .... ??? "distributive category" ... ?? ....
?? idea that at derived level, there really is more surviving ... ???? 4/4 ??? .... ??? symmetric monoidal topos ???? ..... ???? ..... .....
?? extra left adjoint of essential diagonal acting as categorified comultiplication with tensor product acting as categorified multiplication .... ??? vs ... ??? "maxwell" ... ??? .... [right adjoint of cocontinuous extension of tensor product] being also a left adjoint of something, so that it can act as a categorified comultiplication with cartesian product acting as categorified multiplication .... ?? see above bit about derived level ... ?? where distinction between left and right adjoint maybe more or less vanishes, thus restoring full 4/4 or 6/6 or whatever .... ??? ... ... ??? ....
?? 6/6 ... ??? grothendieck's 6 operations ??? ... ??? .... perhaps not ... ?? ....
?? checking notes from december for fit with current picture ... ?? "generalized day convolution" .... ??? tensor product as "essentialness" of something ..... ????? ....
?? getting more explicit about how the structure described in the non-affine case fits as part of the structure described in the affine case ... the "left-adjoint functors" as the _"essentialness"_ of the corresponding "essential geometric morphisms" ..... ????? .....
?????? extra left adjoint of cartesian product vs extra _second_ right adjoint of tensor product ..... ????? extra right adjoint of .... ??? candidate for certain categorified coproduct ... ?? ....
???? maybe 5/6 ??? ..... ????? .....
?? presheaves on single-object symmetric monoidal category x as same as for x^op ... ??? not so in multi-object case ??????? ..... ????? .... ????? ..... i guess : single-object smc as self-opposite; not so in multi-object case ?? ... ???? ... ??? identity fr as "star-structure" ??? ..... ????? ...... ???? ..... ??? "serre duality" and inverse line bundle ??? .... ????? .....
?? so i'm hopeful that a bunch of stuff (about the algebraic structure on the category tqs(x) of "toric quasicoherent sheaves" over a toric variety x, roughly) is fitting together reasonably well now ... i'll try to describe it here ... it seems to be fitting together well enough to give me that feeling of "why didn't i see it before when it seems so clear now?" (and associated feeling of "everyone (or at least someone) probably knows this already") even though i think that i actually understand some of the reasons that i didn't see it all so clearly before ...
it seems useful to organize the ideas here by first considering the ("natural" in some sense ... though there's some trickiness involved here ...) algebraic structure on tqs(x) when x is affine, and then picking out from among that considerable amount of structure the part that survives to the non-affine case ...
in the affine case, tqs(x) is a symmetric monoidal object in the (cartesian symmetric monoidal) 2-category where the objects are the totally distributive toposes and the morphisms are the essential geometric morphisms ...
(more generally, the category of presheaves on any symmetric monoidal small category is such a symmetric monoidal object ... for pretty straightforward reasons ...)
in the non-affine case, tqs(x) is a symmetric monoidal object in the (symmetric monoidal) 2-category where the objects are the small-cocomplete categories and the morphisms are the left-adjoint functors, and it's simultaneously also a grothendieck topos whose diagonal geometric morphism is essential, with the extra left adjoint of the diagonal acting as a categorified comultiplication engaged in categorified bialgebra compatibility with the tensor product acting as categorified multiplication.
it seems reasonable to take more or less the above description as an axiomatic definition of some sort of "generalized toric category" ... more, in that for example we might want for some (not all ...) purposes to impose the further condition that the double negation topos is a categorified hopf algebra ... less, in that for example, we might want to separate out the categorified bialgebra and the topos as separate categories, conceptually "dual" to each other in some sense ... not sure how that idea might play out ...
there's a bit more of the structure in the affine case that survives to the non-affine case, but it seems not so fundamental offhand ... the totally distributive topos with essential symmetric monoidal structure hangs on just barely as a non-totally-distributive topos with non-essential symmetric semi-monoidal structure ... one of the reasons that this seems less fundamental is because of the (contestable ...) philosophy that "semi-" concepts are less fundamental ... still, you could try incorporating this extra structure in the axiomatic definition ...
?? hmmm .... ?? 3/4 vs ?/6 ??? .... ???? .... essential geometric morphism as made of three parts ... ??? ....
?? stuff about cartesianness of categorified bialgebra as red herring ???? ... ???? .... still ..... ???? .... ???? ...... ??? trying to study cateogrified prop of natural operations here .... ??? "distributive category" ... ?? ....
?? idea that at derived level, there really is more surviving ... ???? 4/4 ??? .... ??? symmetric monoidal topos ???? ..... ???? ..... .....
?? extra left adjoint of essential diagonal acting as categorified comultiplication with tensor product acting as categorified multiplication .... ??? vs ... ??? "maxwell" ... ??? .... [right adjoint of cocontinuous extension of tensor product] being also a left adjoint of something, so that it can act as a categorified comultiplication with cartesian product acting as categorified multiplication .... ?? see above bit about derived level ... ?? where distinction between left and right adjoint maybe more or less vanishes, thus restoring full 4/4 or 6/6 or whatever .... ??? ... ... ??? ....
?? 6/6 ... ??? grothendieck's 6 operations ??? ... ??? .... perhaps not ... ?? ....
?? checking notes from december for fit with current picture ... ?? "generalized day convolution" .... ??? tensor product as "essentialness" of something ..... ????? ....
?? getting more explicit about how the structure described in the non-affine case fits as part of the structure described in the affine case ... the "left-adjoint functors" as the _"essentialness"_ of the corresponding "essential geometric morphisms" ..... ????? .....
?????? extra left adjoint of cartesian product vs extra _second_ right adjoint of tensor product ..... ????? extra right adjoint of .... ??? candidate for certain categorified coproduct ... ?? ....
???? maybe 5/6 ??? ..... ????? .....
?? categorified bialgebra associated to toric variety, from filteredly cocontinuous set-valued functor viewpoint .... ?? ...
?? restriction of filteredly cocontinuous functor of 2 variables along diagonal ... ??? .... ???? cartesian product vs tensor product here ??? .... ??? .... ?? filteredly bicocontinuous .... ???? ..... ?? what happens to cofan categories under cartesian product (?? ...) of toric varieties ?? ... ???? .....
?? restriction of filteredly cocontinuous functor of 2 variables along diagonal ... ??? .... ???? cartesian product vs tensor product here ??? .... ??? .... ?? filteredly bicocontinuous .... ???? ..... ?? what happens to cofan categories under cartesian product (?? ...) of toric varieties ?? ... ???? .....
Saturday, February 18, 2012
?? [idea that "restriction along diagonal" is cocontinuous extension of cartesian product of presheaves] as fitting with [idea about decategorified analog of cartesian product of presheaves] ???? ..... ??? though in these contexts maybe morally better to think of "set-valued functors" than "presheaves" .... ???? ......
?? so then left adjoint of this restriction process as "geometric realization of presheaves as bi-presheaves, taking basic figure to "its square"" .... ???? .... ?? which, again, seems to be fitting with .... ???? naive expectation about categorified comultiplication compatible with tensor product (?? by which i seem to mean "day convolution" here ... ??? ....) as categorified multiplication .... ???? ....
