??so at the moment i'm confused between the orbit stack of a simple algebraic group (or something like that) acting on itself by translation, vs that of just the formal translations acting on it... ??because of confusion about the representations (in some sense...??) of the translation group vs of the formal translation group... ??or something??
??so in trying to straighten this out, maybe we should try to understand the orbit stack constructions here very systematically in the coherent sheaf picture... or something... ??...
Wednesday, December 29, 2010
Tuesday, December 28, 2010
Monday, December 27, 2010
Sunday, December 26, 2010
a silly idea occurred to me while reading tom leinster's brief topos theory introduction... trying to explain the idea of "subobject classifier" / "truth-values object" ... presheaf topos examples, such as "graph" topos... idea that some intermediate truth values embody idea of certain sort of "hanger-on" wrt "in crowd" ... ??i guess that the prototypical example here is just pre-sheaves on the walking arrow...
vague memories of some sort of categorification of fractional linear transformations (os...) involving something about suspension and loop spaces and so forth... os... ???sa how that relates to this stuff about derived cat of a2 quiver reps.... as triangulated category... representing the "1/3" (and other...) translations as
derived morita contexts... os... asf os... ??... ...try to ask baez about this... os...
??also sa "hodge duality" ??
??wa sa "hodge duality" and relationship between "dimensional analysis" and "projective geometry"?? ....
derived morita contexts... os... asf os... ??... ...try to ask baez about this... os...
??also sa "hodge duality" ??
??wa sa "hodge duality" and relationship between "dimensional analysis" and "projective geometry"?? ....
Saturday, December 25, 2010
started trying to respond to ben-zvi, but i might be getting bogged down... i'll try thinking outloud here...
well, i'm pretty confused now, though perhaps mainly just because you're reminding me of questions about d-modules that have had me confused for a long time. in hopes of getting unconfused i'll try thinking outloud here about some of my confusions.
given a cocommutative hopf algebra g acting on a commutative algebra r, we can construct a new (generally noncommutative) algebra from this. (perhaps this is called a "semi-direct product" in some terminology?) a module of the new algebra is essentially "a way of extending the action of g on r to an action of g on the pair (r,m) for some r-module m". if g acts on the triple (r,m1,m2) then it also acts on the pair (r,m1 tensor m2), so the modules of the new algebra form a tensor category.
specializing to the case where r is the coordinate algebra of an affine scheme x and g is the lie algebra of derivations of r, this gives the tensor category of d-modules over x. (is that correct?)
the general construction here is morally similar to taking the "orbit stack" of an action of an algebraic group acting on an affine scheme...
??hmm, but ben-zvi said something about "quotienting by the formal neighborhood of the diagonal" (or something like that...), which maybe has a pretty different flavor from "taking the orbit stack of the action by the formal group generated by the vector fields" (or something like that...) .... the flavor difference having to do with functoriality / variance ...???or something.... ???.... ??so what about how this relates to issues about pullback and pushforward of d-modules??? ... and so forth ... ????....
(???sa history of ideas about "noncommutative geometry" os... ???asf os... ????)
??so what about how this (???...) might tie in with... ???all sorts of stuff about "dg spaces" and so forth... and their relationship to stacks and so forth.... ???? or something??? .... ???hmm, _did_ we think about this before?? maybe even think about resolving certain functoriality paradoxes this way ???? or something???...
what about discrete analogs here???...
??what about morita equivalence here??? .... something about matrix algebras ... ???or something?? .... ??some sort of level slip (or something??) here???
??hmm, so maybe i _have_ been making a silly mistake here for decades... about what d-modules are... ??something about the orbit stack of the action of [the formal translation group of a vector space v] on v ... vs the orbit stack of the action of [the formal diffeomorphism group of v] on v ... ??or something like that?? ... i wonder whether we might have straightened this out before and then forgotten about it... perhaps not...
??of course part of the point is that "orbit stack of the action of [the formal translation group of a vector space v] on v" has much more functoriality than manifestly suggested by that phrase ... ???...
??so how might this affect our attempts to learn other stuff involving d-modules?? for example the alleged relationship to perverse sheaves and so forth... ???.... hmmm.... "beilinson-bernstein localization" ... ??ideas about d-modules and "ayntax/semantics completeness theorem for differential equations" and so forth... ??... ??something about "dirac delta function" and so forth... ??... "distributional solutions" ... ...
