something about ... ??weak dual of the generic object x (wrt the ag doctrine...) as 0 because of lack of any interesting x tensor ? -> 1 ... ???or something?? ... that is, something about... "too little raher than too co-much" ?? ... or something ... ??...
??so have we got _any_ examples where... ???"freely adjoining an adjoint for something causes something else to become that adjoint" ??? or something??
??something about... possible analog... noncommutative monoid... walking one-sided inverse situation ... ???forcing two-sided inverse to exist... ???or something?? ...
???so what _about_ P^1 case??? ....
??theory of epi e : 1+1 -> 1 ... with dual of cokernel of its mate ... st ... e is its mate's cokernel's mate's cokernel????
??punctured plane?? ??as "quasi-affine" ... ???...
??so ... ??consider prop as follows : ... ???...
?? e1,e2 : 1 -> 1 ...
"unit" ... 1 -> ( cok( 1 -> 1+1 ) tensor x ) ... ???maybe problematic??
"co-unit" ... ( cok( 1 -> 1+1 ) tensor x ) -> 1 ...???
?? x+x -> 1 ... ??st x -> x+x -> 1 is zero ... ???or something???
?? cok( 1 -> 1+1 ) tensor x ... ??as cok( x -> x+x ) ... ???
1 -> cok( x -> x+x )
cok( x -> x+x ) -> 1
???maybe we really need the prop for a dual pair of objects, and then... ??apply gabriel-ulmer duality ... ???or something???....
well, so... ???suppose that we have a "quadrant" of modules of k[x,y] ...
?? equipped with "cap" and "cup" operators ... ??...
?? and "permutation" operators ... ???...
???satisfying some hopefully obvious compatibility laws, and so forth... ??...
???and then some additional operators... ??relating to one of the objects being the cokernel object of "(x,y)" : 1 -> 1+1, and ... ??something about mate's cokernel's mate's cokernel ... ???...
Monday, January 31, 2011
??so let me try to describe the "tensor product" of (small) finitely cocomplete k-linear categories C and D somewhat explicitly...
??for some reason i'm suspecting that it's something like "fp bimodules that are flat in each argument separately" ... ???
sorry, "flat" here is prety much a red herring ... ??...
??for some reason i'm suspecting that it's something like "fp bimodules that are flat in each argument separately" ... ???
sorry, "flat" here is prety much a red herring ... ??...
tensor product of finitely cocomplete categories (for martin)
x, y finitely cocomplete categories...
functors (x X y)^op -> set ...
??"compact" objects in the category of these... ???...
functors (x X y)^op -> set ...
??"compact" objects in the category of these... ???...
coherent toposes and stuff (for martin)
i want to try to describe here my understanding of how the tradeoff between working with all small colimits vs just finite colimits works in topos theory... on the grounds that topos theory is similar to what we're doing...
??coherent topos as locally finitely presentable category where the compact objects are closed under finite limits ... ???or something like that?? ...
topos as "geometric theory" ...
??coherent topos as locally finitely presentable category where the compact objects are closed under finite limits ... ???or something like that?? ...
topos as "geometric theory" ...
??what are the formal properties of a map f : r1 X r2 -> r3 corresponding to a homomorphism from the tensor product of r1 and r2 ?? ... and so forth...
??came up because i was thinking about tensor product of finitely cocomplete algebroids, vs "hom-wise" tensor product of their underlying algebroids, vs cartesian product of their underlying algebroids... ??something about representing bi-[cokernel-preserving]-k-linear functor in terms of each of these... ???or something???....
??came up because i was thinking about tensor product of finitely cocomplete algebroids, vs "hom-wise" tensor product of their underlying algebroids, vs cartesian product of their underlying algebroids... ??something about representing bi-[cokernel-preserving]-k-linear functor in terms of each of these... ???or something???....
Sunday, January 30, 2011
message to martin, third draft
hi. i think that we said that we'd try meeting on sunday. i just want to try to mention here some topics that i hope we get to talk about. actually, there are so many topics that i want to talk about, that there's little hope that we'll actually be able to talk about all of them in one day (even if you don't have topics of your own to bring up as well, which seems unlikely).
among the topics that i eventually want to talk about are both:
1:
ways of trying to solve some of the problems that we've been working on so far, such as how to prove some improved version of your theorem about the universal property of the quasicoherent sheaves over P^n as a symmetric monoidal cocomplete k-linear category. this includes trying to answer all of the questions that you've been sending to me that i haven't been able to answer yet.
but then also:
2:
topics that i'm interested in but which we haven't gotten around to talking about yet because of focusing up till now on the problems mentioned above (and also because of my general slowness). that is, i have optimistic dreams that i'd like to tell you about, about developing this work into an ambitious program; whereas if i wait to tell you about these dreams until after the foundation results are carefully established, then i might have to wait forever. (for example many of the ideas that alex and i have worked on are very undeveloped so far; i'm much better at formulating approximate conjectures and sketching the big picture of a program than at carefully proving theorems.)
the division into #1 and #2 above is imprecise, in that some topics lie in the overlap of them both. anyway, i'll try to list here some topics, mainly belonging in #2.
first, there are many additional examples of ag theories that i'd like to talk about:
the theory of an epimorphism from the unit object 1 to the direct sum 1 + 1.
the theory of an object.
the theory of a dualizable object.
the theory of an n-dimensional object.
the theory of a lie algebra object. (you mentioned "algebraic theories" the other day; i wasn't sure whether you were deliberately suggesting that the study of them is similar in many ways to the study of what i've been calling "algebraic-geometric theories". in fact, there's even a significant overlap between these two kinds of theories, as exemplified by this example of "the theory of a lie algebra object". in any case, i'm definitely suggesting a parallelism here, that both of these are examples of what i call a "doctrine", about which more later.)
the theory of an n-dimensional lie algebra object.
the theory of a flagged n-dimensional object.
the theory of a flag on the direct sum of n copies of the unit object.
the theory of a curve of genus g. (this example needs a lot of work to make sense, but it's a very interesting example!)
the theory of right-exact functors from a given finitely cocomplete k-linear category. (in other words, "the free symmetric monoidal finitely cocomplete k-linear category on a given finitely cocomplete k-linear category".)
and so forth; there are many other examples...
second, i'd like to discuss the general idea of what i call "doctrines". roughly, a doctrine is a groupoid-enriched category which is "locally finitely presentable" in an approperiate groupoid-enriched sense. ag theories form a doctrine; other examples include the doctrine of algebraic theories, the doctrine of coherent toposes, and so forth.
one particular idea about doctrines that i'd like to discuss is the idea that, for example, the "big zariski topos" of a sufficiently nice scheme x (or of something somewhat more general, such as an algebraic stack of some kind) can be thought of as "the same theory" for which x is the moduli space, but expressed in a different doctrine (namely the doctrine of coherent ringed toposes, instead of the doctrine of ag theories). i haven't succeeded in getting this to actually work yet, but i'm optimistic that the basic idea makes good sense.
third, i'd like to discuss the idea of studying universal properties of symmetric monoidal homotopy-cocomplete differential graded k-linear categories, including especially the ones arising by taking chain complexes of objects from a symmetric monoidal cocomplete k-linear category. very vaguely, i think that this is a "cohomological" or "higher" analog of what we're doing...
well, i guess that i'll stop here for now... there's other topics that i meant to include but i won't get around to mentioning them all here...
i guess that this message came out pretty disorganized, so maybe it won't be too readable for you, but maybe i can at least use it as a reminder for myself about some topics that i might like to eventually bring up...
among the topics that i eventually want to talk about are both:
1:
ways of trying to solve some of the problems that we've been working on so far, such as how to prove some improved version of your theorem about the universal property of the quasicoherent sheaves over P^n as a symmetric monoidal cocomplete k-linear category. this includes trying to answer all of the questions that you've been sending to me that i haven't been able to answer yet.
but then also:
2:
topics that i'm interested in but which we haven't gotten around to talking about yet because of focusing up till now on the problems mentioned above (and also because of my general slowness). that is, i have optimistic dreams that i'd like to tell you about, about developing this work into an ambitious program; whereas if i wait to tell you about these dreams until after the foundation results are carefully established, then i might have to wait forever. (for example many of the ideas that alex and i have worked on are very undeveloped so far; i'm much better at formulating approximate conjectures and sketching the big picture of a program than at carefully proving theorems.)
the division into #1 and #2 above is imprecise, in that some topics lie in the overlap of them both. anyway, i'll try to list here some topics, mainly belonging in #2.
first, there are many additional examples of ag theories that i'd like to talk about:
the theory of an epimorphism from the unit object 1 to the direct sum 1 + 1.
the theory of an object.
the theory of a dualizable object.
the theory of an n-dimensional object.
the theory of a lie algebra object. (you mentioned "algebraic theories" the other day; i wasn't sure whether you were deliberately suggesting that the study of them is similar in many ways to the study of what i've been calling "algebraic-geometric theories". in fact, there's even a significant overlap between these two kinds of theories, as exemplified by this example of "the theory of a lie algebra object". in any case, i'm definitely suggesting a parallelism here, that both of these are examples of what i call a "doctrine", about which more later.)
the theory of an n-dimensional lie algebra object.
the theory of a flagged n-dimensional object.
the theory of a flag on the direct sum of n copies of the unit object.
the theory of a curve of genus g. (this example needs a lot of work to make sense, but it's a very interesting example!)
the theory of right-exact functors from a given finitely cocomplete k-linear category. (in other words, "the free symmetric monoidal finitely cocomplete k-linear category on a given finitely cocomplete k-linear category".)
and so forth; there are many other examples...
second, i'd like to discuss the general idea of what i call "doctrines". roughly, a doctrine is a groupoid-enriched category which is "locally finitely presentable" in an approperiate groupoid-enriched sense. ag theories form a doctrine; other examples include the doctrine of algebraic theories, the doctrine of coherent toposes, and so forth.
one particular idea about doctrines that i'd like to discuss is the idea that, for example, the "big zariski topos" of a sufficiently nice scheme x (or of something somewhat more general, such as an algebraic stack of some kind) can be thought of as "the same theory" for which x is the moduli space, but expressed in a different doctrine (namely the doctrine of coherent ringed toposes, instead of the doctrine of ag theories). i haven't succeeded in getting this to actually work yet, but i'm optimistic that the basic idea makes good sense.
third, i'd like to discuss the idea of studying universal properties of symmetric monoidal homotopy-cocomplete differential graded k-linear categories, including especially the ones arising by taking chain complexes of objects from a symmetric monoidal cocomplete k-linear category. very vaguely, i think that this is a "cohomological" or "higher" analog of what we're doing...
well, i guess that i'll stop here for now... there's other topics that i meant to include but i won't get around to mentioning them all here...
i guess that this message came out pretty disorganized, so maybe it won't be too readable for you, but maybe i can at least use it as a reminder for myself about some topics that i might like to eventually bring up...
Saturday, January 29, 2011
message to martin
hi. i think that we said that we'd try meeting tomorrow, sunday that is. i just want to try to mention here some topics that i hope we get to talk about. actually, there are so many topics that i want to talk about, that there's little hope that we'll actually be able to talk about all of them tomorrow (even if you don't have topics of your own to bring up as well, which seems unlikely).
among the topics that i want to talk about are both:
1:
ways of resolving some of the technical difficulties(?? os??? asf os...) problems that we've been trying to solve so far, such as how to prove some improved version of your theorem about the quasicoherent sheaves over P^n as forming a specific (cocomplete...???os...) ag theory. and also, dealing with questions that you (??and me?? ??os...) have raised in connection with this... os... asf os...
but then also:
2:
topics that i'm very interested in but which we haven't gotten around to talking about yet because of focusing up till now on #1 above... (and also becaause of my general slowness, asf ...????) ... topics which are somewhat more "conceptual" and less "technical" (os...) ... ????sa interesting questions that raise themselves once you get beyond the technical difficulties... ??either by actually resolving those technical difficulties, or by optimistic thinking... ??os... ??sa fe things that alex and i have worked on, that you asked about at one point, if i remember correctly...
