concerning idea of extra "compatibility" in "combined doctrine" in non-toric case ... ??what about toric analog ??? .... ?? ...
??also toric analog of "reconciliation between limits and colimits at derived category level" ??? ... and so forth ... ??
Friday, April 15, 2011
??so what about ideals whose underlying modules are flat?? ... and so forth ... ???... ??something about filtration more general than ideal power filtration ... ??and so forth ... ?? ...
??what about "geometric interpretation" of ext (and so forth....) between coherent sheaves?? ... and so forth ... ?? ... ???in terms of ... ???how "supports" (or something) of coherent sheaves intersect... ??and so forth ... ????...
???something about "intersection theory" here ???? .... ???and so forth ??? ....
??what about "geometric interpretation" of ext (and so forth....) between coherent sheaves?? ... and so forth ... ?? ... ???in terms of ... ???how "supports" (or something) of coherent sheaves intersect... ??and so forth ... ????...
???something about "intersection theory" here ???? .... ???and so forth ??? ....
Thursday, April 14, 2011
??so let's try to describe some nice simple example of "non-formalness" ... ???_have_ we ever really tried this before??? ... hmmm... (??also something about "non-conicalness" ??? ... ??don't we have familiar examples where "associated graded is non-isomorphic to original filtered ..." ??? ... ???or something??? ... ...) ...
??so we want some really simple example of "ext^2" (or something...) being non-trivial ... ???...
???something about polynomials in 2 variables x,y ... or something ?? ... ???....
[a|ax=ay=0] ... free resolution ...
??trying to make this into non-trivial 2-place chain complex .... ????....
???what about "minimality" here ??? .... ...or something ...
place 0 ... < a > ...
place 1 ... < b,c,e >
place 2 ... < f >
d(b)=ax
d(c)=ay
d(e)=0
d(f)=by-cx+e ??
h^0 = < a >/< ax,ay >
h^1 = < by-cx,e >/< by-cx-e >
h^2 = < > ???
???not quite making sense to me yet ??? ....
???what about something about "yoneda _de_composition" ??? ... ???...
??so we want some really simple example of "ext^2" (or something...) being non-trivial ... ???...
???something about polynomials in 2 variables x,y ... or something ?? ... ???....
[a|ax=ay=0] ... free resolution ...
??trying to make this into non-trivial 2-place chain complex .... ????....
???what about "minimality" here ??? .... ...or something ...
place 0 ... < a > ...
place 1 ... < b,c,e >
place 2 ... < f >
d(b)=ax
d(c)=ay
d(e)=0
d(f)=by-cx+e ??
h^0 = < a >/< ax,ay >
h^1 = < by-cx,e >/< by-cx-e >
h^2 = < > ???
???not quite making sense to me yet ??? ....
???what about something about "yoneda _de_composition" ??? ... ???...
??so, temporarily assuming that the comparison morphism from a topos t to the topos corresponding to its filteredly cocomplete category of t-models isn't always an equivalence ...
is there some nice way to recognize when a grothendieck topology gives a topos t for which the comparison morphism _is_ an equivalence?? ... ??...
is there some nice way to recognize when a grothendieck topology gives a topos t for which the comparison morphism _is_ an equivalence?? ... ??...
Wednesday, April 13, 2011
what about the idea that localization inverting f doesn't seem very "flat" at zero of f??... ??or something...
??some vague idea that "you're not supposed to look from that point because it's been excised" or something ... ?? but... ???something about module vs commutative algebra ... ???problems either way??? ... ???? ...
??some vague idea that "you're not supposed to look from that point because it's been excised" or something ... ?? but... ???something about module vs commutative algebra ... ???problems either way??? ... ???? ...
i'm going to try to sketch for todd here some ideas about non-toric analogs of some of the ideas that we've been trying to develop in the toric case...
so suppose that we take the topos x of "toric quasicoherent sheaves" on a "reasonable" toric variety, and then take its category c of models. then we conjecture that the comparison geometric morphism to (??or is it from??) the topos of filteredly cocontinuous set-valued functors on c is an equivalence.
then let's try working out a non-toric analog here...
the quasicoherent sheaves over a reasonable variety form an abelian category x... ??then the exact functors from x to modules over the base ring play the role of the mnodels here...