??so is it true that d-modules over the underlying affine scheme of an affine algebraic group can be thought of as .... ???modules of the "quantum double" (or something...) of something... ???or something???? ... not sure i said that anywhere close to correct yet... what i'm trying to get at here is that the idea that d-modules over a vector space v can be thought of as quasicoherent sheaves over the orbit stack of the formal translation group of v seems like it shouldn't depend on the abeliannes of v (thinking of a vector space as a sort of abelian affine algebraic group.... or something...) ... ???try to work out more details here...
??ok, so not "quantum double" here, i guess, because that's about the conjugation action... we actually want the translation action here, i think....
??hmm, so what about something about... ???contrast to quantum double here?? ... something about... quantum double as hopf algebra over base field (or something) vs this other (??) orbit stack as not hopf algebra over commutative algebra...??? and so forth ... ?? ... braidedness of tensor product.... ??_is_ there another tensor product around here, symmetric instead of braided???....
??anyway, is it actually obvious at a "concrete" level that this orbit stack is independent of the algebraic group structure... ??? ...or something like that .. ??... ??hmm, perhaps directly related to that similar question ...(??not to be confused with slightly dissimilar question... ??about "duflo isomorphism" or something??)... ??about poincare-birkhoff-witt theorem??...
???so what about the idea of "d-modules over a dg space"?? ... and so forth??? ...??what _about_ relationship to stuff like "perverse sheaves" and so forth???....
??so now that we may have straightened out some confusions about d-modules here, can we get any idea of what ben-zvi is talking about about some sort of "non-affine behavior" here????? ....
this is what ben-zvi said:
"The main technical problem with D-modules from our point of view is that pushforward is essentially never conservative (except for finite maps) - that’s the sense in which D-modules behave non-affinely even on affine varieties..and which is why on stacks with affine diagonal Tannakian constructions with D-modules – eg description of sheaves on a fiber product as a categorical tensor product of categories of sheaves – fail dramatically.. though they DO hold for schemes."
??so what in the world are they talking about?? ... i don't get it yet... ??very vaguely though it does remind me of something about... how geometric pullbacks and equalizers (or something like that... hope that i didn't get the arrows backwards here...) seem screwed up for cocommutative coalgebras... ???maybe dual to geometric colimits being screwed up for commutative algebras?? ??? or something ... ??...
??while searching for "non-affine" in the "alg geom for category theorists" thread i accidentally came across earlier parts of the dicussion from 2009... possibly interesting evolution (and/or lack thereof) in viewpoints of some participants...
??so what about tensor product for perverse sheaves compared to for d-modules??
...
well, i'm pretty confused now, though perhaps mainly just because you're reminding me of questions about d-modules that have had me confused for a long time. in hopes of getting unconfused i'll try thinking outloud here about some of my confusions.
given a cocommutative hopf algebra g acting on a commutative algebra r, we can construct a new (generally noncommutative) algebra from this. (perhaps this is called a "semi-direct product" in some terminology?) a module of the new algebra is essentially "a way of extending the action of g on r to an action of g on the pair (r,m) for some r-module m". if g acts on the triple (r,m1,m2) then it also acts on the pair (r,m1 tensor m2), so the modules of the new algebra form a tensor category.
specializing to the case where r is the coordinate algebra of an affine scheme x and g is the lie algebra of derivations of r, this gives the tensor category of d-modules over x. (is that correct?)
the general construction here is morally similar to taking the "orbit stack" of an action of an algebraic group acting on an affine scheme...
??hmm, but ben-zvi said something about "quotienting by the formal neighborhood of the diagonal" (or something like that...), which maybe has a pretty different flavor from "taking the orbit stack of the action by the formal group generated by the vector fields" (or something like that...) .... the flavor difference having to do with functoriality / variance ...???or something.... ???.... ??so what about how this relates to issues about pullback and pushforward of d-modules??? ... and so forth ... ????....
(???sa history of ideas about "noncommutative geometry" os... ???asf os... ????)
??so what about how this (???...) might tie in with... ???all sorts of stuff about "dg spaces" and so forth... and their relationship to stacks and so forth.... ???? or something??? .... ???hmm, _did_ we think about this before?? maybe even think about resolving certain functoriality paradoxes this way ???? or something???...
what about discrete analogs here???...