???and actually the above division into #1 and #2 is somewhat artificial, there being lots of overlap... some of the alleged "technical" questions being actually conceptually very interesting, when you think about them the right way... ??...anyway, i'll proceed to try to list some items from #1 and/or from #2, hopefully making clear which of #1 and/or #2 i'm placing it in... ??os...
#1
???finitely cocomplete vs cocomplete ....
#2
???lots of examples of ag theories ... working both from "syntactic presentation" end (??maybe really "semantic" end??? ) and "syntactic category" end ... ???os???
epimorphism from 1 to 1+1
maps a,b : 1 -> 1 which are each other's cokernel ...
n-dimensional object (??variations???)
dualizable object (adjoint pair of objects)
object
epimorphism
free ag theory on a finitely cocomplete algebroid ...
string of good embeddings starting with 0, with each cokernel invertible...
???something about ag theory of n-tuple of line bundles equipped with (for example) lie alg structure... and so forth ...
n-dimensional lie algebra object ...
???something about cuboquadratic algebra ... ???...
?????sa doctrines.... ????
??something about big zariski topos....
finitely cocomplete algebroid ...???
????something about cohomology... derived categories... and so forth ... ???....
???something about "renormalization" and so forth... filtered... associated graded ... and so forth... normal bundle... ???...
martin mentioned something about "algebraic theories" today... so i think that i should try incorporating this into the discussion... something about "doctrines" ...
blending together various viewpoints... algebraic theory as parallel to ag theory ... and so forth ... ??
??also something about "universal property as tool for reducing abstraction level" ... and so forth... ??...
???i guess that this message came out even more disorganized than i expected... so maybe it won't be too readable for you, but maybe i can at least use it as a reminder for myself about some topics that i might like to eventually bring up...
among the topics that i want to talk about are both:
1:
ways of resolving some of the technical difficulties(?? os??? asf os...) problems that we've been trying to solve so far, such as how to prove some improved version of your theorem about the quasicoherent sheaves over P^n as forming a specific (cocomplete...???os...) ag theory. and also, dealing with questions that you (??and me?? ??os...) have raised in connection with this... os... asf os...
but then also:
2:
topics that i'm very interested in but which we haven't gotten around to talking about yet because of focusing up till now on #1 above... (and also becaause of my general slowness, asf ...????) ... topics which are somewhat more "conceptual" and less "technical" (os...) ... ????sa interesting questions that raise themselves once you get beyond the technical difficulties... ??either by actually resolving those technical difficulties, or by optimistic thinking... ??os... ??sa fe things that alex and i have worked on, that you asked about at one point, if i remember correctly...
???and actually the above division into #1 and #2 is somewhat artificial, there being lots of overlap... some of the alleged "technical" questions being actually conceptually very interesting, when you think about them the right way... ??...anyway, i'll proceed to try to list some items from #1 and/or from #2, hopefully making clear which of #1 and/or #2 i'm placing it in... ??os...
#1
???finitely cocomplete vs cocomplete ....
#2
???lots of examples of ag theories ... working both from "syntactic presentation" end (??maybe really "semantic" end??? ) and "syntactic category" end ... ???os???
epimorphism from 1 to 1+1
maps a,b : 1 -> 1 which are each other's cokernel ...
n-dimensional object (??variations???)
dualizable object (adjoint pair of objects)
object
epimorphism
free ag theory on a finitely cocomplete algebroid ...
string of good embeddings starting with 0, with each cokernel invertible...
???something about ag theory of n-tuple of line bundles equipped with (for example) lie alg structure... and so forth ...
n-dimensional lie algebra object ...
???something about cuboquadratic algebra ... ???...
?????sa doctrines.... ????
??something about big zariski topos....
finitely cocomplete algebroid ...???
????something about cohomology... derived categories... and so forth ... ???....
???something about "renormalization" and so forth... filtered... associated graded ... and so forth... normal bundle... ???...
martin mentioned something about "algebraic theories" today... so i think that i should try incorporating this into the discussion... something about "doctrines" ...
blending together various viewpoints... algebraic theory as parallel to ag theory ... and so forth ... ??
??also something about "universal property as tool for reducing abstraction level" ... and so forth... ??...
???i guess that this message came out even more disorganized than i expected... so maybe it won't be too readable for you, but maybe i can at least use it as a reminder for myself about some topics that i might like to eventually bring up...
Friday, January 28, 2011
consider the ag theory of .... ???an epimorphism from the standard line to itself ... ??? ???something about k-module equipped with monic linear operator ... ???
how about, for example, epimorphism from standard line to direct sum of two copies of it?? ???k-module equipped with pair of linear operators which are "jointly monic" or something??? ???given x there's no more than a single pair y,z st f(y) + g(z) = x ?? ??might as well take x = 0 ??... f(y) + g(z) is never 0 unless both y and z are???
??certainly seems like a weird theory; what "use" could it possibly have???
-xg f + xf g = 0 .... ????must set (-xg,xf) to zero then??? ??but then must also set xf to zero, and thus also (x,0), and thus also x ... ???or something???
??if this is correct, then maybe it's somewhat promising that such a weird theory is syntactically inconsistent ... ???or something ??...
??but wait, what about taking natural numbers as basis, and having f map each basis element n to 2n, and g map it to 2n+1 ??? ??or something??? ... ???...
??oh, but that f and g don't commute ... hmm, interesting ... ???something about ... ??you can have a non-trivial parallel pair of endo-maps whose images intersect trivially, but not if they commute ... ??or something...
how about, for example, epimorphism from standard line to direct sum of two copies of it?? ???k-module equipped with pair of linear operators which are "jointly monic" or something??? ???given x there's no more than a single pair y,z st f(y) + g(z) = x ?? ??might as well take x = 0 ??... f(y) + g(z) is never 0 unless both y and z are???
??certainly seems like a weird theory; what "use" could it possibly have???
-xg f + xf g = 0 .... ????must set (-xg,xf) to zero then??? ??but then must also set xf to zero, and thus also (x,0), and thus also x ... ???or something???
??if this is correct, then maybe it's somewhat promising that such a weird theory is syntactically inconsistent ... ???or something ??...
??but wait, what about taking natural numbers as basis, and having f map each basis element n to 2n, and g map it to 2n+1 ??? ??or something??? ... ???...
??oh, but that f and g don't commute ... hmm, interesting ... ???something about ... ??you can have a non-trivial parallel pair of endo-maps whose images intersect trivially, but not if they commute ... ??or something...
notes for next discussion with derek
??group acting on short exact sequence... ??equivalent to short exact sequence in category of representations of the group ... ??special case of affine vector space as short exact sequence...
???something about filtered vector space with extra structure ... ???something about green convolution here???? or something????.... ??something about filtered lie algebra with filtered rep ... ???and so forth .... ???....
???something about filtered vector space with extra structure ... ???something about green convolution here???? or something????.... ??something about filtered lie algebra with filtered rep ... ???and so forth .... ???....
Wednesday, January 26, 2011
talking to adam katz about cyclic operads and di-operads again...
??so what _about_ viewing both of these as symmetric monoidal functors in certain way?? ??relationship to certain ideas of clement berger and michael batanin and so forth??? ??that is, something about similar ways of thinking about other things... ???but then what about also relationship to other ways of thinking about same thing???...
???this idea of using forgetful functor from sym mon cat where operations (??or composeable trees of them, os???) are di-graphs (or something...) to one where they're graphs (or something...) to get di-operad from cyclic operad... ...???something about whether this forgetful functor is "fibration" in certain sense ... ??in which case what's "fiber functor" (or something) like?? ... hmmmm.... ??hmm, maybe not a fibration ... ???...
(??hmm, so _is_ there something rather "opetopic" going on here??? ???...)
??so what _about_ viewing both of these as symmetric monoidal functors in certain way?? ??relationship to certain ideas of clement berger and michael batanin and so forth??? ??that is, something about similar ways of thinking about other things... ???but then what about also relationship to other ways of thinking about same thing???...
???this idea of using forgetful functor from sym mon cat where operations (??or composeable trees of them, os???) are di-graphs (or something...) to one where they're graphs (or something...) to get di-operad from cyclic operad... ...???something about whether this forgetful functor is "fibration" in certain sense ... ??in which case what's "fiber functor" (or something) like?? ... hmmmm.... ??hmm, maybe not a fibration ... ???...
(??hmm, so _is_ there something rather "opetopic" going on here??? ???...)
??so... ??at the moment i'm imagining that there's some nice way of getting from an ag theory to a "dg scheme" ... or something like that... ???hmm, i guess actually from something like... a family of models of a dg theory... (??something about "groupoid object" ... ???in where????.... ??hmm, suppose that family is parameterized by model spectrum of another ag theory ... where models may be non-rigid ... ??? ... ??and so forth ???...) to a dg scheme ... ???but ... ??is this going to be as "interesting" as i'm sort of hoping?? ???something about ... ??ag theory as only a groupoid, rather than a higher groupoid .... ????hmmm, but on the other hand maybe something about ... ??topological groupoid as giving arbitrary htpy type ... ???.... ???still confusing here????.....
???rational homotopy theory of a dg scheme / space .... ???....
???....
??ao what _about_ that "reverse engineering" approach that i took??? .... hmmm ... ???... ????....
???so what _about_ dgla vs dgca ??? ..... and so forth .... ???making some sense now ??? ..... ????..... hmmmm .... ?????? .....
???so what _about_ "higher obstructions" ... ???... and so forth .... ???? .....
????? ???... ??_not_ quite making sense yet, but maybe should ??? .... ??something about grading vs filtration ... ??? ???something about forgetting grading ... "super-geometry" ... ??"spectrum of cohomology ring" ... ...kleinian singularity ... ??? and so forth ... ???.... ?????..... and so forth .... ????....
???hmmm... ???so what abotu something the way a perfectly ordinary group, finite for example, may have higher cohomology ??? .... ???... hmmm... ???... ???also something about "bar construction" here??? .... and so forth ... ??? ??something abot "etale fundamental group" and so forth ??? .... ????....
??still lots of confusion here... ???discrete vs codiscrete ... ??isomorphism vs path ... ??bundle vs bundle with connection .... "moduli space/stack" vs "classifying space/stack" ... ???? .... and so forth ... ???something about de rham dgca and fundamental infinity-groupoid ... and so forth ... ???....
??something about leray-serre and "cohomology with coefficients in a shef of cohomology groups" and so forth.... ??something about certain factorization of geometric morphisms... something about "codiscrete ..." here ... ???.... and so forth ... ????....
???something about "why deformations can be expressed (or something...) in terms of cocycles of some sort; that is, in terms of something that in principle is homotopy-theoretic??" ... ????....
???is there some idea here that... ????when you're "doing deformation theory at a scaling limit" (??or "multi-scaling limit" or something???) (??or something????), you should be... ???thinking of the tangent cone as the substrate for a projective (?or multi-projective??...) variety ... ???or something??? .... and so forth ... ???? ???or something?????..... ??hmm, maybe there _is_ something like this??? seems like i was leanign in that direction myself... ???...though... ???maybe not very clearly spelled out... ???can't remember anyone else offhand describing it this way??? ??or any other similar way??? .... ??hmmm, or maybe... ???maybe there was some way of thinking about this stuff that did fit naturally with... ???something about whole complex of ideas connected with...???associated graded object of ideal power filtration, intepreted as something about "normal bundle"... and so forth ... ???or something ????.... ???something about ..."tangent space" (??and/or more general sort of "infinitesimal analysis" / "renormalization" ???...) at point of "stack" .... ????....