??? so go ahead and try investigating "non-affine" case here ... ??? ....
?? idea of ... ??? question of whether diagonal is automatically essential ... ??? try to identify diagonal geometric morphism and/or its extra left adjoint as part of "covariant algebraic structure on _set_" ... ???? but seems very degenerate from this point of view .... ?????? ...... diagonal of terminal object as very degenerate .... ???? ..... identity functor ... so yeah it's left adjoint exists too, but ... ??? .... ?? ...
?? non-toric analog of "object as function wrt topos structure, measure wrt categorified bialgebra structure" ??? .... ??? maybe makes some sense ??? .... ?? quasicoherent as function wrt abelian category structure, measure wrt symetric monoidal cocomplete algebroid structure ????? ..... ????? ...... ?? is this telling us about some sort of "exact pullback (?? ...)" here ???? ... ?? .... ???? confusion ... ??? .... ??? further confusion ..... ???? sense ("moduli stack" ... ???? ...) in which quasicoherent sheaves are definitely (?? ...) supposed to be like functions wrt smcca structure .... ??? ..... ???? ... ??? "intensive vs extensive" ... ??? as very confusing / messed up / complicated when "spectrum" carries further algebraic (?? ...) structure ...
?? so then left adjoint of this restriction process as "geometric realization of presheaves as bi-presheaves, taking basic figure to "its square"" .... ???? .... ?? which, again, seems to be fitting with .... ???? naive expectation about categorified comultiplication compatible with tensor product (?? by which i seem to mean "day convolution" here ... ??? ....) as categorified multiplication .... ???? ....
??? so go ahead and try investigating "non-affine" case here ... ??? ....
?? idea of ... ??? question of whether diagonal is automatically essential ... ??? try to identify diagonal geometric morphism and/or its extra left adjoint as part of "covariant algebraic structure on _set_" ... ???? but seems very degenerate from this point of view .... ?????? ...... diagonal of terminal object as very degenerate .... ???? ..... identity functor ... so yeah it's left adjoint exists too, but ... ??? .... ?? ...
?? non-toric analog of "object as function wrt topos structure, measure wrt categorified bialgebra structure" ??? .... ??? maybe makes some sense ??? .... ?? quasicoherent as function wrt abelian category structure, measure wrt symetric monoidal cocomplete algebroid structure ????? ..... ????? ...... ?? is this telling us about some sort of "exact pullback (?? ...)" here ???? ... ?? .... ???? confusion ... ??? .... ??? further confusion ..... ???? sense ("moduli stack" ... ???? ...) in which quasicoherent sheaves are definitely (?? ...) supposed to be like functions wrt smcca structure .... ??? ..... ???? ... ??? "intensive vs extensive" ... ??? as very confusing / messed up / complicated when "spectrum" carries further algebraic (?? ...) structure ...
?? binary connective truth table vs binary game payoff matrix ... ??
?? measure vs function ... ???? ... possi-/probability distribution vs "intensity function" .... ?? "filter" .... ???
?? "threat" ... "equilibrium" ... ??? .... "focus point", "acceptable (?? ...) alternatives", irrelevant/unexpected/perverse (???) possibility ....
??? "unless" ... ??? vs "(?i)or" ??? .... "you will give me all of your money unless i beat you up" .... ??? .... ?? a unless b ... "b as only escape from a" ... "if not b then a" ... "b or a" .... ???? ....
?? "give me all of your money or i will beat you up" ... "here is all my money, please take it and then beat me up" .... ???? .... "was that an exclusive or then?" ... "i don't know, i hadn't gotten that far" ....
?? measure vs function ... ???? ... possi-/probability distribution vs "intensity function" .... ?? "filter" .... ???
?? "threat" ... "equilibrium" ... ??? .... "focus point", "acceptable (?? ...) alternatives", irrelevant/unexpected/perverse (???) possibility ....
??? "unless" ... ??? vs "(?i)or" ??? .... "you will give me all of your money unless i beat you up" .... ??? .... ?? a unless b ... "b as only escape from a" ... "if not b then a" ... "b or a" .... ???? ....
?? "give me all of your money or i will beat you up" ... "here is all my money, please take it and then beat me up" .... ???? .... "was that an exclusive or then?" ... "i don't know, i hadn't gotten that far" ....
Friday, February 17, 2012
?? but ... ?? _left_ adjoint of cartesian product is simply diagonal ....... ????? .... ??? .... ???? .....
??? funny interrelationship among .... ?? site diagonal, presheaf topos diagonal, and presheaf topos codiagonal ..... ???? ....
?? "essentialness" of site diagonal as ... ?? presheaf topos diagonal, almost sort of ?? ... ???? .....
?? bimonoid compatibility with diagonal functor of cat c as automatic wrt tensor product = cartesian product .... ??? non-automatic and very different (??? seemingly maybe "opposite" sometimes ???? ..... ??? .... "fourier duality" .... ????) wrt tensor product = "tensor product" ...... ???? ....
??? level slips ... ?? ....
?? decategorified analog ??? ..... ????? ......
??? funny interrelationship among .... ?? site diagonal, presheaf topos diagonal, and presheaf topos codiagonal ..... ???? ....
?? "essentialness" of site diagonal as ... ?? presheaf topos diagonal, almost sort of ?? ... ???? .....
?? bimonoid compatibility with diagonal functor of cat c as automatic wrt tensor product = cartesian product .... ??? non-automatic and very different (??? seemingly maybe "opposite" sometimes ???? ..... ??? .... "fourier duality" .... ????) wrt tensor product = "tensor product" ...... ???? ....
??? level slips ... ?? ....
?? decategorified analog ??? ..... ????? ......
?? so let's try getting somewhat of a feel for left adjoint of cocontinuous extension of cartesian product in presheaf topos .... ???? ....
?? first, just plain cocontinuous extension of cartesian product ....
?? graph ....
?? "bi-graph" ...
?? "realize bi-graph as graph by using vertex for vertex-vertex, vertex pair for edge-vertex and vertex-edge, and ... edge^2 for edge-edge" .... ???? ...
?? right adjoint here as .... ?? ....
universal (?? generic ?? ...) bicocontinuous followed by cocontinuous = bicocontinuous .... ???? ... ??? adjoints to these ??? ....
aXb -> a#b -> c
?? level slip here ... "cartesian product" ... as functor with domain a cartesian product ... ??? ....
_graph_ X _graph_ -> _bigraph_ -> _graph_
?? "restrict bigraph along :
walking graph category -> walking bigraph category
to get graph" .... ??? ....
?? is this coming out the same as what we get from other description above ?? .... ??? what's going on here ??? .... ??? vaguely reminds me of ... ?? some stuff people say about simplicial objects ... ??? ....
?? first, just plain cocontinuous extension of cartesian product ....
?? graph ....
?? "bi-graph" ...
?? "realize bi-graph as graph by using vertex for vertex-vertex, vertex pair for edge-vertex and vertex-edge, and ... edge^2 for edge-edge" .... ???? ...