??what about morita equivalence here??? .... something about matrix algebras ... ???or something?? .... ??some sort of level slip (or something??) here???
??hmm, so maybe i _have_ been making a silly mistake here for decades... about what d-modules are... ??something about the orbit stack of the action of [the formal translation group of a vector space v] on v ... vs the orbit stack of the action of [the formal diffeomorphism group of v] on v ... ??or something like that?? ... i wonder whether we might have straightened this out before and then forgotten about it... perhaps not...
??of course part of the point is that "orbit stack of the action of [the formal translation group of a vector space v] on v" has much more functoriality than manifestly suggested by that phrase ... ???...
??so how might this affect our attempts to learn other stuff involving d-modules?? for example the alleged relationship to perverse sheaves and so forth... ???.... hmmm.... "beilinson-bernstein localization" ... ??ideas about d-modules and "ayntax/semantics completeness theorem for differential equations" and so forth... ??... ??something about "dirac delta function" and so forth... ??... "distributional solutions" ... ...
??so is it true that d-modules over the underlying affine scheme of an affine algebraic group can be thought of as .... ???modules of the "quantum double" (or something...) of something... ???or something???? ... not sure i said that anywhere close to correct yet... what i'm trying to get at here is that the idea that d-modules over a vector space v can be thought of as quasicoherent sheaves over the orbit stack of the formal translation group of v seems like it shouldn't depend on the abeliannes of v (thinking of a vector space as a sort of abelian affine algebraic group.... or something...) ... ???try to work out more details here...
??ok, so not "quantum double" here, i guess, because that's about the conjugation action... we actually want the translation action here, i think....
??hmm, so what about something about... ???contrast to quantum double here?? ... something about... quantum double as hopf algebra over base field (or something) vs this other (??) orbit stack as not hopf algebra over commutative algebra...??? and so forth ... ?? ... braidedness of tensor product.... ??_is_ there another tensor product around here, symmetric instead of braided???....
??anyway, is it actually obvious at a "concrete" level that this orbit stack is independent of the algebraic group structure... ??? ...or something like that .. ??... ??hmm, perhaps directly related to that similar question ...(??not to be confused with slightly dissimilar question... ??about "duflo isomorphism" or something??)... ??about poincare-birkhoff-witt theorem??...
???so what about the idea of "d-modules over a dg space"?? ... and so forth??? ...??what _about_ relationship to stuff like "perverse sheaves" and so forth???....
??so now that we may have straightened out some confusions about d-modules here, can we get any idea of what ben-zvi is talking about about some sort of "non-affine behavior" here????? ....
this is what ben-zvi said:
"The main technical problem with D-modules from our point of view is that pushforward is essentially never conservative (except for finite maps) - that’s the sense in which D-modules behave non-affinely even on affine varieties..and which is why on stacks with affine diagonal Tannakian constructions with D-modules – eg description of sheaves on a fiber product as a categorical tensor product of categories of sheaves – fail dramatically.. though they DO hold for schemes."
??so what in the world are they talking about?? ... i don't get it yet... ??very vaguely though it does remind me of something about... how geometric pullbacks and equalizers (or something like that... hope that i didn't get the arrows backwards here...) seem screwed up for cocommutative coalgebras... ???maybe dual to geometric colimits being screwed up for commutative algebras?? ??? or something ... ??...
??while searching for "non-affine" in the "alg geom for category theorists" thread i accidentally came across earlier parts of the dicussion from 2009... possibly interesting evolution (and/or lack thereof) in viewpoints of some participants...
??so what about tensor product for perverse sheaves compared to for d-modules??
...
Thursday, December 23, 2010
Wednesday, December 22, 2010
are the following two projective curves equivalent (in the "strong" sense... including equivalence of their tautological line bundles...)?:
start with a 2-dot dynkin diagram... and some dot in it, and the favorite point of the corresponding grassmanian...
1 treat that point as a line in the other grassmanian, and treat that other grassmanian as a a projective variety via generalized pluecker embedding (or something...)
2 take the 2d schubert variety of that point, and take its tangent cone at the basepoint; then take the corresponding projective curve
...or something like that...
seems like these should be obviously the same, but i'm not quite seeing the obviousness yet...