??what _about_ macroscopic dgca here???.... not settling for tangent cone ... ???what _about_ relationship between "formal" and "conical" and so forth ???? .....
???rational homotopy theory of a dg scheme / space .... ???....
???....
??ao what _about_ that "reverse engineering" approach that i took??? .... hmmm ... ???... ????....
???so what _about_ dgla vs dgca ??? ..... and so forth .... ???making some sense now ??? ..... ????..... hmmmm .... ?????? .....
???so what _about_ "higher obstructions" ... ???... and so forth .... ???? .....
????? ???... ??_not_ quite making sense yet, but maybe should ??? .... ??something about grading vs filtration ... ??? ???something about forgetting grading ... "super-geometry" ... ??"spectrum of cohomology ring" ... ...kleinian singularity ... ??? and so forth ... ???.... ?????..... and so forth .... ????....
???hmmm... ???so what abotu something the way a perfectly ordinary group, finite for example, may have higher cohomology ??? .... ???... hmmm... ???... ???also something about "bar construction" here??? .... and so forth ... ??? ??something abot "etale fundamental group" and so forth ??? .... ????....
??still lots of confusion here... ???discrete vs codiscrete ... ??isomorphism vs path ... ??bundle vs bundle with connection .... "moduli space/stack" vs "classifying space/stack" ... ???? .... and so forth ... ???something about de rham dgca and fundamental infinity-groupoid ... and so forth ... ???....
??something about leray-serre and "cohomology with coefficients in a shef of cohomology groups" and so forth.... ??something about certain factorization of geometric morphisms... something about "codiscrete ..." here ... ???.... and so forth ... ????....
???something about "why deformations can be expressed (or something...) in terms of cocycles of some sort; that is, in terms of something that in principle is homotopy-theoretic??" ... ????....
???is there some idea here that... ????when you're "doing deformation theory at a scaling limit" (??or "multi-scaling limit" or something???) (??or something????), you should be... ???thinking of the tangent cone as the substrate for a projective (?or multi-projective??...) variety ... ???or something??? .... and so forth ... ???? ???or something?????..... ??hmm, maybe there _is_ something like this??? seems like i was leanign in that direction myself... ???...though... ???maybe not very clearly spelled out... ???can't remember anyone else offhand describing it this way??? ??or any other similar way??? .... ??hmmm, or maybe... ???maybe there was some way of thinking about this stuff that did fit naturally with... ???something about whole complex of ideas connected with...???associated graded object of ideal power filtration, intepreted as something about "normal bundle"... and so forth ... ???or something ????.... ???something about ..."tangent space" (??and/or more general sort of "infinitesimal analysis" / "renormalization" ???...) at point of "stack" .... ????....
??what _about_ macroscopic dgca here???.... not settling for tangent cone ... ???what _about_ relationship between "formal" and "conical" and so forth ???? .....
Tuesday, January 25, 2011
??so what about the relationship between ag theory of "n-dimensional object" (or something... ??issues about invertibleness of n .... and so forth ... ???) and ... ???exotic representation theory of gl(n) ??? .... and so forth... ???....
representation of alg gp ct of "instantiation" thereof ... ??... and so forth... ???
representation of alg gp ct of "instantiation" thereof ... ??... and so forth... ???
Monday, January 24, 2011
Sunday, January 23, 2011
reply to martin
1 attitude towards "coherent" ... free cocomplete ag theory on ag theory... "coherent core" ... or something... ??any reason to define such core in non-coherent case?? and so forth... ??sa "finitely presented" ... ??no claim about any alleged meaning of concept in "sheaf" case ... ?? ... ??something not really studying a different doctrine; just studying specially simple examples... ??though also with constraint on morphisms... ??...
2 proof of left-universal property as ag theory of coherent sheaves over P^1 ...
????something about cohomology...
???something about ag theory of n-tuple of line bundles equipped with (for example) lie alg structure... and so forth ...
???something about "renormalization" and so forth... filtered... associated graded ... and so forth... normal bundle... ???...
2 proof of left-universal property as ag theory of coherent sheaves over P^1 ...
????something about cohomology...
???something about ag theory of n-tuple of line bundles equipped with (for example) lie alg structure... and so forth ...
???something about "renormalization" and so forth... filtered... associated graded ... and so forth... normal bundle... ???...
Saturday, January 22, 2011
notes for next discussion with kenji
??should really start fresh with clearer idea of how to proceed... ??especially now that i've got the book... ??or something...
Friday, January 21, 2011
notes for (some future) discussion with martin
martin mentioned something about "algebraic theories" today... so i think that i should try incorporating this into the discussion... something about "doctrines" ...
blending together various viewpoints... algebraic theory as parallel to ag theory ... and so forth ... ??
??also something about "universal property as tool for reducing abstraction level" ... and so forth... ??...
blending together various viewpoints... algebraic theory as parallel to ag theory ... and so forth ... ??
??also something about "universal property as tool for reducing abstraction level" ... and so forth... ??...
notes for next discussion with kenji
associahedron... generalized associativity as something about connectedness of associahedron 1-skeletons or something?? ... and so forth... ??degrees of "partial generalized associativity" (case of single n-tuple vs entire binary operation???...??...), maybe trying to understand them in terms of associahedron combinatorics... and so forth... ???... ??to some extent belaboring the problem that i always had with "generalized associativity", that the trick is less how to prove it than how to state it... or something... ??....
??something about trying to develop exact formula for catalan numbers ... hmmm... ??...
??something about trying to develop exact formula for catalan numbers ... hmmm... ??...
notes for next discussion with chris rogers
??try aksing about how stuff like symplectic geometry and poisson algebras (??and so forth???....) fits into theory of "dg spaces" ???...
ok, i asked, and he said that the (or at least one) idea is that a poisson structure on a space x can be thought of as a special way of taking the gca of multi-vector fields on x and putting a d on it to make it a dgca...
??so ... ??idle guess as to what sort of lie algebroid (or something...) this might correspond to ... ???maybe something about poisson manifold as foliated by symplectic manifolds... each with almost canonical connectioned line bundle, or something... ???something about getting lie algberoid fomr this, maybe ??? ....
??something about having nice simple examples to work with of lie algebroids whose representations we pretty concretely understand... ??for this purpose use those "semi-direct product algebras" ... ??"action lie algberoids" ... ?? ... including case of d-modules over an affine algebraic group ... or soemthing... and so forth... ??...
??the general program to relate lie algebroid reps to dg modules... ??something about confusion between co-simplicial vs simplicial here ???... and so forth...
ok, i asked, and he said that the (or at least one) idea is that a poisson structure on a space x can be thought of as a special way of taking the gca of multi-vector fields on x and putting a d on it to make it a dgca...
??so ... ??idle guess as to what sort of lie algebroid (or something...) this might correspond to ... ???maybe something about poisson manifold as foliated by symplectic manifolds... each with almost canonical connectioned line bundle, or something... ???something about getting lie algberoid fomr this, maybe ??? ....
??something about having nice simple examples to work with of lie algebroids whose representations we pretty concretely understand... ??for this purpose use those "semi-direct product algebras" ... ??"action lie algberoids" ... ?? ... including case of d-modules over an affine algebraic group ... or soemthing... and so forth... ??...
??the general program to relate lie algebroid reps to dg modules... ??something about confusion between co-simplicial vs simplicial here ???... and so forth...
Thursday, January 20, 2011
??so... ??have we got an example here of an ag theory with an epi between dualizable objects, with its mate's cokernel object also dualizable, but where the original epi is not its mate's cokernel's mate's cokernel?? ??namely, the category of Z-filtered fp k-modules (with everything in some filtration stage) with the epi from "k born at 0" to "k born at -1"... it's mate's cokernel object is 0, because it's mate is epi too... but that makes it's mates's cokernel's mate a zero map, and the cokernel of that is an isomorphism, whereas the original epi is not iso here... ???or something??
??and then we've also got examples of the sort-of complementary phenomenon... ...??where the mate's cokernel object isn't dualizable... ??right?? ... namely... the filtered module example above is about P^0, whereas with P^1 it seems that the epi's mate's cokernel object isn't dualizable... i think ...
??so ... the condition on an epi between dualizable objects of being "good" does seem to break down into a couple of non-automatic, non-vacuous stages, where the failure can occur at either stage... first, the mate's cokernel object might not be dualizable; then if it is, the original epi might still not be the mate's cokernel's mate's cokernel ... ??...
...assuming that i didn't miscalculate too badly in these examples...
??and then we've also got examples of the sort-of complementary phenomenon... ...??where the mate's cokernel object isn't dualizable... ??right?? ... namely... the filtered module example above is about P^0, whereas with P^1 it seems that the epi's mate's cokernel object isn't dualizable... i think ...
??so ... the condition on an epi between dualizable objects of being "good" does seem to break down into a couple of non-automatic, non-vacuous stages, where the failure can occur at either stage... first, the mate's cokernel object might not be dualizable; then if it is, the original epi might still not be the mate's cokernel's mate's cokernel ... ??...
...assuming that i didn't miscalculate too badly in these examples...
a couple of times recently i've found myself getting confused about the relationship between various concepts of dual... now i'm thinking that there's a very simple relationship between two of them that i've understood in the past and really shouldn't have forgotten about, which might be the relationship that i was searching for in some of the recent instances...
so let's see, consider adjoint 1-cells in a 2-category... ??and consider the yoneda embedding of the 2-category... ???any 2-functor preserves adjoints... ??...
??maybe some variance / level confusion here??... flip/slip ... ??...
??so for example with adjoint objects in a monoidal category, tensoring with an adjoint of an object x provides an adjoint to tensoring with x, thus giving an "internal homming from x" functor ... ?? so then [x,1] = 1 tensor adjoint(x) = adjoint(x) ... ??...
?????so what were the recent instances?? one was in discussing "absolute colimits" (or something...) with mike shulman on the n-category cafe... "half-exact functor induces homotopy-exact ..." ... i should work this stuff out...
there's something suggestive here about absolute colimits and ... ??adjoint objects as a strengthened form of a certain sort of internal hom ... ???maybe really close connection here???....
in that discussion with shulman (and also that guy richard something-or-other...) i also found myself bumping into ideas that i tried to learn from todd... ??related to things like "cartesian bicategories" which as a matter of fact i'm again trying to learn from him...
also, this business about line objects and so forth... "good embedding" / "good epi" in an algebraic-geometric theory... "quasi-regular epi" ...??...
so let's see, consider adjoint 1-cells in a 2-category... ??and consider the yoneda embedding of the 2-category... ???any 2-functor preserves adjoints... ??...
??maybe some variance / level confusion here??... flip/slip ... ??...
??so for example with adjoint objects in a monoidal category, tensoring with an adjoint of an object x provides an adjoint to tensoring with x, thus giving an "internal homming from x" functor ... ?? so then [x,1] = 1 tensor adjoint(x) = adjoint(x) ... ??...
?????so what were the recent instances?? one was in discussing "absolute colimits" (or something...) with mike shulman on the n-category cafe... "half-exact functor induces homotopy-exact ..." ... i should work this stuff out...
there's something suggestive here about absolute colimits and ... ??adjoint objects as a strengthened form of a certain sort of internal hom ... ???maybe really close connection here???....
in that discussion with shulman (and also that guy richard something-or-other...) i also found myself bumping into ideas that i tried to learn from todd... ??related to things like "cartesian bicategories" which as a matter of fact i'm again trying to learn from him...
also, this business about line objects and so forth... "good embedding" / "good epi" in an algebraic-geometric theory... "quasi-regular epi" ...??...