?? right adjoint here as .... ?? ....
universal (?? generic ?? ...) bicocontinuous followed by cocontinuous = bicocontinuous .... ???? ... ??? adjoints to these ??? ....
aXb -> a#b -> c
?? level slip here ... "cartesian product" ... as functor with domain a cartesian product ... ??? ....
_graph_ X _graph_ -> _bigraph_ -> _graph_
?? "restrict bigraph along :
walking graph category -> walking bigraph category
to get graph" .... ??? ....
?? is this coming out the same as what we get from other description above ?? .... ??? what's going on here ??? .... ??? vaguely reminds me of ... ?? some stuff people say about simplicial objects ... ??? ....
[clump 1 ...
?? idea of .... ???? bicocontinuous extension of cartesian product having extra left adjoint acting as categorified comultiplication for tensor product acting as categorified multiplication ?? ....
?? basic object of study in toric geometry as sym mon cat which is also topos with essential diagonal with "the essentialness" (?? ...) acting as comult for the tensor product as mult .... ??? .... ??? _is_ this at all parsing (?? ...) now ??? .... ?? affine case ??? ....
?? mayhem ... ??? same category acting as both measures and functions .... ??? .... ?? topos as topos _of_ categorified functions on categorified space .... ?? ... toric quasicoherent sheaves as ... ??? categorified measures on ... ??? symmetric monoidal cat, in affine case ??? .... ????? ..... ?? if diagonal of topos is non-essential (?? when _does/n't_ this happen, if n/ever ?? .... ??? .... ?? confusion about relationship to "totally distributive" and/or non-/affine ?? ... ??? ....), then ..... ????? somewhat hopeless to think of the topos objects as measures ??? ....
???? "measure vs op-measure" here ???? ..... ???? ..... ???? ....
??? cartesian product in topos as _the_ left adjoint of the "diagonal" geometric morphism ... (?? consider decategorified analog ... ?? ....) .... ?? and that left adjoint is backwards to the diagonal arrow itself, so without essentialness of diagonal, topos objects seem to be behaving as functions rather than as measures ... ??? but now .... with essentialness of diagonal (?? so as to have topos objects able to behave as measures, so as to get a "comultiplication" left adjoint from diagonal, so as to use this as an actual comultiplication with bimonoid compatibility with tensor product .... ????? ...) then we should be getting an additional left adjoint, right ?? ... ?? but where / what is that extra left adjoint ?? ... is it something that we already had but was only a right adjoint, or is it itself new and its right adjoint was previously existing ??? .... ??? essentialness means there's an extra .... hmmm, i was going to say extra right adjoint ... ?? but maybe that's wrong, and moreover good that it's wrong ???? .... ?? the correct idea being that there's an extra _left_ adjoint, which makes "the" left adjoint of the geometric morphism into a right adjoint as well ... ???? ..... ??? so we're hoping that .... left adjoint of cocontinuous extension (??? hmmm, seems like i've been saying "bicocontinuous extension of cocontinuous" for "cocontinuous extension of bicocontinuous" a bit ...) of cartesian product really does exist and act as comultiplication for tensor product as multiplication, even / especially in non-affine case .... ??? .... ??? and we're still pretty confused about "fourier dual" of this idea .... ????? ......
??? antidote / synthesis to .... ??? .... level slips (?? ...) involving ... "bimonoid compatibility with diagonal comultiplication as roughly corresponding to living sharply on basis of point-like elts wrt diagonal comultiplication" ... ???? pun (?? or _is_ it just common sense ??? ....) on "diagonal" / "diagonalized" here ???? .... ?? when cartesian product of representables is/n't representable .... ??? ....
?? toric geometry as generalized fourier duality and todd's question about what accidental topos classifies ... ??? .....
]
[clump 2 ....
?? correspondences ... and / or "partial maps" .... ?? vs alternative tensor product ??? ..... monoids wrt these various alternatives .... ???? .... ??? stacky vs non-stacky case ??? ... ?? funny relationship to toric geometry ???? ...... non-affine case .... ????? ...
?? post-doctrinal logic ... ?? weirdness thereof ?? ... ?? lack of automatic "agglomeration process" .... "sketch" .... ??? .... ?? progression from .... ?? theory where model cat is nicely closed under certain agglomeration processes (?? "positiveness" .... ??? ... ??? ....) to where model cat is not but theory 2-cat is ... ?? to case where not even (...) theory 2-cat is .... ??? .... ??? ....
]
?? "vicarious empathy" .... ?? ... (?? "even if you can't experience real empathy ...")
maxwell ... electromagnetism ... "restoration of symmetry between electricity and magnetism" ... analogy here .... ?? extra left adjoint to bicocontinuous extension of tensor product ??? ..... ?? problems ??? .... ?? usual one of back to danger of getting monoidal topos in non-affine case .... ???? .... ?? dual categorified bialgebra ... ??? .... ?? good (?? ... ?? "representative" ?? ... ???? ....) categorified linear functional .... ????? boolean alg ... opposite category .... commutativity ..... ???? .....
?? idea of ordinary quasicoherent sheaves on toric variety as closer than toric quasicoherent sheaves to achieving maxwell symmetry ideal ?? .... ?? but maybe also losing distinctive toric quality ... ??? i mean, in that non-toruic examples may exhibit same pattern ... ???? ..... .... ??? bug / feature ... ?? ....
??? 3/4 ??? .... ???
?? level slip (??? ...) concerning ... ?? automaticness (?? ...) of compatibility with diagonal comultiplication .... from ordinary commutative monoid to bialg of measures and/or cateogrified bialg of categorified measures .... ??? ..... ??? ....
?? idea of .... ???? bicocontinuous extension of cartesian product having extra left adjoint acting as categorified comultiplication for tensor product acting as categorified multiplication ?? ....
?? basic object of study in toric geometry as sym mon cat which is also topos with essential diagonal with "the essentialness" (?? ...) acting as comult for the tensor product as mult .... ??? .... ??? _is_ this at all parsing (?? ...) now ??? .... ?? affine case ??? ....
?? mayhem ... ??? same category acting as both measures and functions .... ??? .... ?? topos as topos _of_ categorified functions on categorified space .... ?? ... toric quasicoherent sheaves as ... ??? categorified measures on ... ??? symmetric monoidal cat, in affine case ??? .... ????? ..... ?? if diagonal of topos is non-essential (?? when _does/n't_ this happen, if n/ever ?? .... ??? .... ?? confusion about relationship to "totally distributive" and/or non-/affine ?? ... ??? ....), then ..... ????? somewhat hopeless to think of the topos objects as measures ??? ....
???? "measure vs op-measure" here ???? ..... ???? ..... ???? ....