...also generalize this...
start with a 2-dot dynkin diagram... and some dot in it, and the favorite point of the corresponding grassmanian...
1 treat that point as a line in the other grassmanian, and treat that other grassmanian as a a projective variety via generalized pluecker embedding (or something...)
2 take the 2d schubert variety of that point, and take its tangent cone at the basepoint; then take the corresponding projective curve
...or something like that...
seems like these should be obviously the same, but i'm not quite seeing the obviousness yet...
...also generalize this...
Monday, December 6, 2010
so is mike shulman hinting that over v = _cocomplete poset_, it's not strict kernel but some sort of lax kernel that can be re-expressed as a weighted colimit?? or something??
"lax kenrel" sounds disappointingly "trivial", though... whether "lax" or "op-lax", i think... ??though maybe it's not so trivial over v = _cocomplete category_ ?? ...
"lax kenrel" sounds disappointingly "trivial", though... whether "lax" or "op-lax", i think... ??though maybe it's not so trivial over v = _cocomplete category_ ?? ...
notes for next meeting with alex
so we worked out the total derived functor of "kernel", more or less... seeing how it begins to mix the concepts of kernel and cokernel together... but then we want to flesh this out a lot... how it relates to "long exact homology sequence" and so forth... something about weighted colimits and limits, and adjoint weights ... ??and so forth... ???
alex also brought up the idea of analogs of "flat ag morphism" in topos theory... or something like that... ??...
alex also brought up the idea of analogs of "flat ag morphism" in topos theory... or something like that... ??...
so consider the process of passing from a symmetric monoidal finitely cocomplete algebroid to a symmetric monoidal finitely homotopy-cocomplete dg algebroid (or something like that) by taking chain complexes...
to what extent is this "full and faithful" or something?? ... and so forth...
similarly for just finitely cocomplete algebroid, and so forth... ??relevance of "reflection functors" involving quiver representations here??...
to what extent is this "full and faithful" or something?? ... and so forth...
similarly for just finitely cocomplete algebroid, and so forth... ??relevance of "reflection functors" involving quiver representations here??...
so is richard garner essentially saying that ... ??for example... ??quantaloid modules always have adjoints?? or something like that??? ... if so then how does this relate to the discussion where todd warned me about confusion between "converse" and "adjoint" ??? ... or something...
i'm still having some trouble with certain level slip here... that i noticed quite a while ago but wasn't sure how badly it was confusing me...
categories enriched over truth values, vs categories enriched over complete semilattices... and so forth... ???.....
i'm still having some trouble with certain level slip here... that i noticed quite a while ago but wasn't sure how badly it was confusing me...
categories enriched over truth values, vs categories enriched over complete semilattices... and so forth... ???.....
questions for henriques
1 is it true that the "homotopy category" of all (??bounded or something??) chain complexes of objects from an abelian category is essentially the derived category of (again, bounded or something...) chain complexes of objects from another abelian category? or something like that?
2 how to understand "complete intersection" ??... geometric intuition... ??something about not necessarily "reduced" (or something) case...
3 what about some sort of "dual" of a dg module of a dg algebroid?? ...and so forth...
2 how to understand "complete intersection" ??... geometric intuition... ??something about not necessarily "reduced" (or something) case...
3 what about some sort of "dual" of a dg module of a dg algebroid?? ...and so forth...
hmmm... in our discussion on saturday about "homotopy kernel as the total derived functor of kernel" (or something... and so forth... automatic blending together of limits and coliits at homotopy level...), alex and i may have accidentally come very close to solving our other problem... about how to categorify kaleidoscope reflections into reflection functors in the form of derived morita contexts... ??
??so... a "projective embedding" sort of corresponds to a line bundle... ??so then for example what sort of "system of line bundles" does an "embedding into a grassmanian of 2d linear subspaces" correspond to?? ... (or something like that... ??what about "what sort of system of line bundles does a bundle of 2d vector spaces correspond to?" ?? .... ???) ...in a way this seems like a very straightforward question, but... ??nevertheless it seems like thinking about it should be illuminating in certain ways... relationships between "algebraic geometry" and "homotopy theory" or something... ??and so forth??...
??something about "representation theory" too ... ??...
??what about also "multi-projective embedding" here?? ... and so forth... ??or something... ???....