??so let's consider the "separated coherent presheafs over p^1" ...(not very systematic nonce terminology here...)
??or should we maybe even try p^0 instead? ...
let's take the cokernel of the mate of the obvious epi...
hmmm... ??what about the idea that if (for example...) we're looking for objects in [the theory of a line object equipped with an epi from 1] that must become adjoint to the cokernel of the mate of the epi, one possibility to check might be the internal hom [cokernel of mate of the epi,1] ?? ... ??or something?? ... on the other hand there's also the adjoint in the quasicoherent sheaf category, included back via fully faithful right adjoint ... ???....
??the obvious epi is the change-of-filtration morphism... 1 = "all born at 0", L = "all born at -1" ... ?? ...
??then the mate of this is the change-of-filtration from "all born at 1" to "all born at 0" ?? .... ??so then what's the cokernel of this?? ??hmm, shouldn't it be 0???? .... because the mate is again epi .... ???? ...
???doesn't it seem like something's screwed up here???.....
??like i'm almost claiming that there's an "equation" that holds in the free example but that doens't occur in some other example ... ???or something?? ...
???something about ... ??"mate for life" ??? ...??...
oh wait a minute... this is the p^0 example... ??maybe the mate really is epi in this example ... ????... ??for life?? ... ??yes, i think so...
??is that going to screw up the purpose of the example, so we might have to use p^1 instead of p^0 ?? or what??
??or should we maybe even try p^0 instead? ...
let's take the cokernel of the mate of the obvious epi...
hmmm... ??what about the idea that if (for example...) we're looking for objects in [the theory of a line object equipped with an epi from 1] that must become adjoint to the cokernel of the mate of the epi, one possibility to check might be the internal hom [cokernel of mate of the epi,1] ?? ... ??or something?? ... on the other hand there's also the adjoint in the quasicoherent sheaf category, included back via fully faithful right adjoint ... ???....
??the obvious epi is the change-of-filtration morphism... 1 = "all born at 0", L = "all born at -1" ... ?? ...
??then the mate of this is the change-of-filtration from "all born at 1" to "all born at 0" ?? .... ??so then what's the cokernel of this?? ??hmm, shouldn't it be 0???? .... because the mate is again epi .... ???? ...
???doesn't it seem like something's screwed up here???.....
??like i'm almost claiming that there's an "equation" that holds in the free example but that doens't occur in some other example ... ???or something?? ...
???something about ... ??"mate for life" ??? ...??...
oh wait a minute... this is the p^0 example... ??maybe the mate really is epi in this example ... ????... ??for life?? ... ??yes, i think so...
??is that going to screw up the purpose of the example, so we might have to use p^1 instead of p^0 ?? or what??
talking to adam katz about cyclic operads and di-operads and stuff...
??commutative operad and lie operad as cyclic operads... their cyclic algebras being resp frobenius algebras and [lie algebras with invariant bilinear form] ?? or something...
??finding cyclic operads weird at the moment... ??because you can only straightforwardly get a di-operad from them instead of a prop??? ...??or something??
??maybe something about...
prop : prop under "bilinear from prop" (??or something???) ::
di-operad : cyclic operad
???? .... ???or something?? ...
??situations where duals exist, vs where things are self-dual ...???...??...
??and so forth?? ... ??lots of possibilities here??...
??some of these things i've probably gotten confused about before... ???....
??something about ... ??"natural structure of underlying operad of prop under prop p" ... ??and so forth ... ??...
??commutative operad and lie operad as cyclic operads... their cyclic algebras being resp frobenius algebras and [lie algebras with invariant bilinear form] ?? or something...
??finding cyclic operads weird at the moment... ??because you can only straightforwardly get a di-operad from them instead of a prop??? ...??or something??
??maybe something about...
prop : prop under "bilinear from prop" (??or something???) ::
di-operad : cyclic operad
???? .... ???or something?? ...
??situations where duals exist, vs where things are self-dual ...???...??...
??and so forth?? ... ??lots of possibilities here??...
??some of these things i've probably gotten confused about before... ???....
??something about ... ??"natural structure of underlying operad of prop under prop p" ... ??and so forth ... ??...
Wednesday, January 19, 2011
??walking good embedding ag theory?? ... and so forth ...
??consider objects in ag theory of "line object l equipped with epi from 1^[n+1]"
that go to dual of cokernel of mate of epi in coherent sheaves over projective n-space... ???and so forth?? ...
??maybe some parallel idea involving filtered vector spaces... ??....
??does dualizableness of certain cokernel guarantee goodness of epi, or is that maybe automatic and the critical part is elsewhere ??... and so forth ... ??...
??consider objects in ag theory of "line object l equipped with epi from 1^[n+1]"
that go to dual of cokernel of mate of epi in coherent sheaves over projective n-space... ???and so forth?? ...
??maybe some parallel idea involving filtered vector spaces... ??....
??does dualizableness of certain cokernel guarantee goodness of epi, or is that maybe automatic and the critical part is elsewhere ??... and so forth ... ??...
so let e : x -> y be an epi between dualizable objects (or something...) in a symmetric monoidal finitely cocomplete k-linear category... ??...
??then consider... the mate of e... ??and the cokernel of that... and the mate of that... ??... and the cokernel of that... and the mate of that... ??? and whether that returns to the original?? or something?? ??is there a comparison map here to invert?? ??or something?? ...
??consider smfcc k-linear cat where all adjoints exist (or something...) and the loop always closes that way... ???...
??how well-defined is this loop-closure property, in general?? ... or something...
??then consider... the mate of e... ??and the cokernel of that... and the mate of that... ??... and the cokernel of that... and the mate of that... ??? and whether that returns to the original?? or something?? ??is there a comparison map here to invert?? ??or something?? ...
??consider smfcc k-linear cat where all adjoints exist (or something...) and the loop always closes that way... ???...
??how well-defined is this loop-closure property, in general?? ... or something...
notes for next discussion with todd
??should try to introduce "toric" versions of concepts?? ...
??proof of certain universal property of coherent sheaves over P^1 ...
??issue of ... ??forcing strong dual to exist... ??whether this promotes weak dual... ??and so forth...
??walking adjunction... ??and/or "walking bosonic adjunction" ... ???....
???something about quasitopos and so forth... ??"zariski quasitopos" ... ??...
??hecke bicategory as monoidal and part of tricategory?? ...
??proof of certain universal property of coherent sheaves over P^1 ...
??issue of ... ??forcing strong dual to exist... ??whether this promotes weak dual... ??and so forth...
??walking adjunction... ??and/or "walking bosonic adjunction" ... ???....
???something about quasitopos and so forth... ??"zariski quasitopos" ... ??...
??hecke bicategory as monoidal and part of tricategory?? ...
Tuesday, January 18, 2011
??so what about this quasitopos analogy (or something) we seem to be running into here??
???something about... ??topos has subobject classifier but pulling back along geometric morphism doesn't preserve it, maybe analogous to good tensor category has kernels but pulling back along geometric morphism doesn't preserve it... ???or something?? .... ??something about kernels and "obstruction theory" ... ?????hmmm.... ????....
??so what about distributivity condition satisfied by toposes but not by quasitoposes????? or something.... ???....
??so what _about_ "zariski quasitopos of a smfcca" ??? ... and so forth...
is the category of modules of a ring object in a quasitopos generally a finitely cocomplete algebroid but not an abelian category?? (??relevance for smooth spaces?? ...?? ??also consider examples like simplicial complexes... ???...)
??so what about quasi-locales???? (or something...)
??so what about "the logic of quasitoposes" ??? might it be something about ... ???some one particular "modal operator" or something???? something about comparable pair of grothendieck topologies... ??something about "ghost points" ... (??relationship to "voodoo mathematics" ??...) ??... ??is it really just "split-level" like this?? ??if so then how do you recover an injective geometric morphism of toposes from a quasitopos?? ... or something... ???... ???what _about_ relationship to enrichment in categories ct in groupoids?? ... and so forth...
??so what _about_ coherent sheaves over the (zariski) line, separated for the topology which removes the origin ?? ... and so forth...
??also simply sheaves over it, separated for that topology ...
?now realizing that a lot of my vague memories about "quasitopos" are tangled together with idea of "concrete quasitopos" ...
??locale as quasitopos where all points are ghost points ??? ... or something??
something about "distributive" aspect of topos vs of locale ... and so forth... ??...
??free quasitopos on ...?? ... ???... and so forth...
??"quasi-geometric theory" ... ???...
??quasi-topos as mix of topos and locale?? or something?? ...
???something about... ??topos has subobject classifier but pulling back along geometric morphism doesn't preserve it, maybe analogous to good tensor category has kernels but pulling back along geometric morphism doesn't preserve it... ???or something?? .... ??something about kernels and "obstruction theory" ... ?????hmmm.... ????....
??so what about distributivity condition satisfied by toposes but not by quasitoposes????? or something.... ???....
??so what _about_ "zariski quasitopos of a smfcca" ??? ... and so forth...
is the category of modules of a ring object in a quasitopos generally a finitely cocomplete algebroid but not an abelian category?? (??relevance for smooth spaces?? ...?? ??also consider examples like simplicial complexes... ???...)
??so what about quasi-locales???? (or something...)
??so what about "the logic of quasitoposes" ??? might it be something about ... ???some one particular "modal operator" or something???? something about comparable pair of grothendieck topologies... ??something about "ghost points" ... (??relationship to "voodoo mathematics" ??...) ??... ??is it really just "split-level" like this?? ??if so then how do you recover an injective geometric morphism of toposes from a quasitopos?? ... or something... ???... ???what _about_ relationship to enrichment in categories ct in groupoids?? ... and so forth...
??so what _about_ coherent sheaves over the (zariski) line, separated for the topology which removes the origin ?? ... and so forth...
??also simply sheaves over it, separated for that topology ...
?now realizing that a lot of my vague memories about "quasitopos" are tangled together with idea of "concrete quasitopos" ...
??locale as quasitopos where all points are ghost points ??? ... or something??
something about "distributive" aspect of topos vs of locale ... and so forth... ??...
??free quasitopos on ...?? ... ???... and so forth...
??"quasi-geometric theory" ... ???...
??quasi-topos as mix of topos and locale?? or something?? ...
??filtered k-modules as walking line object with epi section smfcca ??
??martin says that wrt exact tensor functors (or something...) the coherent sheaves over projective n-space are the walking line object with jointly epi [n+1]-tuple of sections... ??or something?? ...
hmm... so consider the smfcca of k[x]-modules where x acts injectively ... ??? ... ??????.... ??hmm, might there be something going on here about category-valued vs groupoid-valued universal properties??
??martin says that wrt exact tensor functors (or something...) the coherent sheaves over projective n-space are the walking line object with jointly epi [n+1]-tuple of sections... ??or something?? ...
hmm... so consider the smfcca of k[x]-modules where x acts injectively ... ??? ... ??????.... ??hmm, might there be something going on here about category-valued vs groupoid-valued universal properties??
Monday, January 17, 2011
??almost seems like underlying finitely cocomplete algebroid of an abelian category obeys some extra "equational law" (or something...), but that's probably not correct... ??so then what _is_ going on?? ... ...??something about tensor product as well, or ... ???... ??hmm, what _about_ structure not preserved by the morphisms, but which might somehow affect the structure that _is_ preserved?? ... in general... ??doesn't it seem like this could happen, in general?? ... ??or something???... ??any conspicuous examples??? ...