??except that here i should probably make up my mind for now as to whether i'll work with finite colimits or arbitrary ones... ?? ...
so suppose that we take the topos x of "toric quasicoherent sheaves" on a "reasonable" toric variety, and then take its category c of models. then we conjecture that the comparison geometric morphism to (??or is it from??) the topos of filteredly cocontinuous set-valued functors on c is an equivalence.
then let's try working out a non-toric analog here...
the quasicoherent sheaves over a reasonable variety form an abelian category x... ??then the exact functors from x to modules over the base ring play the role of the mnodels here...
??except that here i should probably make up my mind for now as to whether i'll work with finite colimits or arbitrary ones... ?? ...
notes for next discussion with todd
??"non-toric analog" of whole bunch of stuff arising out of recent discussion about toposes of filteredly cocontinuous set-valued functors ... ??starting with "accidental abelian category" as non-toric analog of "accidental topos" ?? ... (something about accidental topos as (thus? ...) not-so-accidental ... and so forth ....)
toric and non-toric "combined doctrine" ... and so forth ... ??struggling to connect with semi-famous grothendieck topologies ... and so forth ... ??...
??something about example where toric quasicoherent sheaves over "toric pre-stack" _don't_ form a to(r)pos ?? ... and so forth ... ???....
...todd's stuff about kock-zoeberlein and co-inverter, and so forth ...
??maybe ending (or something...) with non-toric analog of "topos for which comparison geometric morphism to topos of filteredly cocontinuous set-valued functors on classical model category _isn't_ an equivalence" ...??... and so forth ...
(??no obvious concept of "deliberate abelian category" ??.....)
???hmm, something about moduli stack of elliptic curves as toric stack?? ??and looking at its accidental topos ??? .... and so forth ... ??... ???hmmm, so what _about_ whether it's of form "filt(c,_set_)" (or something...), and what this might say about "stacky analog of fan" and so forth ???....
??something about concrete geometric aspect of "toric geometry" ... 2-(maybe also 3- ??...)place chain complexes... and so forth ... ??... ??maybe also something about taking "dimension" aspect of toric dimensional theory seriously ... and so forth ... ???....
???something about issue of colimit preservation by passage from dimensional theory to ag theory ... and so forth ... ???toric analog too?? ...
check other semi-recent notes ... ??...
toric and non-toric "combined doctrine" ... and so forth ... ??struggling to connect with semi-famous grothendieck topologies ... and so forth ... ??...
??something about example where toric quasicoherent sheaves over "toric pre-stack" _don't_ form a to(r)pos ?? ... and so forth ... ???....
...todd's stuff about kock-zoeberlein and co-inverter, and so forth ...
??maybe ending (or something...) with non-toric analog of "topos for which comparison geometric morphism to topos of filteredly cocontinuous set-valued functors on classical model category _isn't_ an equivalence" ...??... and so forth ...
(??no obvious concept of "deliberate abelian category" ??.....)
???hmm, something about moduli stack of elliptic curves as toric stack?? ??and looking at its accidental topos ??? .... and so forth ... ??... ???hmmm, so what _about_ whether it's of form "filt(c,_set_)" (or something...), and what this might say about "stacky analog of fan" and so forth ???....
??something about concrete geometric aspect of "toric geometry" ... 2-(maybe also 3- ??...)place chain complexes... and so forth ... ??... ??maybe also something about taking "dimension" aspect of toric dimensional theory seriously ... and so forth ... ???....
???something about issue of colimit preservation by passage from dimensional theory to ag theory ... and so forth ... ???toric analog too?? ...
check other semi-recent notes ... ??...
??so what about topos t for which comparison geometric morphism from/to (???) [t' given by filteredly cocontinuous set-valued functors on the classical model category of t] is _not_ an equivalence?? ??also what about "non-toric analog" here??