??what about the idea of "harmless extra structure" here?? ...or something like that... ???....
??so ... ??we have a sort of quasi-algorithm that given a line bundle produces a projective embedding: something like: take the vector space of sections of the bundle, and embed into it in the hopefully obvious way... ??or something?? (something about divisor of a point here?? something about "divisor" vs "ideal" ??? .... ???....) ??so then what's the corresponding quasi-algorithm that, given a 2d vector bundle, produces an embedding into a grassmanian of 2d linear subspaces?? ... well, so somehow we should obtain a vector space whose grassmanian of 2d vector spaces would be just the thing to contain an embedded copy of our base space... ???so what _is_ this how??... ??how about the vector space of all sections of the 2d vector bundle?? or something?? so given a point of our base space, assign to it the subspace of sections that vanish at that point... ??what dimension should that subspace be??
at the moment we seem to be doing a fairly good job of making things that should (i think...) be familiar seem exotic... ???...
???what about sections of a vector bundle that vanish on some higher-dim variety?? ....
??....
??"divisor" as "holomorphic structure" on (??standard??) "meromorphic line bundle" .... ????...
???so what about something about "birational geometry" and "flatness" ??? .... or something ... ???...
??what's a "holomorphic structure on the standard meromorphic 2d vector bundle" ??? ... ??or something???.....
??something about "representation theory" too ... ??...
??what about also "multi-projective embedding" here?? ... and so forth... ??or something... ???....
??what about the idea of "harmless extra structure" here?? ...or something like that... ???....
??so ... ??we have a sort of quasi-algorithm that given a line bundle produces a projective embedding: something like: take the vector space of sections of the bundle, and embed into it in the hopefully obvious way... ??or something?? (something about divisor of a point here?? something about "divisor" vs "ideal" ??? .... ???....) ??so then what's the corresponding quasi-algorithm that, given a 2d vector bundle, produces an embedding into a grassmanian of 2d linear subspaces?? ... well, so somehow we should obtain a vector space whose grassmanian of 2d vector spaces would be just the thing to contain an embedded copy of our base space... ???so what _is_ this how??... ??how about the vector space of all sections of the 2d vector bundle?? or something?? so given a point of our base space, assign to it the subspace of sections that vanish at that point... ??what dimension should that subspace be??
at the moment we seem to be doing a fairly good job of making things that should (i think...) be familiar seem exotic... ???...
???what about sections of a vector bundle that vanish on some higher-dim variety?? ....
??....
??"divisor" as "holomorphic structure" on (??standard??) "meromorphic line bundle" .... ????...
???so what about something about "birational geometry" and "flatness" ??? .... or something ... ???...
??what's a "holomorphic structure on the standard meromorphic 2d vector bundle" ??? ... ??or something???.....
Wednesday, December 1, 2010
notes for discussion with baez this evening
there's lots of interesting digressions that ideally i'd like to explore here... not sure whether i should instead try to stick to the main path, towards the relationship between homotopy limits and homotopy colimits in stable homotopy theory...
not sure how much i should bring up the "abelian analog of diaconescu's theorem" idea here... (and other ideas about "flat modules" ... algebraico-geometric interpretation of flatness...) perhaps try looking up that n-category cafe thread about flat presheaves (or something...) to see how explicitly the analogy is spelled out there...
perhaps i should try to get pretty quickly to the adjunction between cokernel and kernel in an abelian category, and how this is a prelude to homotopy cokernel and homotopy kernel being inverse in stable homotopy theory... or something like that... (digress about relationship to snake lemma??)
then introduce the issue of "weights" for diagram schemes... motivated in part by idea that homotopy cokernel and homotopy kernel aren't just inverse to each other, but are "the same" up to a specification of weight and co-weight... or something like that... develop the analogy between the colimit of a weighted diagram and the expectation of a random variable... (digress about "calculus of co-ends" and/or "calculus of weighted colimits" ??...)
develop interpretation of weighted homotopy limits in chain complex world as weak homming from a dg module of a dg algebroid, and of weighted homotopy colimits as weak tensoring with a dg op-module of a dg algberoid... then develop equivalence between weighted homotopy limits and weighted homotopy colimits in terms of appropriate kind of "weak duality" between a dg module and a dg op-module...