??try to develop simplicial approach to rep of groupoid, then adapt to lie algebroid case??? ???or something?? ???something about... ???for each j, we should have a vector bundle over the set of j-simplexes in the nerve of the groupoid... with the fiber over a particular j-simplex x being the vector space of "coherent sections" of the representation over x ... ??or something ...
???hmm, so what about dg modules of a dgca as special (??in just what way???) simplicial modules of its dold-kan correspondent ?? ... or something...
??idea that "koszul duality" (or something...) in "lie algebra case" (or something...) might work a bit differently, or perhaps rather be capable of being thought of a bit differently, from the way that i was thinking about it ... ??i was thinking something like... "a lie algebra as a vector space together with some sort of alternative dgca structure on its exterior algebra ..." .... ????or something????...... ???whereas now it seems like it should be something like... ??"a lie algebra as a certain kind (meaning with just plain property...) of dgca" ... ???or something???
??try to develop simplicial approach to rep of groupoid, then adapt to lie algebroid case??? ???or something?? ???something about... ???for each j, we should have a vector bundle over the set of j-simplexes in the nerve of the groupoid... with the fiber over a particular j-simplex x being the vector space of "coherent sections" of the representation over x ... ??or something ...
???hmm, so what about dg modules of a dgca as special (??in just what way???) simplicial modules of its dold-kan correspondent ?? ... or something...
??idea that "koszul duality" (or something...) in "lie algebra case" (or something...) might work a bit differently, or perhaps rather be capable of being thought of a bit differently, from the way that i was thinking about it ... ??i was thinking something like... "a lie algebra as a vector space together with some sort of alternative dgca structure on its exterior algebra ..." .... ????or something????...... ???whereas now it seems like it should be something like... ??"a lie algebra as a certain kind (meaning with just plain property...) of dgca" ... ???or something???
notes for next discussion with martin
??toric case?? ??or do we not understand it well enough to get into that now?? ??so what _about_ the analog of the main confusions (...??...) in the toric case???? ....
??return (??...) to the case of pure smfcca ... try to state universal property of coherent sheaves wrt this doctrine nicely ... ??then also consider whether the "extra" holds automotically in an "abelian" context ... ??or something??? ... ??try to "systematize" ??? ....
??something about _not_ getting rid of irrelevant point... ??getting rid of relevant one instead?? ...???maybe something about "separated presheaf" ????? ..... and so forth .... ???...
??return (??...) to the case of pure smfcca ... try to state universal property of coherent sheaves wrt this doctrine nicely ... ??then also consider whether the "extra" holds automotically in an "abelian" context ... ??or something??? ... ??try to "systematize" ??? ....
??something about _not_ getting rid of irrelevant point... ??getting rid of relevant one instead?? ...???maybe something about "separated presheaf" ????? ..... and so forth .... ???...
notes for next discussion with derek
??something about relationship between bruhat classes and nilpotent lie algebras... ??what were some of the ideas here???.... ??something about nilradicals... ??also something about... bijectiveness of exponential map for nilpotent lie algebra, and bruhat classes (or something...) as shaped like vector spaces... or something... ??something about ... bruhat class dimension numerology and convex cones in root diagram... and so forth... ???also something about how weird it seems for tangent space of homogneeous space to be a lie algebra; what might that mean??? .... and so forth... oh yeah, and then there's all that stuff about correspondence between graded nilpotent lie algebras and... ??well, something about some sort of distributions ... ?? and so forth ...
??something about how to boost by a gimbal configuration?? ... or wait, maybe those are those idealized impossible boosts... ??... ??"sub-isometry" then??? ??"linear contraction" or something?? ...
??conformal completion of 2+1 flat spacetime... something about change of favorite event... interlocking bruhat classifications ... ??and so forth...
??answer to puzzle... light cone... basepoint singularity... zariski tangent space...
??something about how to boost by a gimbal configuration?? ... or wait, maybe those are those idealized impossible boosts... ??... ??"sub-isometry" then??? ??"linear contraction" or something?? ...
??conformal completion of 2+1 flat spacetime... something about change of favorite event... interlocking bruhat classifications ... ??and so forth...
??answer to puzzle... light cone... basepoint singularity... zariski tangent space...
Saturday, January 15, 2011
martin e-mailed me about a difficulty in the attempted proof that the coherent sheaves over the projective space of v is the "walking invertible quotient of v" symmetric monoidal finitely cocomplete algebroid... i'll try thinking outloud about it here... i'm not surprised that a difficulty shows up here, but i'm still somewhat optimistic about getting past it...
hmm... been chatting about it with martin a bit... wouldn't be surprised if i've been making some really stupid mistake here...
??to compel the cokernel of f to be zero is to invert the comparison map from...
??well perhaps either from 0 to the cokernel or vice versa, but seems like one of those makes better conceptual sense... the cokernel is always a quotient object of the codomain... ??so maybe it makes better sense to say that the comparison (??if that's an appropriate name for it... ??...) map from the cokernel to 0 is invertible...
??then maybe we should consider inverting all of the morphisms from bounded objects to zero ... ??or something like that??? .... hmmm, maybe not even that's justified ... ???or something?? ... i was going to say though that maybe ... well, some coinverters are easy to "explicitly calculate" and some less so... not sure how easy this (??...) one might be ... ??...
??so suppose that we invert the map from x to 0 in a cocontinuous way, and suppose
also that we have a map f : y -> x st cok(f) = x ... ???then does cok(f) then necessarily become 0 ?? .... hmm, at the moment i seem to be in danger of thinking that that's trivially true... ???or something??? ... ok, yes, that seems trivially true, but it's not the relevant question ... so then what _is_ the relevant question??
well, one question is whether f becomes invertible... but of course the answer to that is pretty obviously no... but perhaps just to nail it into the ground i should describe some universal or at least prototypical example ... or something...
well, so let's consider the "walking map" ... hmmm... this is reminding me now of my discussions with julie bergner about serre subcategories and so forth in the context of quiver representations... but that was pretty certainly before i was clear about what doctrine i should be working in, right?? ... actually i'm even a bit unsure about whether my motivation in those discussions had to do with coherent sheaves at all; i'm not certain that i really knew anything about coherent sheaves by that time... ??...
anyway, let's consider representations of the a2 quiver, and let's cocontinuously compel ....
hmmm... this could be interesting... i'm pretty sure that the discussions with julie were before i'd realized about those nice simple examples of finitely cocomplete algebroids that aren't abelian categories... which are probably showing up here...
so maybe this is hinting to us that there _is_ some pretty easily explicitly
describable localization (but also subcategory...) of the graded modules that has the universal property that i've been (presumably mistakenly) ascribing to the coherent sheaves ... and this localization is not an abelian category... or something ... ??so is there a really obvious guess to make here??
??so 0->1 included into 1->1 is mono ... but it's also the generic morphism in the cocomplete context... or something... and cocontinuously forcing its cokernel to be zero gives you the generic epi... which lives in the walking epi, which is the "fg monos" ... ??so it seems that in the fg monos, the cokernel of the inclusion of 0->1 into 1->1 is, of course not the old cokernel which is 1->0 and not mono, but rather the "monicization" of this... which is perhaps 0->0 ??? or something??
??let's try testing that... ??a morphism from the mono 1->1 to the mono
m : x >-> y
(aka "an element of domain(m)" ??) which has the property that [preceding it by the inclusion from 0->1 to 1->1 gives zero] (aka "it gets taken to zero by m"??) is essentially ... ???just zero?? ??that's sort of the whole point of monos, that the only thing they take to zero is zero ... ???or something??...
right... i think... so 0->1 included into 1->1 is epi in the fg monos... or something...
??but what's the lesson supposed to be here???
??that... ??well, for one thing, 0->1 included into 1->1 certainly isn't invertible in the fg monos ... it's just epi ... ?? or something??
??so what about the graded modules where ... ???????? ....
??so _did_ we miss something about "thick subcategory" being kernel of _exact_ functor ?? ??or something?? ... if so then sounds like pretty stupid mistake ... ???... still not sure about this... ??.... ??something about finite vs arbitrary co/limits?? or something???
??so what about attempted analogy (of _some_thing ... ???...) to grothendieck topology.... ????? ... ???what about tensor products vs cartesian products ? ...
... ???...
??so what about idea that if we think somewhat systematically about all the sorts of semantically (or something...) motivated examples that we've tried to develop in doctrine of symmetric monoidal finitely cocomplete algebroids then we ought to be able to figure out what's wrong... ??or something??? ... ???is it already somewhat near-obvious that what's wrong has something to do with... abelianness vs mere finite cocompleteness... ??what _about_ whether there's some nice way to "get abelianness" (or something...) without demanding (too much...) more than "right-exactness" of the tensor functors and without losing "presentableness" (or something...) ?? .... ????...
hmm... ??if we're not supposed to require exactness, then... ??why _do_ we always seem to mod out by a thick subcategory when passing to a subscheme.... ???or something??... ... ????... ???
??hmmm... "evaluation at a point" isn't flat... ??but then it's not a localization... (??why do you think they call it "localization" ????..... ??...) ...??so what about the "localization part" of such an evaluation?? ... hmm, usual localization/conservative factorization ?? or something??? localization (??or something??) is flat; is conservative "anti-flat" or something??? ....
??any relationship to some puzzle that we had about "artin-wraith glueing" in algebraic geometry?? or something ... ??something about coherent sheaves ... ???...
so what _about_ universal property wrt _exact_ tensor functor?? .... ??despite problematicness of compatibility between tensor product and kernels?? or something?? ... ????..... ... ???....
hmm... ??so consider... the objects that think that a given morphism (or collection thereof) is epi ... ???or something...
??some sort of level slip involving "exact tensor functor" and "flat" ... ??or something?? ... ??something about... ??a line object seems already about as flat as it can get, but .... ???or something???
so what _about_ the universal property of the coherent sheaves over projective space as a smfcca ??... as martin suggests... ??... ??seems somewhat straightforward?? ... but... ??do we at all understand how to look at it systematically?? ... ???.... ??any possibility of expressing it in terms of systematically expressed type of (co-)"embedding"?? ... enhanced epi ... ??or something?? ??or is there some sort of "context-sensitiveness" (or something...) that screws this up?? ... ???hmmm... ???...
??what about sheaf vs separated presheaf here... ??not too clear yet why it might be showing up... ??hmm... ??when we force an otherwise completely generic morphism to become epi, then... ??maybe it makes sense that in that case the corresponding "sheaf condition" (vs "separated presheaf condition" ... ??...) amounts to making the morphism invertible ... ?? ... ??because .... ??? ... ????.....
??is there some nice systematic relationship between universal properties wrt flat ag morphisms and wrt general ag morphisms, allowing maybe to understand the latter in terms of the former?? ??or something like that??...
??relationship between martin's idea about element-wise proofs for tensor categories, and... ??some "doctrine" ideas... ??or something??...
[cut from skype-hat with martin:
one vague idea that i have that i might like to discuss eventually is that because of the importance of abelian categories as compared to mere finitely cocomplete additive categories, it might turn out to be a good idea to place more emphasis on exact tensor functors than on merely right-exact ones, even though right-exact ones ... because of "incocompleteness" issue ... ?? ...]
??so consider ... ???line object L tw sections s,t st ... ?? (s,t) is the cokernel of a certain morphism from ... ?? ...
hmm... been chatting about it with martin a bit... wouldn't be surprised if i've been making some really stupid mistake here...
??to compel the cokernel of f to be zero is to invert the comparison map from...
??well perhaps either from 0 to the cokernel or vice versa, but seems like one of those makes better conceptual sense... the cokernel is always a quotient object of the codomain... ??so maybe it makes better sense to say that the comparison (??if that's an appropriate name for it... ??...) map from the cokernel to 0 is invertible...