??? cartesian product in topos as _the_ left adjoint of the "diagonal" geometric morphism ... (?? consider decategorified analog ... ?? ....) .... ?? and that left adjoint is backwards to the diagonal arrow itself, so without essentialness of diagonal, topos objects seem to be behaving as functions rather than as measures ... ??? but now .... with essentialness of diagonal (?? so as to have topos objects able to behave as measures, so as to get a "comultiplication" left adjoint from diagonal, so as to use this as an actual comultiplication with bimonoid compatibility with tensor product .... ????? ...) then we should be getting an additional left adjoint, right ?? ... ?? but where / what is that extra left adjoint ?? ... is it something that we already had but was only a right adjoint, or is it itself new and its right adjoint was previously existing ??? .... ??? essentialness means there's an extra .... hmmm, i was going to say extra right adjoint ... ?? but maybe that's wrong, and moreover good that it's wrong ???? .... ?? the correct idea being that there's an extra _left_ adjoint, which makes "the" left adjoint of the geometric morphism into a right adjoint as well ... ???? ..... ??? so we're hoping that .... left adjoint of cocontinuous extension (??? hmmm, seems like i've been saying "bicocontinuous extension of cocontinuous" for "cocontinuous extension of bicocontinuous" a bit ...) of cartesian product really does exist and act as comultiplication for tensor product as multiplication, even / especially in non-affine case .... ??? .... ??? and we're still pretty confused about "fourier dual" of this idea .... ????? ......
??? antidote / synthesis to .... ??? .... level slips (?? ...) involving ... "bimonoid compatibility with diagonal comultiplication as roughly corresponding to living sharply on basis of point-like elts wrt diagonal comultiplication" ... ???? pun (?? or _is_ it just common sense ??? ....) on "diagonal" / "diagonalized" here ???? .... ?? when cartesian product of representables is/n't representable .... ??? ....
?? toric geometry as generalized fourier duality and todd's question about what accidental topos classifies ... ??? .....
]
[clump 2 ....
?? correspondences ... and / or "partial maps" .... ?? vs alternative tensor product ??? ..... monoids wrt these various alternatives .... ???? .... ??? stacky vs non-stacky case ??? ... ?? funny relationship to toric geometry ???? ...... non-affine case .... ????? ...
?? post-doctrinal logic ... ?? weirdness thereof ?? ... ?? lack of automatic "agglomeration process" .... "sketch" .... ??? .... ?? progression from .... ?? theory where model cat is nicely closed under certain agglomeration processes (?? "positiveness" .... ??? ... ??? ....) to where model cat is not but theory 2-cat is ... ?? to case where not even (...) theory 2-cat is .... ??? .... ??? ....
]
?? "vicarious empathy" .... ?? ... (?? "even if you can't experience real empathy ...")
maxwell ... electromagnetism ... "restoration of symmetry between electricity and magnetism" ... analogy here .... ?? extra left adjoint to bicocontinuous extension of tensor product ??? ..... ?? problems ??? .... ?? usual one of back to danger of getting monoidal topos in non-affine case .... ???? .... ?? dual categorified bialgebra ... ??? .... ?? good (?? ... ?? "representative" ?? ... ???? ....) categorified linear functional .... ????? boolean alg ... opposite category .... commutativity ..... ???? .....
?? idea of ordinary quasicoherent sheaves on toric variety as closer than toric quasicoherent sheaves to achieving maxwell symmetry ideal ?? .... ?? but maybe also losing distinctive toric quality ... ??? i mean, in that non-toruic examples may exhibit same pattern ... ???? ..... .... ??? bug / feature ... ?? ....
??? 3/4 ??? .... ???
?? level slip (??? ...) concerning ... ?? automaticness (?? ...) of compatibility with diagonal comultiplication .... from ordinary commutative monoid to bialg of measures and/or cateogrified bialg of categorified measures .... ??? ..... ??? ....
Thursday, February 16, 2012
?? is there a tensor product of commutative rings given by ... ?? adjoining new unit elements, tensoring normally, then taking augmentation ideal ..... ????? .... ??? ...
?? maybe boring ??? .... if works at all ... ??? ...
?? affine toric varieties as "boring" .... ?? in somewhat different (?? ...) but maybe related way ... ??? .... commutative monoidal variety ..... ??? .....
?? stacky (?? ...) analog ?? .... ??? ....
?? maybe boring ??? .... if works at all ... ??? ...
?? affine toric varieties as "boring" .... ?? in somewhat different (?? ...) but maybe related way ... ??? .... commutative monoidal variety ..... ??? .....
?? stacky (?? ...) analog ?? .... ??? ....
Wednesday, February 15, 2012
my state of confusion at the moment is such that i've got lots of questions but have to struggle to intelligibly articulate them ... anyway i'm going to take a stab at articulating one here ....
given a bistable bimonoidal object y in a stable monoidal 2-category x, it seems reasonable (to me at the moment) to talk about y having an extra "cartesian" structure, amounting to an adjunction p between the nullary multiplication and comultiplication operations on y, together with another adjunction q between the binary multiplication and comultiplication operations on y, together with axioms saying that "for each n, there's essentially just one adjunction between the n-ary multiplication and comultiplication operations on y built from p and q".
(the case where x = the stable monoidal 2-category of categories, with cartesian product of categories as the tensor product, is supposed to explain why this terminology is reasonable; my intent is that in this case a bistable bimonoidal object y amounts to essentially just a symmetric monoidal category (with the diagonal y -> yXy as the comultiplication of the bimonoidal object), and a cartesian structure on y amounts to the property of y being cartesian (that is, that the unit object 1 in y is terminal, and that the putative projections defined using the terminalness of 1 in fact form product cones).)
whereas the 2-cells in the "walking bistable bimonoid" stable monoidal 2-category are supposed to be the isomorphisms between parallel finite spans between finite sets, the 2-cells in the "walking cartesian bistable bimonoid" stable monoidal 2-category are supposed to be the general morphisms between them, instead of just the isomorphisms. (i'm being a bit sloppy here about "cartesian vs co-cartesian" and similar distinctions, but hopefully i can get away with it for the moment without causing excessive confusion.)
(it seems to be turning out that cartesianness of bistable bimonoids is relevant to toric geometry in a somewhat different way than i'd been suspecting earlier, but hopefully i can ignore that for now ...)
ok, so if the set-up so far isn't too hopelessly screwed up yet, then i can try to ask my question here: suppose that we take x to be the stable monoidal 2-category where an object is a cocomplete category and a morphism is a cocontinuous functor and the tensor product of objects is "the walking multi-cocontinuous functor out of those objects"; then what does a cartesian bistable bimonoidal object y in x amount to? is it essentially just a cocomplete category with finite cartesian products, with cartesian product being multi-cocontinuous?
given a bistable bimonoidal object y in a stable monoidal 2-category x, it seems reasonable (to me at the moment) to talk about y having an extra "cartesian" structure, amounting to an adjunction p between the nullary multiplication and comultiplication operations on y, together with another adjunction q between the binary multiplication and comultiplication operations on y, together with axioms saying that "for each n, there's essentially just one adjunction between the n-ary multiplication and comultiplication operations on y built from p and q".