...try to work out many examples... non-existence of duals in set-based or vector-space-based case, vs existence of weak duals in chain-complex-based case...
...try to relate this "duality" stuff to "absolute colimits" and so forth... and to algebraic geometry...
not sure how much i should bring up the "abelian analog of diaconescu's theorem" idea here... (and other ideas about "flat modules" ... algebraico-geometric interpretation of flatness...) perhaps try looking up that n-category cafe thread about flat presheaves (or something...) to see how explicitly the analogy is spelled out there...
perhaps i should try to get pretty quickly to the adjunction between cokernel and kernel in an abelian category, and how this is a prelude to homotopy cokernel and homotopy kernel being inverse in stable homotopy theory... or something like that... (digress about relationship to snake lemma??)
then introduce the issue of "weights" for diagram schemes... motivated in part by idea that homotopy cokernel and homotopy kernel aren't just inverse to each other, but are "the same" up to a specification of weight and co-weight... or something like that... develop the analogy between the colimit of a weighted diagram and the expectation of a random variable... (digress about "calculus of co-ends" and/or "calculus of weighted colimits" ??...)
develop interpretation of weighted homotopy limits in chain complex world as weak homming from a dg module of a dg algebroid, and of weighted homotopy colimits as weak tensoring with a dg op-module of a dg algberoid... then develop equivalence between weighted homotopy limits and weighted homotopy colimits in terms of appropriate kind of "weak duality" between a dg module and a dg op-module...
...try to work out many examples... non-existence of duals in set-based or vector-space-based case, vs existence of weak duals in chain-complex-based case...
...try to relate this "duality" stuff to "absolute colimits" and so forth... and to algebraic geometry...
an e-mail i wrote but didn't send
something you said this evening annoyed me (assuming that i heard it correctly, which i'm not sure i did), and i wanted to explain a bit about why... not that i'll necessarily actually send this to you; i also want to explain it to myself, perhaps to help me remember about some of the things that go on...
i was describing this particular long story that i want to tell you, a long story about some math that i've been working on, long because i've been making a lot of progress recently without much of a chance to talk to you about it, and so there's a lot that you missed, making up a long story that i want to tell you about. but i thought that i heard you say something like... that the main reason that you were unsure about whether you'd ever hear the end of the story is because you didn't know whether i'd stay interested in my own story, or instead veer off in some other direction like (you apparently think) i always do...
i didn't respond to that, in part because i wasn't sure i heard you correctly. but if i did respond i'd probably say something like the following...
i stay interested in my own stories much more consistently than you seem to realize. the reason it seems otherwise to you is that you're so absent and uninvolved that you don't see the common thread that makes it still the same story when you come back a few weeks (or months or years) later. stories unfold and evolve (and also sometimes have quick cuts and dissolves and so forth) and sometimes you have to pay attention to follow them.
the stories that i pursue are way bigger and more ambitious than you realize. you just don't see the big picture because you're not paying enough attention. when you pay so little and so infrequent attention then all you can see is little fragments.
if you actually followed the ideas that i'm working on and understood their universal relevance you wouldn't feel such a need to seek some kind of pseudo-relevance in pseudo-"practical" work.
i remember you once expressing dissatisfaction with my work by saying that "it didn't go anywhere" but i wasn't sure about exactly what that criticism was supposed to mean. i think that i'm getting a better idea now that what it meant is that you're just not paying enough attention to follow what i'm doing.
i can't find it now... i looked for it in the n-category cafe thread that jonathan woolf posted to around 2006, but didn't find it there... i'm pretty sure it's somewhere thuogh... where you described some of our work on fundamental n-categories of some kind of "stratified" spaces... and you said something like how this was something you worked on when you were younger, and now you weren't so young anymore, and now it was time to hand it over to those who are young now... i remember being struck when i read that... you showed no understanding that this is a project that i've never abandoned, that i've continued to work on, with pretty slow progress because i can't get you to meaningfully participate in it even if i beg for it...a project that has evolved into other projects that i'm currently deeply involved in, but still recognizably the same project if you'd been paying attention... and it's different for me than for you: you've made your career in significant part off of my work, but you've never been willing to help me in any meaningful way with my career... it's true that i have no desire to retire and hand off the work to the next generation, but i couldn't do that even if i wanted to because i have no career to retire from because you buy into the conventional wisdom that the kind of work that i do isn't worth rewarding...