??then maybe we should consider inverting all of the morphisms from bounded objects to zero ... ??or something like that??? .... hmmm, maybe not even that's justified ... ???or something?? ... i was going to say though that maybe ... well, some coinverters are easy to "explicitly calculate" and some less so... not sure how easy this (??...) one might be ... ??...
??so suppose that we invert the map from x to 0 in a cocontinuous way, and suppose
also that we have a map f : y -> x st cok(f) = x ... ???then does cok(f) then necessarily become 0 ?? .... hmm, at the moment i seem to be in danger of thinking that that's trivially true... ???or something??? ... ok, yes, that seems trivially true, but it's not the relevant question ... so then what _is_ the relevant question??
well, one question is whether f becomes invertible... but of course the answer to that is pretty obviously no... but perhaps just to nail it into the ground i should describe some universal or at least prototypical example ... or something...
well, so let's consider the "walking map" ... hmmm... this is reminding me now of my discussions with julie bergner about serre subcategories and so forth in the context of quiver representations... but that was pretty certainly before i was clear about what doctrine i should be working in, right?? ... actually i'm even a bit unsure about whether my motivation in those discussions had to do with coherent sheaves at all; i'm not certain that i really knew anything about coherent sheaves by that time... ??...
anyway, let's consider representations of the a2 quiver, and let's cocontinuously compel ....
hmmm... this could be interesting... i'm pretty sure that the discussions with julie were before i'd realized about those nice simple examples of finitely cocomplete algebroids that aren't abelian categories... which are probably showing up here...
so maybe this is hinting to us that there _is_ some pretty easily explicitly
describable localization (but also subcategory...) of the graded modules that has the universal property that i've been (presumably mistakenly) ascribing to the coherent sheaves ... and this localization is not an abelian category... or something ... ??so is there a really obvious guess to make here??
??so 0->1 included into 1->1 is mono ... but it's also the generic morphism in the cocomplete context... or something... and cocontinuously forcing its cokernel to be zero gives you the generic epi... which lives in the walking epi, which is the "fg monos" ... ??so it seems that in the fg monos, the cokernel of the inclusion of 0->1 into 1->1 is, of course not the old cokernel which is 1->0 and not mono, but rather the "monicization" of this... which is perhaps 0->0 ??? or something??
??let's try testing that... ??a morphism from the mono 1->1 to the mono
m : x >-> y
(aka "an element of domain(m)" ??) which has the property that [preceding it by the inclusion from 0->1 to 1->1 gives zero] (aka "it gets taken to zero by m"??) is essentially ... ???just zero?? ??that's sort of the whole point of monos, that the only thing they take to zero is zero ... ???or something??...
right... i think... so 0->1 included into 1->1 is epi in the fg monos... or something...
??but what's the lesson supposed to be here???
??that... ??well, for one thing, 0->1 included into 1->1 certainly isn't invertible in the fg monos ... it's just epi ... ?? or something??
??so what about the graded modules where ... ???????? ....
??so _did_ we miss something about "thick subcategory" being kernel of _exact_ functor ?? ??or something?? ... if so then sounds like pretty stupid mistake ... ???... still not sure about this... ??.... ??something about finite vs arbitrary co/limits?? or something???
??so what about attempted analogy (of _some_thing ... ???...) to grothendieck topology.... ????? ... ???what about tensor products vs cartesian products ? ...
... ???...
??so what about idea that if we think somewhat systematically about all the sorts of semantically (or something...) motivated examples that we've tried to develop in doctrine of symmetric monoidal finitely cocomplete algebroids then we ought to be able to figure out what's wrong... ??or something??? ... ???is it already somewhat near-obvious that what's wrong has something to do with... abelianness vs mere finite cocompleteness... ??what _about_ whether there's some nice way to "get abelianness" (or something...) without demanding (too much...) more than "right-exactness" of the tensor functors and without losing "presentableness" (or something...) ?? .... ????...
hmm... ??if we're not supposed to require exactness, then... ??why _do_ we always seem to mod out by a thick subcategory when passing to a subscheme.... ???or something??... ... ????... ???
??hmmm... "evaluation at a point" isn't flat... ??but then it's not a localization... (??why do you think they call it "localization" ????..... ??...) ...??so what about the "localization part" of such an evaluation?? ... hmm, usual localization/conservative factorization ?? or something??? localization (??or something??) is flat; is conservative "anti-flat" or something??? ....
??any relationship to some puzzle that we had about "artin-wraith glueing" in algebraic geometry?? or something ... ??something about coherent sheaves ... ???...
so what _about_ universal property wrt _exact_ tensor functor?? .... ??despite problematicness of compatibility between tensor product and kernels?? or something?? ... ????..... ... ???....
hmm... ??so consider... the objects that think that a given morphism (or collection thereof) is epi ... ???or something...
??some sort of level slip involving "exact tensor functor" and "flat" ... ??or something?? ... ??something about... ??a line object seems already about as flat as it can get, but .... ???or something???
so what _about_ the universal property of the coherent sheaves over projective space as a smfcca ??... as martin suggests... ??... ??seems somewhat straightforward?? ... but... ??do we at all understand how to look at it systematically?? ... ???.... ??any possibility of expressing it in terms of systematically expressed type of (co-)"embedding"?? ... enhanced epi ... ??or something?? ??or is there some sort of "context-sensitiveness" (or something...) that screws this up?? ... ???hmmm... ???...
??what about sheaf vs separated presheaf here... ??not too clear yet why it might be showing up... ??hmm... ??when we force an otherwise completely generic morphism to become epi, then... ??maybe it makes sense that in that case the corresponding "sheaf condition" (vs "separated presheaf condition" ... ??...) amounts to making the morphism invertible ... ?? ... ??because .... ??? ... ????.....
??is there some nice systematic relationship between universal properties wrt flat ag morphisms and wrt general ag morphisms, allowing maybe to understand the latter in terms of the former?? ??or something like that??...
??relationship between martin's idea about element-wise proofs for tensor categories, and... ??some "doctrine" ideas... ??or something??...
[cut from skype-hat with martin:
one vague idea that i have that i might like to discuss eventually is that because of the importance of abelian categories as compared to mere finitely cocomplete additive categories, it might turn out to be a good idea to place more emphasis on exact tensor functors than on merely right-exact ones, even though right-exact ones ... because of "incocompleteness" issue ... ?? ...]
??so consider ... ???line object L tw sections s,t st ... ?? (s,t) is the cokernel of a certain morphism from ... ?? ...
i still have this idea that certain ways of getting a "ringed topos" from a symmetric monoidal finitely cocomplete algebroid should correspond to interpretations of doctrines, but i still haven't worked out the details enough to know whether it really works; the lack of someone to discuss it with is really slowing me down... (a couple of days after i wrote that i started discussing some of this stuff with todd though...)
i've vaguely thought a bit about trying to develop some sort of decategorified analog of this, but at least one attempt to do this seems problematic enough to make me wonder whether it's telling me that it's the wrong idea in the categorified case as well...
consider obtaining a distributive lattice (or "frame" or something...) from a commutative ring in the more or less usual "zariski" way... or something ilke that...
one question is whether this corresponds to something like an interpretation of lex theories... ????....
recently learning a bit about "malcev varieties" from the viewpoint that it has something to do with lattices of congruences being modular... ??thought of as a step on the way to distributiveness?? ... has me wondering about thinking of the decategorified zariski construction this way... ??and about how this compares to the doctrine interpretation idea... ??and also about the possibility of categorifying the malcev variety idea or its "distributive analog" ....
??what about... ??something about general idea of... ??relationships between properties of categories and of subobject lattices here?? ... like "distributiveness" ... ???and maybe "modularness" in some way???... hmmm...??...
??sort of de/categorification here?? ... something about locale vs topos, and so forth... ??...
i've vaguely thought a bit about trying to develop some sort of decategorified analog of this, but at least one attempt to do this seems problematic enough to make me wonder whether it's telling me that it's the wrong idea in the categorified case as well...
consider obtaining a distributive lattice (or "frame" or something...) from a commutative ring in the more or less usual "zariski" way... or something ilke that...
one question is whether this corresponds to something like an interpretation of lex theories... ????....
recently learning a bit about "malcev varieties" from the viewpoint that it has something to do with lattices of congruences being modular... ??thought of as a step on the way to distributiveness?? ... has me wondering about thinking of the decategorified zariski construction this way... ??and about how this compares to the doctrine interpretation idea... ??and also about the possibility of categorifying the malcev variety idea or its "distributive analog" ....
??what about... ??something about general idea of... ??relationships between properties of categories and of subobject lattices here?? ... like "distributiveness" ... ???and maybe "modularness" in some way???... hmmm...??...
??sort of de/categorification here?? ... something about locale vs topos, and so forth... ??...
Friday, January 14, 2011
talking to chris rogers about my recent realizations about mistakes that i'd been making in thinking about d-modules...
??something about ... ?connecting "dg scheme as algebraic stack" with "smfcca as algebraic stack" by figuring out what the "coherent sheaves" over the stack corresponding to a dg scheme are... in the case where the dg scheme is "codiscrete" (the de rham case), these coherent (or i guess quasicoherent actually... or something...) sheaves should be the d-modules... in more general cases it should be interesting and not so difficult to figure out what they are....
??relationship to dg modules of dgcas ??? .....
??does a d-module have some sort of "nerve" or "co-nerve" or something??? ....
and so forth .... ????....
hmm, at this point perhaps the questions we're asking making sense only for those dgcas that are sufficiently "low-dimensional" in a certain sense... i forget to what extent we might have previously worked out what this sense is... ??maybe just something about being generated by generators (and/or relators??? or something???) in low degrees???? or something??... ??hmm, might this give some clues about special dg modules that might correspond to dg modules?? or something??... and so forth...
???something about ... ??"koszul duality" ... "bar construction" ... "chevalley-eilenberg" .... ???? and so forth... ???....
hmm, seems like... might be that with certain "degree constraint" (or something) on dgca, we might be getting basically just stacks corresponding to action groupoids of lie algebra actions on commutative algebras ... ??or something??...
hmm, or maybe just something about groupoids in general ... or something... or i guess that i mean lie algebroids or something... ???....
??something about... ??thinking of a certain degree-constrained sort of dgca as a "lie algebroid", and describing what the "coherent sheaves" should be in terms of the dgca ... ???or something???...
what about ... ??allowing only lower-degree generators, but automatically _imposing_ higher-degree relators?? ... or something ... ????..... ??anything sesqui-clever here?? or something???....
??semi-direct product symbol "x|" ...???
weyl algebra s(v*) x| s(v) ...
induced morphism to s(v*) x| env(polynomial vector fields on v) ...
??but confusion about how it seems like there's almost a morphism going the other way... ??or something??? ... at least, that enveloping algebra maps into the weyl algebra... ???or something????.... ...and so forth...
??something about confusion between "covariant differentiation" and "lie differentiation" ... ???"lie differentiation" as something about s(v*) x| env(polynomial vector fields on v)... "covariant differentiation" as something about s(v*) x| s(v) ... ??or something???.....
maybe i should try to check how standard my usage of "semi-direct product" is here... ??not clear to me offhand how it relates to semi-direct product of group acting on other group ... ??... ??funny how in the case of the weyl algebra there's this sort of symmetry between the actor and the acted-upon ... ??or something??? ... also something about relationships between nilpotent and solvable ... ??and so forth???....