(the case where x = the stable monoidal 2-category of categories, with cartesian product of categories as the tensor product, is supposed to explain why this terminology is reasonable; my intent is that in this case a bistable bimonoidal object y amounts to essentially just a symmetric monoidal category (with the diagonal y -> yXy as the comultiplication of the bimonoidal object), and a cartesian structure on y amounts to the property of y being cartesian (that is, that the unit object 1 in y is terminal, and that the putative projections defined using the terminalness of 1 in fact form product cones).)
whereas the 2-cells in the "walking bistable bimonoid" stable monoidal 2-category are supposed to be the isomorphisms between parallel finite spans between finite sets, the 2-cells in the "walking cartesian bistable bimonoid" stable monoidal 2-category are supposed to be the general morphisms between them, instead of just the isomorphisms. (i'm being a bit sloppy here about "cartesian vs co-cartesian" and similar distinctions, but hopefully i can get away with it for the moment without causing excessive confusion.)
(it seems to be turning out that cartesianness of bistable bimonoids is relevant to toric geometry in a somewhat different way than i'd been suspecting earlier, but hopefully i can ignore that for now ...)
ok, so if the set-up so far isn't too hopelessly screwed up yet, then i can try to ask my question here: suppose that we take x to be the stable monoidal 2-category where an object is a cocomplete category and a morphism is a cocontinuous functor and the tensor product of objects is "the walking multi-cocontinuous functor out of those objects"; then what does a cartesian bistable bimonoidal object y in x amount to? is it essentially just a cocomplete category with finite cartesian products, with cartesian product being multi-cocontinuous?
?? had vague idea that progressively higher spans might arise from / relate to simply (?? ...) "refining the adjunction between multiplication and comultiplication to a progressively higher adjoint adjunction ..." (?? ...) but .... ??? unclear ?? .... ??? as to whether the pattern involves stuff that doesn't really parse nicely for just a lone adjunction .... ???? .... ???? ....
?? cartesian bimonoid x as stable bimonoid x with stable comonoid structure on generic object x -> x ?? ... ??? .... ??? microcosmic parsing requirements here ??? ..... ??? alleged bit about ... ?? needing cartesianness of x in order to define object of natural transformations between given parallel pair of x-enriched functors ... ??? .... ?? other alleged bit about monoidalness of [comonoid y, monoid z] .... ??? hmmm, maybe relevant .... ????? .....
?? "equality becoming isomorphism" vs "morphism becoming arity" .... ??? "x-oidal d-theory" for x theory of doctrine (?? ...) d ?? ...
?? "comultiplication" adjoint (?? ...) to multiplication but without bimonoid compatibility between them .... ??? ....
?? stable bimonoid (?? in stable 2-cat ...).... ??? .... with unit and counit forming adjunction between nullary mult and comult, and another unit and counit forming adjunction between binary mult and comult .... ??? hmmm, and then maybe the axioms to say that "there's essentially just one adjunction for each arity" amount to coassociativity (?? and cocommutativity ???? .... ????) clause for certain comonoid structure .... ???? ...... ... to try to anticipate certain hopefully obvious questions i was going to ask .... ???? .... ??? relationship of all this to naive "diagonal" and "projection" .... ???? .... ??? lurking "codiagonal" and "coprojection" ????? ..... ???? .....
?? "distributive category" ... ???? as cartesian in some nice (?? maybe obvious ??) way ?? ...
????? hopf object in hopf cat vs in cartesian cat ...... ????? ...... ????? .... ???? ...... ???? bimonoid in monoidal cat vs in bimonoidal cat .... ???? .....
?? "stable monoid with stable monoid structure on generic object" ....
?? "stable monoid with stable monoid structure on generic object and stable monoid structure on generic generic object" .......
?? "stable monoid with stable monoid structure on generic object and stable monoid structure on generic generic object and ......" .......
????? ......
??? bicartesian structure on 2-category of cartesian categories ??? .... ???? .....
?? "variable abstract category" ..... ???? ....
?? cartesian bimonoid x as stable bimonoid x with stable comonoid structure on generic object x -> x ?? ... ??? .... ??? microcosmic parsing requirements here ??? ..... ??? alleged bit about ... ?? needing cartesianness of x in order to define object of natural transformations between given parallel pair of x-enriched functors ... ??? .... ?? other alleged bit about monoidalness of [comonoid y, monoid z] .... ??? hmmm, maybe relevant .... ????? .....
?? "equality becoming isomorphism" vs "morphism becoming arity" .... ??? "x-oidal d-theory" for x theory of doctrine (?? ...) d ?? ...
?? "comultiplication" adjoint (?? ...) to multiplication but without bimonoid compatibility between them .... ??? ....
?? stable bimonoid (?? in stable 2-cat ...).... ??? .... with unit and counit forming adjunction between nullary mult and comult, and another unit and counit forming adjunction between binary mult and comult .... ??? hmmm, and then maybe the axioms to say that "there's essentially just one adjunction for each arity" amount to coassociativity (?? and cocommutativity ???? .... ????) clause for certain comonoid structure .... ???? ...... ... to try to anticipate certain hopefully obvious questions i was going to ask .... ???? .... ??? relationship of all this to naive "diagonal" and "projection" .... ???? .... ??? lurking "codiagonal" and "coprojection" ????? ..... ???? .....
?? "distributive category" ... ???? as cartesian in some nice (?? maybe obvious ??) way ?? ...
????? hopf object in hopf cat vs in cartesian cat ...... ????? ...... ????? .... ???? ...... ???? bimonoid in monoidal cat vs in bimonoidal cat .... ???? .....
?? "stable monoid with stable monoid structure on generic object" ....
?? "stable monoid with stable monoid structure on generic object and stable monoid structure on generic generic object" .......
?? "stable monoid with stable monoid structure on generic object and stable monoid structure on generic generic object and ......" .......
????? ......
??? bicartesian structure on 2-category of cartesian categories ??? .... ???? .....
?? "variable abstract category" ..... ???? ....
Tuesday, February 14, 2012
?? so for example, consider the co-bisimplicial object in _simplicial set_ whose "realization" functor is cartesian prouct ... in other words consider cartesian product of representable simplicial sets ... ??? .... realizing a bi-simplex as the cartesian product in _simplicial set_ of the corresponding pair of simplexes ... ??? ...
?? then analog for actions of commutative monoid m ... tri-module ... just one representable m-set, namely m with cayley action .... cartesian product with itself as m^2 ... co-bi-m-set object in _m-set_ ... ??isn't this real close to part of categorified bialgebra structure that we've been looking at .... ???? ....
?? then analog for actions of commutative monoid m ... tri-module ... just one representable m-set, namely m with cayley action .... cartesian product with itself as m^2 ... co-bi-m-set object in _m-set_ ... ??isn't this real close to part of categorified bialgebra structure that we've been looking at .... ???? ....
?? extra structure on categorified bialgebra arising from being categorified measures on categorified monoid where just _objects_ are invertible ... ???? ..... (?? contravariant functoriality of inverse ... ??? ....) ??? .... ???? non-toric analog ??? .... "triviality" of "affine" case ??? .... ???? .... ?? "serre duality" / "off-center center" / "twisting by line bundle" ?? .... ??? ....
?? "cartesian bimonoid(-al abstract category)" .... ??? mutltiplication and comultiplication as adjoint _to each other_ .... ???? ..... ??? adjointness and property-vs-structure(-vs-...) games ... ??? .... ?? generalizations .... span, span, span, ... ?? .....