while i was looking for the post where i remember you saying that, i also happened across this:
|Thanks! As so often the case, this grandiose vision was developed
|jointly by James Dolan and me, but he's not to blame for my
|description of it.
this sounds to me like your attempt to pay lip service to something i once asked from you. i asked that you stop talking about me, stop attributing any ideas to me unless you also make it clear to people that if they want to find out my actual ideas then they'll have to talk to me personally. that's completely different from a formulaic assumption of responsibility for something you wrote. i asked for what i asked for because i want the opportunity to communicate my own ideas to people in my own way, not with the distortions introduced by your inadequacies as an expositor, but with the distortions introduced by my own inadequacies as an expositor. i've repeatedly heard from people that they thought that i preferred for you to speak for me, and shock when i told them that nothing could be farther from the truth.
i guess that you just don't get it... it would be funny if you actually did the things that you do out of malice, but i really don't see the evidence for that.
i was describing this particular long story that i want to tell you, a long story about some math that i've been working on, long because i've been making a lot of progress recently without much of a chance to talk to you about it, and so there's a lot that you missed, making up a long story that i want to tell you about. but i thought that i heard you say something like... that the main reason that you were unsure about whether you'd ever hear the end of the story is because you didn't know whether i'd stay interested in my own story, or instead veer off in some other direction like (you apparently think) i always do...
i didn't respond to that, in part because i wasn't sure i heard you correctly. but if i did respond i'd probably say something like the following...
i stay interested in my own stories much more consistently than you seem to realize. the reason it seems otherwise to you is that you're so absent and uninvolved that you don't see the common thread that makes it still the same story when you come back a few weeks (or months or years) later. stories unfold and evolve (and also sometimes have quick cuts and dissolves and so forth) and sometimes you have to pay attention to follow them.
the stories that i pursue are way bigger and more ambitious than you realize. you just don't see the big picture because you're not paying enough attention. when you pay so little and so infrequent attention then all you can see is little fragments.
if you actually followed the ideas that i'm working on and understood their universal relevance you wouldn't feel such a need to seek some kind of pseudo-relevance in pseudo-"practical" work.
i remember you once expressing dissatisfaction with my work by saying that "it didn't go anywhere" but i wasn't sure about exactly what that criticism was supposed to mean. i think that i'm getting a better idea now that what it meant is that you're just not paying enough attention to follow what i'm doing.
i can't find it now... i looked for it in the n-category cafe thread that jonathan woolf posted to around 2006, but didn't find it there... i'm pretty sure it's somewhere thuogh... where you described some of our work on fundamental n-categories of some kind of "stratified" spaces... and you said something like how this was something you worked on when you were younger, and now you weren't so young anymore, and now it was time to hand it over to those who are young now... i remember being struck when i read that... you showed no understanding that this is a project that i've never abandoned, that i've continued to work on, with pretty slow progress because i can't get you to meaningfully participate in it even if i beg for it...a project that has evolved into other projects that i'm currently deeply involved in, but still recognizably the same project if you'd been paying attention... and it's different for me than for you: you've made your career in significant part off of my work, but you've never been willing to help me in any meaningful way with my career... it's true that i have no desire to retire and hand off the work to the next generation, but i couldn't do that even if i wanted to because i have no career to retire from because you buy into the conventional wisdom that the kind of work that i do isn't worth rewarding...
while i was looking for the post where i remember you saying that, i also happened across this:
|Thanks! As so often the case, this grandiose vision was developed
|jointly by James Dolan and me, but he's not to blame for my
|description of it.
this sounds to me like your attempt to pay lip service to something i once asked from you. i asked that you stop talking about me, stop attributing any ideas to me unless you also make it clear to people that if they want to find out my actual ideas then they'll have to talk to me personally. that's completely different from a formulaic assumption of responsibility for something you wrote. i asked for what i asked for because i want the opportunity to communicate my own ideas to people in my own way, not with the distortions introduced by your inadequacies as an expositor, but with the distortions introduced by my own inadequacies as an expositor. i've repeatedly heard from people that they thought that i preferred for you to speak for me, and shock when i told them that nothing could be farther from the truth.
i guess that you just don't get it... it would be funny if you actually did the things that you do out of malice, but i really don't see the evidence for that.
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