??vague memory of ...???being surprised by some sort of semi-direct product asepct (??in group sense maybe??) of weyl algebra... ?? ...??semi-recently?? ... ??what was_that about??... ??how closely did it tie in with old idea about quantum mechanics and semi-direct product and "measurement process disturbing the quantity being measured"?? ... and so forth... ??might it have been based on incorrect belief that i've recently been trying to recover from??? .... ??again of course, what about relationship to "semi-direct product of hopf algebra acting on commutative algebra" ?? ... and so forth... ???...
??questions (or something...) for chris rogers:
what exactly is a "lie algebroid"?? ... (not worrying about "lie n-algebroid" yet...)
can we think of lie algebroids as forming a reflective (or something) full subcategory of "dg manifolds" or something like that? (allow variations on "manifold" here of course...) exactly how?? some sort of "degree cutoff" or something??...
do we have a clear concept of "quasicoherent sheaf over a lie algebroid", so to speak?? (mainly in "algebraic" case, perhaps?? ... though maybe not exclusively ...) ???... if so, then can we think of these as something like special dg modules of the alleged corresponding dgca?? ... ??...
...and so forth...
??what about trying to relate diff eq aspect of d-module to that of dg module and/or of dgca ??? .... and so forth... ???....
??so what _about_ diff eqs involving infinitesimal simplexes... ??? ... ??"pfaffian..." .... ????....
??something about ... ?connecting "dg scheme as algebraic stack" with "smfcca as algebraic stack" by figuring out what the "coherent sheaves" over the stack corresponding to a dg scheme are... in the case where the dg scheme is "codiscrete" (the de rham case), these coherent (or i guess quasicoherent actually... or something...) sheaves should be the d-modules... in more general cases it should be interesting and not so difficult to figure out what they are....
??relationship to dg modules of dgcas ??? .....
??does a d-module have some sort of "nerve" or "co-nerve" or something??? ....
and so forth .... ????....
hmm, at this point perhaps the questions we're asking making sense only for those dgcas that are sufficiently "low-dimensional" in a certain sense... i forget to what extent we might have previously worked out what this sense is... ??maybe just something about being generated by generators (and/or relators??? or something???) in low degrees???? or something??... ??hmm, might this give some clues about special dg modules that might correspond to dg modules?? or something??... and so forth...
???something about ... ??"koszul duality" ... "bar construction" ... "chevalley-eilenberg" .... ???? and so forth... ???....
hmm, seems like... might be that with certain "degree constraint" (or something) on dgca, we might be getting basically just stacks corresponding to action groupoids of lie algebra actions on commutative algebras ... ??or something??...
hmm, or maybe just something about groupoids in general ... or something... or i guess that i mean lie algebroids or something... ???....
??something about... ??thinking of a certain degree-constrained sort of dgca as a "lie algebroid", and describing what the "coherent sheaves" should be in terms of the dgca ... ???or something???...
what about ... ??allowing only lower-degree generators, but automatically _imposing_ higher-degree relators?? ... or something ... ????..... ??anything sesqui-clever here?? or something???....
??semi-direct product symbol "x|" ...???
weyl algebra s(v*) x| s(v) ...
induced morphism to s(v*) x| env(polynomial vector fields on v) ...
??but confusion about how it seems like there's almost a morphism going the other way... ??or something??? ... at least, that enveloping algebra maps into the weyl algebra... ???or something????.... ...and so forth...
??something about confusion between "covariant differentiation" and "lie differentiation" ... ???"lie differentiation" as something about s(v*) x| env(polynomial vector fields on v)... "covariant differentiation" as something about s(v*) x| s(v) ... ??or something???.....
maybe i should try to check how standard my usage of "semi-direct product" is here... ??not clear to me offhand how it relates to semi-direct product of group acting on other group ... ??... ??funny how in the case of the weyl algebra there's this sort of symmetry between the actor and the acted-upon ... ??or something??? ... also something about relationships between nilpotent and solvable ... ??and so forth???....
??vague memory of ...???being surprised by some sort of semi-direct product asepct (??in group sense maybe??) of weyl algebra... ?? ...??semi-recently?? ... ??what was_that about??... ??how closely did it tie in with old idea about quantum mechanics and semi-direct product and "measurement process disturbing the quantity being measured"?? ... and so forth... ??might it have been based on incorrect belief that i've recently been trying to recover from??? .... ??again of course, what about relationship to "semi-direct product of hopf algebra acting on commutative algebra" ?? ... and so forth... ???...
??questions (or something...) for chris rogers:
what exactly is a "lie algebroid"?? ... (not worrying about "lie n-algebroid" yet...)
can we think of lie algebroids as forming a reflective (or something) full subcategory of "dg manifolds" or something like that? (allow variations on "manifold" here of course...) exactly how?? some sort of "degree cutoff" or something??...
do we have a clear concept of "quasicoherent sheaf over a lie algebroid", so to speak?? (mainly in "algebraic" case, perhaps?? ... though maybe not exclusively ...) ???... if so, then can we think of these as something like special dg modules of the alleged corresponding dgca?? ... ??...
...and so forth...
??what about trying to relate diff eq aspect of d-module to that of dg module and/or of dgca ??? .... and so forth... ???....
??so what _about_ diff eqs involving infinitesimal simplexes... ??? ... ??"pfaffian..." .... ????....
Thursday, January 13, 2011
notes for discussion with martin
1 strictification issues .... ??? and so forth... ??maybe something about todd's proof??...
2 the universal property... ?? ... serre... serre subcategory ... serre ideal subcategory... ??as maybe wrong idea?? ... limits not involved in the doctrine... ??... ??inversion of morphisms ... quotient vs sub ... and so forth... ???grothendieck topology?? ....
3 other universal properties (but same doctrine...) ...
4 ??other doctrines?? ... abelian categories, toposes, ringed toposes .... ???various versions of "tensor category" ... ???something about chain complexes and so forth...
2 the universal property... ?? ... serre... serre subcategory ... serre ideal subcategory... ??as maybe wrong idea?? ... limits not involved in the doctrine... ??... ??inversion of morphisms ... quotient vs sub ... and so forth... ???grothendieck topology?? ....
3 other universal properties (but same doctrine...) ...
4 ??other doctrines?? ... abelian categories, toposes, ringed toposes .... ???various versions of "tensor category" ... ???something about chain complexes and so forth...
Wednesday, January 12, 2011
john huerta and i were thinking about projective light cones last night (in connection with g2 incidence geometry) and we guessed that the tangent space of a point x of one can be naturally identified with "x perp, mod x" ... which vaguely reminded me of the same formulation arising in other contexts... particularly in symplectic geometry, where however i don't offhand see an interpretation as a tangent space... maybe there is such an interpretation, but it doesn't seem to work the same way as in the "quadratic" case... i also can't remember whether i've seen this formulation in the quadratic case before.... ???... hmm, also, what about when x is null of higher dimension, or non-null?? ...
Tuesday, January 11, 2011
e-mail to martin brandenburg
hi. i've started reading the math overflow discussion, and i can see now that there are a lot of issues to discuss. so many that i'm not quite sure where to start...
also i noticed suggestions about carrying out our discussion in a place where other people can follow it, which probably seems like a reasonable idea, though i'm not sure of the best way to implement it...
also i've already tried to warn about my slowness at writing. i should emphasize this again. this slowness is making me wonder whether it might make sense to try to carry out some of our discussion via speaking (over skype, for example). of course this might conflict to some extent with the previous suggestion to carry out the discussion in a place where others can follow it (presumably by
reading)...
let me start by trying to describe perhaps the simplest version of an equivalence between the concepts of "dimensional category" and of "commutative algebra graded by an abelian group" ("graded commutative algebra" for short). in this version we get an equivalence of 1-categories from the category of graded commutative algebras to the category where an object is a "strict" dimensional category and a morphism is a "strict" dimensional functor.
(there's a tradeoff between the usefulness of omitting the strictness requirements in many contexts, and the complication of dealing with an equivalence of 2-categories rather than of 1-categories.)
"strict" here can be taken to mean that the equations that might be weakened to natural isomorphisms (associativity and commutativity of tensor product and of the unit and inverse laws for tensor product in dimensional categories, and preservation of tensor product and unit objects and inverse objects by dimensional functors) are actually required to be equations.
the equivalence is now given (as ben webster indicated) by taking the objects in the dimensional category to be the grades in the graded commutative algebra and the elements in the hom-space [x,y] to be the elements in the grade y-x, and the inverse equivalence by taking the grades to be the objects and the elements in x to be the elements in [1,x].
(i hope that describing the inverse equivalence as above might be enough to make the details clear, but if not then i can try to give them more explicitly.)
this construction gives an equivalence from "graded commutative monoids in x" to "strict x-enriched dimensional categories" under fairly mild conditions on the symmetric monoidal category x. for example when x is the category of sets with tensor product given by cartesian product, a graded commutative monoid in x is a commutative monoid equipped with a homomorphism to an abelian group, and an x-enriched dimensional theory is a "toric dimensional theory".
by the way, notice that the dimensional category corresponding to a graded commutative algebra x can be non-equivalent (even as just a plain category) to the dimensional category of invertible graded modules of x (for example if x is an ungraded commutative algebra with non-trivial ideal class group).
i'll stop here for now, in the interest of moving the discussion along and trying not to get bogged down by my slowness of writing. i'm eager to resolve questions about the universal properties of tensor categories of coherent sheaves (and so forth), in part because i often find myself working in isolation (exacerbated by my difficulties in understanding many parts of the standard literature) and wind up
reinventing (with roughened edges aka lots of mistakes) things that are already known; i hope that this discussion helps me in understanding many things that are already known.
also i noticed suggestions about carrying out our discussion in a place where other people can follow it, which probably seems like a reasonable idea, though i'm not sure of the best way to implement it...
also i've already tried to warn about my slowness at writing. i should emphasize this again. this slowness is making me wonder whether it might make sense to try to carry out some of our discussion via speaking (over skype, for example). of course this might conflict to some extent with the previous suggestion to carry out the discussion in a place where others can follow it (presumably by
reading)...
let me start by trying to describe perhaps the simplest version of an equivalence between the concepts of "dimensional category" and of "commutative algebra graded by an abelian group" ("graded commutative algebra" for short). in this version we get an equivalence of 1-categories from the category of graded commutative algebras to the category where an object is a "strict" dimensional category and a morphism is a "strict" dimensional functor.
(there's a tradeoff between the usefulness of omitting the strictness requirements in many contexts, and the complication of dealing with an equivalence of 2-categories rather than of 1-categories.)
"strict" here can be taken to mean that the equations that might be weakened to natural isomorphisms (associativity and commutativity of tensor product and of the unit and inverse laws for tensor product in dimensional categories, and preservation of tensor product and unit objects and inverse objects by dimensional functors) are actually required to be equations.
the equivalence is now given (as ben webster indicated) by taking the objects in the dimensional category to be the grades in the graded commutative algebra and the elements in the hom-space [x,y] to be the elements in the grade y-x, and the inverse equivalence by taking the grades to be the objects and the elements in x to be the elements in [1,x].
(i hope that describing the inverse equivalence as above might be enough to make the details clear, but if not then i can try to give them more explicitly.)
this construction gives an equivalence from "graded commutative monoids in x" to "strict x-enriched dimensional categories" under fairly mild conditions on the symmetric monoidal category x. for example when x is the category of sets with tensor product given by cartesian product, a graded commutative monoid in x is a commutative monoid equipped with a homomorphism to an abelian group, and an x-enriched dimensional theory is a "toric dimensional theory".
by the way, notice that the dimensional category corresponding to a graded commutative algebra x can be non-equivalent (even as just a plain category) to the dimensional category of invertible graded modules of x (for example if x is an ungraded commutative algebra with non-trivial ideal class group).
i'll stop here for now, in the interest of moving the discussion along and trying not to get bogged down by my slowness of writing. i'm eager to resolve questions about the universal properties of tensor categories of coherent sheaves (and so forth), in part because i often find myself working in isolation (exacerbated by my difficulties in understanding many parts of the standard literature) and wind up
reinventing (with roughened edges aka lots of mistakes) things that are already known; i hope that this discussion helps me in understanding many things that are already known.