?? doesn't this make it a lot more plausible that "unstable cartesian" might make sense ?? .... ??? ...
??? mystical idea that all higher coherence secretly arises from adjointness ?? ... ??
?? kock-zoeberlein here ??? .....level slips .... ??? .... ?? x,y having adjoints vs being adjoint to each other ..... ???? ..... bi-monoid and/or frobenius .... ???? ....
?? "spans forever" (... ?? ... ?? or at least up to a point ... cutoff subtleties ... ??? ...) as "walking object" _something_ ??? ... but what ??? .... ???? and what about span/co-span confusion (?? ...) here ??? .... bi-monoid and/or hopf vs frobenius .... ??? ..... ??? ambijunction vs certain sort of degenerate such ??? ....
??? microcosm .... ???? ..... "morphism becomes arity" ... ?? ...
?? no (?? ...) span-map manifestation evident between "toroic span bimodules" ??? ... ?? general idea of ... ?? .... well, like i almost said, retreating (?? ...) from suspecting x and y are adjoint to suspecting that they each (?? ...) have adjoints ...
?? automaticness of being able to cartesianize a smc by taking stable comonoids ... ?? counterpoint to idea of special "coincidence" .... ????? ......
???but ... "model diagonal" of topos .... ???? ...... ?? or of "distributive category" ??? .... .... ??? compatibility between topos structure and categorified bialgebra structure expressed in terms of this .... ??? ....
?? danger that bimonoid compatibility between "model diagonal" and "tensor product" will be vacuous ?? ... ??? .... ?? tensor product as .... ??? .... only significantly un-property-like structure / stuff ??? ... ???? ....
?? categorified bialgebra structure (...) on ordinary quasicoherent sheaves over toric variety .... ???? ....
?? arbitrary grothendieck topos has model diagonal ... ?? .... ??? ...
??? categorified bialgebra structure and that confusion about ... ??? ambiguity (?? ...) of "point of toric stack over comm monoid" ?? ..... ???? ..... ???? .....
?? trying to "untangle" / "straighten out" (?? ...) "mayhem" by ... ??? stuff about "what adjoint of f preserves in relation to what f preserves" .... ??? ... ??? limit-preservation vs colimit-preservation ... ??? ..... ????? ..... ?? some simple stuff here but maybe much trickier stuff too ?? ... "preserve" ... "commute with" ... ??? "beck-chevalley" ... ??? ... ?? adjoint functor preserving structure embodied by non-adjoint functor .... ??? .... ??? .... ??? ....
?? "cartesian bimonoid(-al abstract category)" .... ??? mutltiplication and comultiplication as adjoint _to each other_ .... ???? ..... ??? adjointness and property-vs-structure(-vs-...) games ... ??? .... ?? generalizations .... span, span, span, ... ?? .....
?? doesn't this make it a lot more plausible that "unstable cartesian" might make sense ?? .... ??? ...
??? mystical idea that all higher coherence secretly arises from adjointness ?? ... ??
?? kock-zoeberlein here ??? .....level slips .... ??? .... ?? x,y having adjoints vs being adjoint to each other ..... ???? ..... bi-monoid and/or frobenius .... ???? ....
?? "spans forever" (... ?? ... ?? or at least up to a point ... cutoff subtleties ... ??? ...) as "walking object" _something_ ??? ... but what ??? .... ???? and what about span/co-span confusion (?? ...) here ??? .... bi-monoid and/or hopf vs frobenius .... ??? ..... ??? ambijunction vs certain sort of degenerate such ??? ....
??? microcosm .... ???? ..... "morphism becomes arity" ... ?? ...
?? no (?? ...) span-map manifestation evident between "toroic span bimodules" ??? ... ?? general idea of ... ?? .... well, like i almost said, retreating (?? ...) from suspecting x and y are adjoint to suspecting that they each (?? ...) have adjoints ...
?? automaticness of being able to cartesianize a smc by taking stable comonoids ... ?? counterpoint to idea of special "coincidence" .... ????? ......
???but ... "model diagonal" of topos .... ???? ...... ?? or of "distributive category" ??? .... .... ??? compatibility between topos structure and categorified bialgebra structure expressed in terms of this .... ??? ....
?? danger that bimonoid compatibility between "model diagonal" and "tensor product" will be vacuous ?? ... ??? .... ?? tensor product as .... ??? .... only significantly un-property-like structure / stuff ??? ... ???? ....
?? categorified bialgebra structure (...) on ordinary quasicoherent sheaves over toric variety .... ???? ....
?? arbitrary grothendieck topos has model diagonal ... ?? .... ??? ...
??? categorified bialgebra structure and that confusion about ... ??? ambiguity (?? ...) of "point of toric stack over comm monoid" ?? ..... ???? ..... ???? .....
?? trying to "untangle" / "straighten out" (?? ...) "mayhem" by ... ??? stuff about "what adjoint of f preserves in relation to what f preserves" .... ??? ... ??? limit-preservation vs colimit-preservation ... ??? ..... ????? ..... ?? some simple stuff here but maybe much trickier stuff too ?? ... "preserve" ... "commute with" ... ??? "beck-chevalley" ... ??? ... ?? adjoint functor preserving structure embodied by non-adjoint functor .... ??? .... ??? .... ??? ....
Monday, February 13, 2012
?? relationship between topos of filteredly cocontinuous set-valued functors and categorified cocommutative coalgebra ?? ....
?? but ... idea that "cartesianness" doesn't parse unless both mult and comult are present .... ??? .....
?? opposite (?? ...) category of a categorified bi-/co-/algebra ?? .... ???? ...
?? but ... idea that "cartesianness" doesn't parse unless both mult and comult are present .... ??? .....
?? opposite (?? ...) category of a categorified bi-/co-/algebra ?? .... ???? ...
?? double-speed dynamical system ... ??? ... ?? "n-fold acceleration" ... ??? ....
?? "half-speed" ... ??? left vs right adjoint ... ??? ....
?? "history" ... ???? ....
?? adjoint pair of bimodules ... ?? essential geometric morphism .... ??? .....
?? adjoint bimodule and absolute weight(-ed colimit) ??? ..... ??? ....
??? absolute weight and essential geometric morphism ??? .... ???? "cauchy completion" .... ???? ..... "morita/karoubi saturation" .... ???? .....
?? "essential geometric morphism as means of reconstructing (?? as best possible ?? ...) functors between sites ...." ... ??? then "ghost" / "virtual" functors .... ??? ....
??? but .... ???? all those adjoint bimodules coming from actual functors ..... ???? .... what sort of alleged "absolute weights" do these correspond to ???? .... ???? .... ??? variable absolute weights ??? ..... ?? plain module (with adjoint ... ???) as "constant" case .... ??? ..... so adjoint module as "virtual object" of site ??? ..... ??? ... ??? seems to fit ??? .... ????? ......
?? maybe also fit with speculation we had about "accidental infinity-topos" and geometric morphism involving such ??? .... ????? ..... ???? .... ??? essential ... ??? .... ???? .... ???? not sure .... ??? .... ???? ....