(??"non-toric analog" of lots of things as big theme for next discussion with todd? ...)
??when is the category of filteredly cocontinuous functors on a category c no more than "ordinarily big" ??? .... and so forth ...
(??"non-toric analog" of lots of things as big theme for next discussion with todd? ...)
??when is the category of filteredly cocontinuous functors on a category c no more than "ordinarily big" ??? .... and so forth ...
Tuesday, April 12, 2011
??so what about the category of Z-graded sets as a torpos corresponding to a "toric stack" ?? ... and so forth ... ??
??what about such alleged toric stack at "toric dimensional theory" level vs at "tag theory" level?? ... and so forth ... ????...
what about something about the way it somehow struck me as funny here for "sequence of vector spaces" (or sets, or something ...??...) to give free _co_-completion ... ???and so forth ... ???some vague intuition, possibly level-slippery, about limit vs co-limit ... and so forth ... ???....
??what about such alleged toric stack at "toric dimensional theory" level vs at "tag theory" level?? ... and so forth ... ????...
what about something about the way it somehow struck me as funny here for "sequence of vector spaces" (or sets, or something ...??...) to give free _co_-completion ... ???and so forth ... ???some vague intuition, possibly level-slippery, about limit vs co-limit ... and so forth ... ???....
so what about "pfaffian system" and/or dgca as something like "interlocking system of de's, one each with solution candidates being n-simplexes in space x, with face of solution as solution of previous one ..." or something ... ???subject to complications involving "syntactic vs semantic consistency" and so forth ... ???....
???so what about .... the "models" of the abelian category of quasicoherent sheaves over a nice algebraic variety ??? ... or something ... ???as analogous to the models of the topos of toric quasicoherent sheaves over a toric variety ... ???... hmmmm..... ??so what about affine case here??? .... and so forth ... ???
??so consider for example tag theory given by pairs of N-sets, vs one given by N^2-sets ... and so forth .... ??something about evident lack of favorite model in former case ... and so forth ...
??also something about toric dimensional theory and plain dimensional theory, and so forth ... nature (??something about "stucture on" and so forth ...) of moduli stack of models ... and so forth ...
??something about ... ???? dim th : enveloping ag th :: abelian group : its group algebra ... ???something about "group algebra" functor (valued in commutative algebras...) as preserving algebraic (weak...) colimits but not algebraic limits ... ???? thus geometric limits but not geometric colimits ... ??? ???hmm ...??seems slightly strange, since... ??i associate "dimensional theories" with "projective geometry" (more or less...) and i think of projective geometry as (of course ...??...) lying on the "good" side as far as geometric colimits are concerned ...
so then what _about_ the geometric interpretation of algebraic products of dimensional theories??? .... and so forth ...
???let's try... the algebraic product of [the usual dimensional theory of the projective line] with itself ...
... 0 0 0 *1* 2 3 4 5 ...
2 3 4 5 6 7 ....
3 4 5 6 7 8 ....
4 5 6 7 8 9 ....
5 6 7 8 9 10....
6 7 8 9 10 11....
.
.
.
??so then what about how graded modules of the above compare to pairs of graded modules.... ????... and so forth ...
??well, so what _is_ this the dimensional theory of ??? ... ??an idempotent number, together with a pair of line objects ...
???what about something about ... ???over P^1 + P^1, taking the line bundles given by "dual tautological over the first line and trivial over the second" and "trivial over the first and dual tautological over the second" ???...
???"an idempotent number z, together with a pair x,y of line objects and sections x#,x1,x2,y#,y1,y2 such that when z=0 then x#=0 and y# is invertible and y1=y2=0 whereas when z=1 then x# is invertible and x1=x2=0 and y#=0" ??? or something??? ???does that actually make any sense??? ... and so forth ...
??zx=x ... z*y1=y1 , z*y2=y2 ... (1-z)y=y , (1-z)x1=x1 , (1-z)x2=x2 ...
??but what about the idea of trying to get things to be invertible here??? ...??does that make any sense ?????.....