Monday, January 10, 2011
??something about baez-kim and symmetric algebra as left adjoint to grade 1 ...
??something about ... ??hermitian structure on line bundle as corresponding to hilbert structure on space of sections?? or something?? ...but with emphasis on projectivization of space of sections as target of corresponding projective embedding ... ???or something...
??what about the "extra" stuff as maybe turning the adjunction into something trickier than an adjunction?? or something??...
??relationship to doctrine viewpoint?? ... and so forth...
..."kaehler variety" ... ??... symplectic space and line bundle vs projective variety and line bundle... overlap ... and so forth... ??... ???something about associated graded and re-scaling and gibbs-boltzmann and symplectic/poisson ... ???... and so forth ... ???...
??something about ... ??hermitian structure on line bundle as corresponding to hilbert structure on space of sections?? or something?? ...but with emphasis on projectivization of space of sections as target of corresponding projective embedding ... ???or something...
??what about the "extra" stuff as maybe turning the adjunction into something trickier than an adjunction?? or something??...
??relationship to doctrine viewpoint?? ... and so forth...
..."kaehler variety" ... ??... symplectic space and line bundle vs projective variety and line bundle... overlap ... and so forth... ??... ???something about associated graded and re-scaling and gibbs-boltzmann and symplectic/poisson ... ???... and so forth ... ???...
Friday, January 7, 2011
??so what about the idea that the split real form is the real form most like the complex form (in a certain way...) and the compact real form is the one least like it, and the others are in between these extremes?? .... vs the other way around...
??for an intermediate form, does each grassmanian manifest as either real or complex?? ??or something?? ... and so forth... ??what _about_ "symmetric spaces" here?? ... and so forth ... ??...
??for an intermediate form, does each grassmanian manifest as either real or complex?? ??or something?? ... and so forth... ??what _about_ "symmetric spaces" here?? ... and so forth ... ??...
notes copied from airline ticket...
1
??conceptual loop (or something...) ...
"renormalization"...
"gibbs-boltzmann deformation" ...
"poisson geometry" ...
"associated graded vector space of a filtered vector space" ...
2
??answer shulman?? ??still sounds 2-stage monadic?? ??also maybe add correction/clarification about abelian category ... starting from ringoid rather than category...
3
some rambling calculations relating to "malcev variety" and "kernel" and so forth... ??...
??something about "equational refinement of horn theory" ... ???...
??? "[x=y] <=> [f(x,y)=k]" ...
?? =>
f(x,x)=k
??vs "f(x,x,y)=y" ?? or something??? ... and so forth... ???...
??? <= ... ???... f(x)=g(x) => h(x)=j(x) ... ???and so forth...
h=fk, j=gk ... ???and so forth... ??...
1
??conceptual loop (or something...) ...
"renormalization"...
"gibbs-boltzmann deformation" ...
"poisson geometry" ...
"associated graded vector space of a filtered vector space" ...
2
??answer shulman?? ??still sounds 2-stage monadic?? ??also maybe add correction/clarification about abelian category ... starting from ringoid rather than category...
3
some rambling calculations relating to "malcev variety" and "kernel" and so forth... ??...
??something about "equational refinement of horn theory" ... ???...
??? "[x=y] <=> [f(x,y)=k]" ...
?? =>
f(x,x)=k
??vs "f(x,x,y)=y" ?? or something??? ... and so forth... ???...
??? <= ... ???... f(x)=g(x) => h(x)=j(x) ... ???and so forth...
h=fk, j=gk ... ???and so forth... ??...
Thursday, January 6, 2011
maybe the (2,1)-category of dimensional theories is "abelian"-flavored as long as you don't use too coarse a concept of "morita equivalence", in which case it becomes more "topos"-flavored?? or something?? ... ??or is "morita equivalence" not so "drastic" here??? ???.....
??and what about "morita equivalence" in "dimensional analysis"?? ... ??...
??and what about "morita equivalence" in "dimensional analysis"?? ... ??...
notes for next discussion with baez
d-modules and coherent sheaves over certain orbit stacks... lie derivative vs covariant derivative... and so forth... big commutative diagram ... ??...
??algebraic geometric theory of "affine line's worth of line objects, equipped with nice way to tensor them riding the addition structure" ... ??also case of abelian variety here (...) ??? and so forth???....
??possibly also stuff about passage to derived category... from "doctrine" viewpoint... and so forth ... ??...
??maybe something about limits vs colimits of dimensional theories?? ... and so forth...
??algebraic geometric theory of "affine line's worth of line objects, equipped with nice way to tensor them riding the addition structure" ... ??also case of abelian variety here (...) ??? and so forth???....
??possibly also stuff about passage to derived category... from "doctrine" viewpoint... and so forth ... ??...
??maybe something about limits vs colimits of dimensional theories?? ... and so forth...
Monday, January 3, 2011
??factorization system involving cohesive (or something) geometric morphism??...
???something about "sequent" vs "sentence" and so forth... ??confusion about factorization systems here... ??and so forth... ???something about whether sequents give formulas in _intuitionistic_ (or something...) logic... ???something about interaction between "conjunction" at different "levels"...
exploitation of cryptomorphism... secondary operation ... ??horn logic ... ??asf??? ... ??something about sketches vs formulas... "multidimensional algebra" .. ???...
"diaconescu's theorem" for various doctrines... ???...
zombie chesire cat... pincushion...
??something about serre ideal subcategories and so forth... ??... ??"viable sub-lizard" and so forth...
???something about "sequent" vs "sentence" and so forth... ??confusion about factorization systems here... ??and so forth... ???something about whether sequents give formulas in _intuitionistic_ (or something...) logic... ???something about interaction between "conjunction" at different "levels"...
exploitation of cryptomorphism... secondary operation ... ??horn logic ... ??asf??? ... ??something about sketches vs formulas... "multidimensional algebra" .. ???...
"diaconescu's theorem" for various doctrines... ???...
zombie chesire cat... pincushion...
??something about serre ideal subcategories and so forth... ??... ??"viable sub-lizard" and so forth...
Sunday, January 2, 2011
??trying to work out some simple topos theory examples following n-category cafe discussion... getting a bit confused...
"1-skeletal simplicial sets" ... "reflexive graphs" ...
??alleged "algebraic-sub"-topos given by those where all edges are loops... ??or something??...
??corresponding in some hopefully obvious way to monoid homomorphism from walking pair of constants to walking constant (= walking idempotent ??...) ... ???...
??but when i try to work this out in certain way i get confused... ??...
hmmm....actually the way that we just phrased it sounds suggestive... fold the two constants together... ??though be careful about universal quantifier hidden in definition of "constant"?? ... or something.... ???... ??so then how _did_ i think that there was some ambiguity (or something...??..) here???.... ??maybe just a silly mistake?? ..which i might not be able to reconstruct now... ??...
??another issue...
??double-negation topology on topos of presheaves over walking loop... ??geometric morphism from ... to ... ??...
???something about leinster's question and dense vs closed ... ???or something???...
??i'm also a bit confused at the moment about the relationship between lawvere-tierney topologies, and heyting algebras as forming a malcheff variety, and heyting algebras vs distributive lattices... and so forth...
"1-skeletal simplicial sets" ... "reflexive graphs" ...
??alleged "algebraic-sub"-topos given by those where all edges are loops... ??or something??...
??corresponding in some hopefully obvious way to monoid homomorphism from walking pair of constants to walking constant (= walking idempotent ??...) ... ???...
??but when i try to work this out in certain way i get confused... ??...
hmmm....actually the way that we just phrased it sounds suggestive... fold the two constants together... ??though be careful about universal quantifier hidden in definition of "constant"?? ... or something.... ???... ??so then how _did_ i think that there was some ambiguity (or something...??..) here???.... ??maybe just a silly mistake?? ..which i might not be able to reconstruct now... ??...
??another issue...
??double-negation topology on topos of presheaves over walking loop... ??geometric morphism from ... to ... ??...
???something about leinster's question and dense vs closed ... ???or something???...
??i'm also a bit confused at the moment about the relationship between lawvere-tierney topologies, and heyting algebras as forming a malcheff variety, and heyting algebras vs distributive lattices... and so forth...
so consider the orbit stack of the trivial action of the additive group of the affine line... ???then i think that there's a certain sheaf over this stack... that i want to know whether it's "coherent" or "quasicoherent" or something ...??...
so how do i formulate this... ???...
??something about "algebraic fourier dual" ... ???....
so how do i formulate this... ???...
??something about "algebraic fourier dual" ... ???....
Saturday, January 1, 2011
??so consider "the algebraic-geometric theory of an affine line's worth of invertible vector spaces, equipped with a nice way of tensoring them riding addition"?? ...
...and so forth ... ???....
...also the "multiplication" case ... ???....
??"invertible module over the affine line, symmetric monoidal wrt additive convolution" ... ???or something...
??to what extent _is_ all this nicely and unambiguously expressible in the doctrine (and/or in some related doctrines... ??...) ?? ...
??how does this relate to geometric interpretation of representations of borel subgroups??... and so forth... ??d-modules, and formal translation group vs actual translation group ... and so forth...
??how does this relate to "representations" (...) of some kind of commutative and/or cocommutative hopf algebras??? ... and so forth...
??how does this relate to "the regular representation picture of a g-torsor"?? ... and so forth ... ???...
...??something about "compactness" issues?? ... or something... ... "coherent..." ... ??....
??something about preservation of "[affine line]-indexed sums"?? ... and so forth...
??something about relationship between ["theory of an x-indexed family of..." for x the model stack of an algebraic-geometric theory] and [internal hom between algebraic-geometric theories... and so forth...]??... ??and... "algebraic stacks" ... ??something about geometric right-universal property of an algebraic stack... involving family indexed by another geometric stack... or something... ??...
??what about decategorified analogs here of.. ??situations where you sort of hope to have internal homs in some generality but they turn out to exist only in rather special cases... ??or something?? something about exponentiable affine schemes and so forth??... something about expressibility of ... ???....
...and so forth ... ???....
...also the "multiplication" case ... ???....
??"invertible module over the affine line, symmetric monoidal wrt additive convolution" ... ???or something...
??to what extent _is_ all this nicely and unambiguously expressible in the doctrine (and/or in some related doctrines... ??...) ?? ...
??how does this relate to geometric interpretation of representations of borel subgroups??... and so forth... ??d-modules, and formal translation group vs actual translation group ... and so forth...
??how does this relate to "representations" (...) of some kind of commutative and/or cocommutative hopf algebras??? ... and so forth...
??how does this relate to "the regular representation picture of a g-torsor"?? ... and so forth ... ???...
...??something about "compactness" issues?? ... or something... ... "coherent..." ... ??....
??something about preservation of "[affine line]-indexed sums"?? ... and so forth...
??something about relationship between ["theory of an x-indexed family of..." for x the model stack of an algebraic-geometric theory] and [internal hom between algebraic-geometric theories... and so forth...]??... ??and... "algebraic stacks" ... ??something about geometric right-universal property of an algebraic stack... involving family indexed by another geometric stack... or something... ??...
??what about decategorified analogs here of.. ??situations where you sort of hope to have internal homs in some generality but they turn out to exist only in rather special cases... ??or something?? something about exponentiable affine schemes and so forth??... something about expressibility of ... ???....
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