??? "absolute" .... "stable" .... ???? ....
??? stable paradise .... ????? lots of virtual objects ??? .... ???? .... ????? ..... ?? try to straighten this out ... ??? ... ??? "derived-level cauchy completion of single-object enriched category as including "all" of its modules" .... ?????? ...... ?????? ..... ??? "karoubi ..." .... projectives ..... ???? .... ???? .... ???? .... "projectiveness as vacuous at derived level" ...... ???? ..... ???? ..... ?? familiar ??? .... ???? ... ???? .....
?? "morita-equivalence" .... "morita-map/functor" ... ?? adjoint bi-module ... ???? .... essential geometric morphism ..... ???? .......
?? "ambijunction" ... ???? .... ....... ??????? .......
?? "cartesian bicategory" .... ???? ....
?? "half-speed" ... ??? left vs right adjoint ... ??? ....
?? "history" ... ???? ....
?? adjoint pair of bimodules ... ?? essential geometric morphism .... ??? .....
?? adjoint bimodule and absolute weight(-ed colimit) ??? ..... ??? ....
??? absolute weight and essential geometric morphism ??? .... ???? "cauchy completion" .... ???? ..... "morita/karoubi saturation" .... ???? .....
?? "essential geometric morphism as means of reconstructing (?? as best possible ?? ...) functors between sites ...." ... ??? then "ghost" / "virtual" functors .... ??? ....
??? but .... ???? all those adjoint bimodules coming from actual functors ..... ???? .... what sort of alleged "absolute weights" do these correspond to ???? .... ???? .... ??? variable absolute weights ??? ..... ?? plain module (with adjoint ... ???) as "constant" case .... ??? ..... so adjoint module as "virtual object" of site ??? ..... ??? ... ??? seems to fit ??? .... ????? ......
?? maybe also fit with speculation we had about "accidental infinity-topos" and geometric morphism involving such ??? .... ????? ..... ???? .... ??? essential ... ??? .... ???? .... ???? not sure .... ??? .... ???? ....
??? "absolute" .... "stable" .... ???? ....
??? stable paradise .... ????? lots of virtual objects ??? .... ???? .... ????? ..... ?? try to straighten this out ... ??? ... ??? "derived-level cauchy completion of single-object enriched category as including "all" of its modules" .... ?????? ...... ?????? ..... ??? "karoubi ..." .... projectives ..... ???? .... ???? .... ???? .... "projectiveness as vacuous at derived level" ...... ???? ..... ???? ..... ?? familiar ??? .... ???? ... ???? .....
?? "morita-equivalence" .... "morita-map/functor" ... ?? adjoint bi-module ... ???? .... essential geometric morphism ..... ???? .......
?? "ambijunction" ... ???? .... ....... ??????? .......
?? "cartesian bicategory" .... ???? ....
Sunday, February 12, 2012
on categorified bialgebras (for todd)
first let me try to give a reasonable definition of "bistable bimonoid in a symmetric monoidal 2-category". i'd guess that this is fairly standard (because it's a straightforward categorification of the concept of "bicommutative bimonoid in a symmetric monoidal 1-category") ...
define wbb ("walking bistable bimonoid") as the symmetric monoidal 2-category where a 0-cell is a finite set, a 1-cell is a finite span, a 2-cell is an isomorphism of parallel spans, and the rest of the structure is hopefully straightforward. a bistable bimonoid in a symmetric monoidal 2-category x is defined to be a symmetric monoidal 2-functor wbb -> x.
define a "bistable categorified bialgebra" to be a bistable bimonoid in the symmetric monoidal 2-category of small-cocomplete categories (with small-cocontinuous functors as 1-cells and natural transformations as 2-cells). define such a bistable categorified bialgebra to be "special" just in case its underlying small-cocomplete category is a presheaf category (so that small-cocontinuous functors are given by bi-presheaves, and natural transformations between them by bi-presheaf morphisms).
given a small symmetric monoidal category m (we'll be especially interested in the case where m has only a single object), the category of presheaves on it is a special bistable categorified bialgebra in a straightforward way (actually in two "opposite" straightforward ways). the corresponding symmetric monoidal 2-functor (finite set, finite span, span isomorphism) -> (small cat, bi-presheaf, bi-presheaf morphism) takes finite set s to the small category m^s, and takes a finite span s <- a -> t to the bi-presheaf given as follows ...
[to be continued]
[note: delineation of the categorified bialgebra structure may seem like conceptual overkill in the case of an affine toric variety (that is, the case of presheaves on a single-object symmetric monoidal category), but my suspicion is that it will turn out to be the right concept in the case of non-affine toric varieties.]
define wbb ("walking bistable bimonoid") as the symmetric monoidal 2-category where a 0-cell is a finite set, a 1-cell is a finite span, a 2-cell is an isomorphism of parallel spans, and the rest of the structure is hopefully straightforward. a bistable bimonoid in a symmetric monoidal 2-category x is defined to be a symmetric monoidal 2-functor wbb -> x.
define a "bistable categorified bialgebra" to be a bistable bimonoid in the symmetric monoidal 2-category of small-cocomplete categories (with small-cocontinuous functors as 1-cells and natural transformations as 2-cells). define such a bistable categorified bialgebra to be "special" just in case its underlying small-cocomplete category is a presheaf category (so that small-cocontinuous functors are given by bi-presheaves, and natural transformations between them by bi-presheaf morphisms).
given a small symmetric monoidal category m (we'll be especially interested in the case where m has only a single object), the category of presheaves on it is a special bistable categorified bialgebra in a straightforward way (actually in two "opposite" straightforward ways). the corresponding symmetric monoidal 2-functor (finite set, finite span, span isomorphism) -> (small cat, bi-presheaf, bi-presheaf morphism) takes finite set s to the small category m^s, and takes a finite span s <- a -> t to the bi-presheaf given as follows ...
[to be continued]
[note: delineation of the categorified bialgebra structure may seem like conceptual overkill in the case of an affine toric variety (that is, the case of presheaves on a single-object symmetric monoidal category), but my suspicion is that it will turn out to be the right concept in the case of non-affine toric varieties.]
?? "allegory" .... ?? with factorization system of relation into co-monic followed by monic .... ??? ... ?? getting allegory (?? ...) from lawvere theory (?? ...) .... ???? .... ?? niceness (?? ...) of such process ?? ...
no wait ... confusion about "joint monic" vs "placewise monic" ... ?? factorization of relation into co-map followed by map .... ???? .... ....
relation as span ... ?? ...
?? matrix treated sesquivariantly : bistable bimonoid : matrix treated bicovariantly : "double stable monoid" .... ??? .... ???? .... just stupid guess ... ?? ...
?? "quantifier sequence" .... ??? arising when both adjoints are available ....
?? when only one is available, then maybe much simpler .... ???? ...
no wait ... confusion about "joint monic" vs "placewise monic" ... ?? factorization of relation into co-map followed by map .... ???? .... ....
relation as span ... ?? ...