??hmm, maybe we left out some generating sections ... ???instead of x#, how about x+ in grade x and x- in grade -x ??... and so forth ... ???.... ???but does that screw up the numerology ??? ...??? ...
???hmmmmm..... ?????..... ??or maybe un-[screw-up] it???...
0 0 0 1 2 3 ....
0 0 0 0 1 2 3
0 0 0 0 1 2 3
0 0 0 0 1 2 3
1 1 1 1 2 3 4
2 2 2 2 3 4 5
3 3 3 3 4 5 6
.
.
.
hmmm....
??so then _is_ there maybe some sort of morita equivalence here, or something ??? .... ??maybe simply (??...) something about ... grade (g1,g2) being the direct sum of grade g1 of the first coordinate module and grade g2 of the second coordinate module ... ??seems pretty likely, i guess ... ???...
??even so, i'm still confused... still seems like... ???progression from dimensional theory to ag theory can't make up its mind (or my mind... or something...) as to whether it preserves geometric sums ... ???or something ??? ...
??maybe it _is_ my mind rather than its own? ... because it's making me think that i should have the same confusion in lots of other contexts as well ... ??or something ...
hmmm, lots of confusion here ... ???"progression from dimensional theory to ag theory" ... as left adjoint part of "doctrine interpretation" ... ??? but could also think of "moduli stack of theory" as stand-in for theory, in which case ... ??well, something about (2,1)-category of such moduli stacks as opposite of corresponding (2,1)-category of theories, so ... ???from this viewpoint what was left adjoint seems like right adjoint ???....
something about ... ???in general, 2 distinct ways of getting right adjoint g from left adjoint f... namely, by taking g = right adjoint of f (??in this context something about "underlying poorer environment" right adjoint to "free richer theory" ...), but then also by taking g = "f^op" ... ???or something ???? ...as above ... ?? ...
(??what _about_ how these relate in case of adjunction coming from morphism of locally presentable categories?? .... and so forth ... ... ????.... hmmm, not particularly "the same" ??? ... ??or something???)
(??something about idea "propositional doctrine" ... ??supposed to mean a doctrine whopse theories are "propositional", sort of ?? ... ??with syntactic and semantic (2,1)-categories being actually just 1-categories.... ??or something ???....)
??anyway... ??so are we saying something like that... the right adjoint (2,1)-functor taking the moduli stack of a dimensional theory to the moduli stack of the resulting ag theory has the extra property not ordinarily expected of a right adjoint that it preserves sums ?? ...???or something ???
(??as opposed to something about whether the right adjoint (2,1)-functor taking an ag environment to its poorer underlying dimensional environment in turn has a right adjont??? ....???also what _about_ something about ... "the (...) decategorified analog that doesn't seem to work" ... ???something about ... nevertheless, there's an example nearby (or something ... ???...) of a functor with both adjoints, namely ... the inclusion functor from _ab gp_ to _comm monoid_
... ???or something??? ....)
??then what _about_ whether it preserves more general (weak...) colimits, and whether it in turn has a right adjoint ??? .... and so forth ... ????....
???then also similar questions about "de-toricization" ?? ...
??any analogy between "basepoint" of toric variety and "degenerate models" of dimensional theories ??? ...???and so forth ???? ....
??also something about toric dimensional theory and plain dimensional theory, and so forth ... nature (??something about "stucture on" and so forth ...) of moduli stack of models ... and so forth ...
??something about ... ???? dim th : enveloping ag th :: abelian group : its group algebra ... ???something about "group algebra" functor (valued in commutative algebras...) as preserving algebraic (weak...) colimits but not algebraic limits ... ???? thus geometric limits but not geometric colimits ... ??? ???hmm ...??seems slightly strange, since... ??i associate "dimensional theories" with "projective geometry" (more or less...) and i think of projective geometry as (of course ...??...) lying on the "good" side as far as geometric colimits are concerned ...
so then what _about_ the geometric interpretation of algebraic products of dimensional theories??? .... and so forth ...
???let's try... the algebraic product of [the usual dimensional theory of the projective line] with itself ...
... 0 0 0 *1* 2 3 4 5 ...
2 3 4 5 6 7 ....
3 4 5 6 7 8 ....
4 5 6 7 8 9 ....
5 6 7 8 9 10....
6 7 8 9 10 11....
.
.
.
??so then what about how graded modules of the above compare to pairs of graded modules.... ????... and so forth ...
??well, so what _is_ this the dimensional theory of ??? ... ??an idempotent number, together with a pair of line objects ...
???what about something about ... ???over P^1 + P^1, taking the line bundles given by "dual tautological over the first line and trivial over the second" and "trivial over the first and dual tautological over the second" ???...
???"an idempotent number z, together with a pair x,y of line objects and sections x#,x1,x2,y#,y1,y2 such that when z=0 then x#=0 and y# is invertible and y1=y2=0 whereas when z=1 then x# is invertible and x1=x2=0 and y#=0" ??? or something??? ???does that actually make any sense??? ... and so forth ...
??zx=x ... z*y1=y1 , z*y2=y2 ... (1-z)y=y , (1-z)x1=x1 , (1-z)x2=x2 ...
??but what about the idea of trying to get things to be invertible here??? ...??does that make any sense ?????.....
??hmm, maybe we left out some generating sections ... ???instead of x#, how about x+ in grade x and x- in grade -x ??... and so forth ... ???.... ???but does that screw up the numerology ??? ...??? ...
???hmmmmm..... ?????..... ??or maybe un-[screw-up] it???...
0 0 0 1 2 3 ....
0 0 0 0 1 2 3
0 0 0 0 1 2 3
0 0 0 0 1 2 3
1 1 1 1 2 3 4
2 2 2 2 3 4 5
3 3 3 3 4 5 6
.
.
.
hmmm....
??so then _is_ there maybe some sort of morita equivalence here, or something ??? .... ??maybe simply (??...) something about ... grade (g1,g2) being the direct sum of grade g1 of the first coordinate module and grade g2 of the second coordinate module ... ??seems pretty likely, i guess ... ???...
??even so, i'm still confused... still seems like... ???progression from dimensional theory to ag theory can't make up its mind (or my mind... or something...) as to whether it preserves geometric sums ... ???or something ??? ...
??maybe it _is_ my mind rather than its own? ... because it's making me think that i should have the same confusion in lots of other contexts as well ... ??or something ...
hmmm, lots of confusion here ... ???"progression from dimensional theory to ag theory" ... as left adjoint part of "doctrine interpretation" ... ??? but could also think of "moduli stack of theory" as stand-in for theory, in which case ... ??well, something about (2,1)-category of such moduli stacks as opposite of corresponding (2,1)-category of theories, so ... ???from this viewpoint what was left adjoint seems like right adjoint ???....
something about ... ???in general, 2 distinct ways of getting right adjoint g from left adjoint f... namely, by taking g = right adjoint of f (??in this context something about "underlying poorer environment" right adjoint to "free richer theory" ...), but then also by taking g = "f^op" ... ???or something ???? ...as above ... ?? ...
(??what _about_ how these relate in case of adjunction coming from morphism of locally presentable categories?? .... and so forth ... ... ????.... hmmm, not particularly "the same" ??? ... ??or something???)
(??something about idea "propositional doctrine" ... ??supposed to mean a doctrine whopse theories are "propositional", sort of ?? ... ??with syntactic and semantic (2,1)-categories being actually just 1-categories.... ??or something ???....)
??anyway... ??so are we saying something like that... the right adjoint (2,1)-functor taking the moduli stack of a dimensional theory to the moduli stack of the resulting ag theory has the extra property not ordinarily expected of a right adjoint that it preserves sums ?? ...???or something ???
(??as opposed to something about whether the right adjoint (2,1)-functor taking an ag environment to its poorer underlying dimensional environment in turn has a right adjont??? ....???also what _about_ something about ... "the (...) decategorified analog that doesn't seem to work" ... ???something about ... nevertheless, there's an example nearby (or something ... ???...) of a functor with both adjoints, namely ... the inclusion functor from _ab gp_ to _comm monoid_
... ???or something??? ....)