?? matrix treated sesquivariantly : bistable bimonoid : matrix treated bicovariantly : "double stable monoid" .... ??? .... ???? .... just stupid guess ... ?? ...
?? "quantifier sequence" .... ??? arising when both adjoints are available ....
?? when only one is available, then maybe much simpler .... ???? ...
?? "whole as sum of parts" ... "big jump as sum of little jumps" ... ??? "jump" ?? ..."step" ?? .... ??? ..... "journey of 1000 miles begins with a single step" ... ?? ?? big steps vs little steps .... ???? .... ?? ....??? .... "longitudinal vs transverse" ??? .... ??? contrast as disappearing in infinitesimal limit ???? ..... ??? vs "wave equations" where it doesn't ?? ... ???? .... .... telescope ... longitudinal .... line-painting truck ... ???? .....
Saturday, February 11, 2012
?? so ... comm monoid as one-object sym mon cat ... ?? functorial operation associated to span s under sym mon structure as suspiciously similar to ordinary operation associated to s under comm monoid structure ... ?? ...
?? bi-module manifestation f# of functor f : x -> y ... f#(x1,y1) = [f(x1),y1]_y ... ??? ....
?? s from a to b ... comm monoid m .... f : m^a -> m^b associated to s under comm monoid / sym mon structure of m .... f#(*,*)=[f(*),*]_m^b .... ???? so m^b as m^a,m^b-bimodule .... vanilla (= "cayley" ?? ...) m^b-module self-structure, m^a-module structure given by matrix / span s ... ???
??? so ... ???? "cartesian" structure here ????? ..... ????? ......
?? adjoint bimodules here ???? ...... affine vs non-affine case ..... ???? ....
??? complex / symplectic // commutative / symmetric .... ???? ....
?? bi-module manifestation f# of functor f : x -> y ... f#(x1,y1) = [f(x1),y1]_y ... ??? ....
?? s from a to b ... comm monoid m .... f : m^a -> m^b associated to s under comm monoid / sym mon structure of m .... f#(*,*)=[f(*),*]_m^b .... ???? so m^b as m^a,m^b-bimodule .... vanilla (= "cayley" ?? ...) m^b-module self-structure, m^a-module structure given by matrix / span s ... ???
??? so ... ???? "cartesian" structure here ????? ..... ????? ......
?? adjoint bimodules here ???? ...... affine vs non-affine case ..... ???? ....
??? complex / symplectic // commutative / symmetric .... ???? ....
?? translating hemingway (or hemingway parody) into german ... ??? .... and/or having hemingway parody contest in german ... ?? ...
?? wiles's description of doing mathematics ... ?? previous room as speck in next room ... ?? ...
?? writing with blue ink and yellow anti-ink on green paper ... ?? or maybe green-tinted glass ... ?? sunglasses ... emerald city ... ??? ... precision-erasing ... ?? ... deconvolution .... ????.....
?? or photographic negative ... superimposed with positive .... ??? .... issue of "neutral" here ??? ... ??? ....
?? positive and negative superimposed with slight shift ... ??? bishop berkeley ... ghost of departed quantity ... ?? cheshire cat ??? ... ??? .....
?? hmmm, gradual / continuous fadeout if you do this with spatially continuous light intensity ... ???? .... "renormalization" .... ??? ..... ?? (somewhat ??) avoiding this renormalization issue with discrete shifts instead of continuous ... ???? ...
?? superimposing sequentially shifted positive copies of superimposition of shifted positive and negative copies ... ?? in effect magnifying the shift ... fundamental theorem of calculus / summing geometric series ... bank account ... sine, cosine ... ??? excessive deposits and withdrawals ... hedge fund .... ??? ....
"big shift as sum of little shifts" ....
?? cross-country skiing on high-relief nazca-like writing ??? .... (?? actual existing examples ?? ...) ...?? "slope" ... positive vs negative ... ??? ... blue and yellow pens yoked together horizontally ... horizontal partial derivative ... or vertically, etc ... horizontal vs vertical bars of various parts of various letters ... in effect more or less disappearing or not disappearing .... ??? .....
?? wiles's description of doing mathematics ... ?? previous room as speck in next room ... ?? ...
?? writing with blue ink and yellow anti-ink on green paper ... ?? or maybe green-tinted glass ... ?? sunglasses ... emerald city ... ??? ... precision-erasing ... ?? ... deconvolution .... ????.....
?? or photographic negative ... superimposed with positive .... ??? .... issue of "neutral" here ??? ... ??? ....
?? positive and negative superimposed with slight shift ... ??? bishop berkeley ... ghost of departed quantity ... ?? cheshire cat ??? ... ??? .....
?? hmmm, gradual / continuous fadeout if you do this with spatially continuous light intensity ... ???? .... "renormalization" .... ??? ..... ?? (somewhat ??) avoiding this renormalization issue with discrete shifts instead of continuous ... ???? ...
?? superimposing sequentially shifted positive copies of superimposition of shifted positive and negative copies ... ?? in effect magnifying the shift ... fundamental theorem of calculus / summing geometric series ... bank account ... sine, cosine ... ??? excessive deposits and withdrawals ... hedge fund .... ??? ....
"big shift as sum of little shifts" ....
?? cross-country skiing on high-relief nazca-like writing ??? .... (?? actual existing examples ?? ...) ...?? "slope" ... positive vs negative ... ??? ... blue and yellow pens yoked together horizontally ... horizontal partial derivative ... or vertically, etc ... horizontal vs vertical bars of various parts of various letters ... in effect more or less disappearing or not disappearing .... ??? .....
Thursday, February 9, 2012
?? "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 ... ??? ....
??? level slip ??? .... ????? ..... ???? ......
?? microcosm ... ??? ....
Wednesday, February 8, 2012
Tuesday, February 7, 2012
?? 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 ...
?? 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 ...
Sunday, February 5, 2012
?? 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.
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.
Saturday, February 4, 2012
?? bracket of observable x with hamiltonian ..... ???? ....
?? heisenberg vs schroedinger picture here .... ???? ....
???? .....
??? vanilla base hamiltonian vs "actual" perturbation .... ??? ....
???? .....
??? ....
??? also "interaction picture" ..... ???? ..... ????? .......
???? .....
?? "momentum" .... ??? symplectic .... ??? ....
?? heisenberg vs schroedinger picture here .... ???? ....
???? .....
??? vanilla base hamiltonian vs "actual" perturbation .... ??? ....
???? .....
??? ....
??? also "interaction picture" ..... ???? ..... ????? .......
???? .....
?? "momentum" .... ??? symplectic .... ??? ....
Friday, February 3, 2012
?? 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_ .... ??? ....
?? ... ??? ....
?? "walking idempotent monoid" monoidal cat =?= "walking monoid" monoidal poset ??? .... ??? ....
?? right adjoint to forgetful 2-fr from _cartesian sym mon poset_ to _sym mon poset_ .... ??? ....
?? ... ??? ....
?? "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 ?? ....
?? "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 ?? ....
Thursday, February 2, 2012
?? 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 ?? ..... ????? .... ??? ...
?? relaitonship to "projection-valued measure" idea ?? ..... ????? .... ??? ...
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