??then what _about_ whether it preserves more general (weak...) colimits, and whether it in turn has a right adjoint ??? .... and so forth ... ????....
???then also similar questions about "de-toricization" ?? ...
??any analogy between "basepoint" of toric variety and "degenerate models" of dimensional theories ??? ...???and so forth ???? ....
Monday, April 11, 2011
weird ... :
??aspects of... ??geometric interpretation of geometric weak colimits of toric stacks ?? ... os ... asf os... ????... ???something about for example trying to g;ue together two lines intersecting at a point, but instead of such singularity (os...) seems to give something sort-of like "direct sum" ... ???asf ?? ???....
???sa ... ???weak colimits of filteredly cocomplete categories??? ... and so forth ... ??...
counterexample ... :
???so... in toric/non-toric case ... ???sa ... trying to contrive example of pre-stack (os...) whose quasicoherent sheaves (defined via "globalization" os... asf os...) aren't a topos/abelian category ... os... by ... sa... ???trying to get non-flat homs involved, os??? .... asf os... ????...
???wa sa ... ????whether flat comm-ring homs are closed under composition ?? ... os... asf os...
prometheus ... :
modern frankenstein ... ???something about filtered-colimit preserving modules of a ringoid, os, asf os????... ???but what about sa where tensor product is coming from ???? .... asf .... ????....
singularity ... :
??so w_a_ singularities of toric varieties??? .... asf .... ???....
??aspects of... ??geometric interpretation of geometric weak colimits of toric stacks ?? ... os ... asf os... ????... ???something about for example trying to g;ue together two lines intersecting at a point, but instead of such singularity (os...) seems to give something sort-of like "direct sum" ... ???asf ?? ???....
???sa ... ???weak colimits of filteredly cocomplete categories??? ... and so forth ... ??...
counterexample ... :
???so... in toric/non-toric case ... ???sa ... trying to contrive example of pre-stack (os...) whose quasicoherent sheaves (defined via "globalization" os... asf os...) aren't a topos/abelian category ... os... by ... sa... ???trying to get non-flat homs involved, os??? .... asf os... ????...
???wa sa ... ????whether flat comm-ring homs are closed under composition ?? ... os... asf os...
prometheus ... :
modern frankenstein ... ???something about filtered-colimit preserving modules of a ringoid, os, asf os????... ???but what about sa where tensor product is coming from ???? .... asf .... ????....
singularity ... :
??so w_a_ singularities of toric varieties??? .... asf .... ???....
??so what about ... given functor f:x->y, considering functor _presheaf(x)_ -> _presheaf(y)_ given by ... "taking formal colimit of x-objects to formal colimit of their y-object f-values" ?? ... or something ... seems obvious ... ???...
??relationship to ... ???forgetful (2,1)-functor taking theory of "combined doctrine" to that of uncombined ... ???and so forth ... ???...
??so what about construeing "combined doctrine" here as simply weak pullback of (2,1)-categories of syntactic theories ?? ... or something ... ??...
??relationship to ... ???forgetful (2,1)-functor taking theory of "combined doctrine" to that of uncombined ... ???and so forth ... ???...
??so what about construeing "combined doctrine" here as simply weak pullback of (2,1)-categories of syntactic theories ?? ... or something ... ??...
Sunday, April 10, 2011
??so consider geometric morphism induced by functor including localization morphisms among more general morphisms between fp commutative rings ??? ... and so forth ...
???more generally, what about image factorization of geometric morphism induced by functor??? ... and so forth ...
???hmm, isn't it something like ... ???full-and-faithful corresponds to "injective" here ??? ... or something ... ???....
???more generally, what about image factorization of geometric morphism induced by functor??? ... and so forth ...
???hmm, isn't it something like ... ???full-and-faithful corresponds to "injective" here ??? ... or something ... ???....
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