Wednesday, December 29, 2010

??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... ??...

Tuesday, December 28, 2010

the definition of "tensor category" that urs mentions... ??is this monadic... ??or something???....
so consider the orbit stack of an affine algebraic group g acting on itself by translation...

morally "framed g-torsor" ... = "nothing" .... ??...

quasicoherent sheaf... module of coordinate algebra of underlying scheme of g... equipped with... ???

??something about co-modules?? ... ???....

Monday, December 27, 2010

does "the zariski topos" live "in" the walking commutative ring topos??

...??in the walking commutative ring topos, consider... ??"zariski locale" object ... ???or something??? .... and so forth... ???

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"?? ....

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??
...

Thursday, December 23, 2010

euclid : synthetic : topos :: descartes : analytic : tensor category

...??

consider "the circle"... "the theory of a unit-length 2-component vector" ...

??what does this translate into in the doctrine of ... ????....

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...

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_ ?? ...

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... ??...
??so what about "natural map from colimit of (weighted??) diagram to limit"?? ...
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??...
general htpy-exact frs between ... ???...
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... ???.....
so... consider... (weighted?) diagram whose colimit is preserved by yoneda embedding... ??relationship to "absolute colimit" or something??? ??does this (...) property depend only on the weighted diagram scheme?? or something??... ??how does this relate to adjoints for modules?? ... 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...
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... ??
reinventing the wheel vs stealing fire from the gods...
??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???.....

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...

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.

Tuesday, November 30, 2010

baez mentioned to me work that some people have done involving the fact that the automorphism group of the convex cone generated by the convex set of states of a fd hilbert space is apparently some familiar group...

this sounded interesting... also slightly surprising that i wouldn't have bumped into it myself ...
when i was talking to baez this evening he asked me a question that confused me...

i'm probably jusy making some silly mistake here...

the question is something like... oh, maybe my mistake was sillier than i'd realized...

when you take the free abelian category on an algebroid, first you freely adjoin limits, and then you freely adjoin colimits... and then for some reason you stop there, even though you _could_ go ahead and adjoin some new limits without disturbing the old ones...

but apparently why i was confused is that i forgot that during the stage where you freely adjoin the colimits, the limits that were previously freely adjoined are of course preserved because the yoneda embedding (the free cocompletion...) always preserves all limits...

notes for discussion with baez this evening

we understand to some extent the important role played by the doctrine of symmetric monoidal finitely cocomplete algebroids in algebraic geometry... but on the other hand, this leaves the role played by finite limits still somewhat mysterious... thus it raises foundational (or something...) questions like: why are people all the time using abelian categories as a tool in algebraic geometry, rather than mere finitely cocomplete algebroids? ...

so roughly i want to try to present an answer to this sort of question... though i'm not sure yet exactly how good an answer it is, and the answer is still somewhat fuzzy...

very roughly, my tentative answer is that when you pass from the doctrine of symmetric monoidal finitely cocomplete algebroids to the infinity-doctrine (or something...) of differential graded such (or something like that...) (now why would you want to do such a thing?? ... well, i'll try to get to that...), the fundamental opposition between limits and colimits is ameliorated by a reconciliation... first of all, in this higher context ordinary limits and colimits get replaced by homotopy limits and homotopy colimits, but moreover, it turns out that the difference between homotopy limits and homotopy colimits evaporates... in a certain sense... and in a certain context... well, there's a lot of qualifications needed here...

the story that i'm trying to tell john here is one of those stories whose telling seems to require an irreducible amount of suspense... the story evaporates if you tell it all at once, because the tension to be resolved in a later chapter needs to be built up in an earlier chapter... perhaps this indicates that the story-teller's own understanding is still lacking, or they'd be able to convey their understanding more directly, without detouring through intermediate stages of partial understanding... certainly my own understanding is lacking in this case... but sometimes that's the way it is, and you need to tell the story in stages like that... i remember william zame, when he wanted to try to tell us such a story in a first-year graduate analysis course, would sometimes, short on time or maybe just patience, try to compress it into a single paragraph or maybe just a single sentence, like "at first you think such-and-such... but then later you realize some-other-such-and-such..."... that's the sort of situation i'm in here, short on time or maybe patience... someone's patience, at least...

the intermediate stage of understanding that i think i need to detour through here, and that i'm suspecting john hasn't had a chance to see much of yet, is the stage where you actually take the doctrine of abelian categories seriously as a doctrine, more specifically as (more or less) a lawvere-style doctrine, a monad (though "weak"...) on the (weak...) 2-category of algebroids... or something like that... apparently it was worked out by peter freyd a very long time ago that this monad can be understood as a composite of the monad for finitely complete algebroids and the one for finitely cocomplete algebroids, with a barr-beck braiding (aka "distributivity law") between the two monads... that is, a monad is a type of monoid, and the tensor product (in this case functor composition) of two monoids isn't generally a monoid unless you have a sufficiently nice "braiding morphism" (aka "yang-baxter operator", although there's no yang-baxter constraint required until three objects are involved instead of just two) between the two monoid objects, either a "nonce" braiding or one coming from a "global" braiding (making the monoidal category in which the monoid objects live braided monoidal). a nonce braiding making two monoid objects tensorable is called a "barr-beck distributivity law", and that's what we have in this case...

ideally, i would like to go into lots of detail about this... exactly how the braiding works (to the extent that i understand it so far... the apparent adjointness between cokernel and kernel, and/or the "commutativity" between them...), and going through some nice simple examples such as "the walking short exact sequence abelian category" and how it relates to "the walking epi finitely cocomplete algebroid"... all directed towards building up the central tension, the opposition between limits and colimits in algebroids... but also planting the seed of the eventual reconciliation of the opposition, which is the striking overlap between limits and colimits in algebroids, namely the way in which finite direct sums qualify as both... i also want to discuss freyd's analogy between abelian categories and toposes (or perhaps better, "coherent geometric theories" or something like that), with a similar barr-beck braiding between limits and colimits, and how this relates to flat modules and "the abelian analog of diaconescu's theorem"...
so what about "flat" (or something...) models of an algebraic-geometric theory in the environment of modules over a local (commutative) ring?? ... or something... ???relationship to stuff like the zariski topos of a scheme (or artin stack, or something...)?? ...
so let's consider the terminal weight on the "walking idempotent" diagram scheme... ??or something like that...
chris rogers and i tried to figure out a nice definition of "abelian category" that makes the "distributivity" nature of the compatibility relation more manifest. we decided that "the cokernel of the kernel is the kernel of the cokernel, via the invertibility of the natural map from the one tpo the other" probably works, and seems pretty nice. so i wonder whether that's a standard alternative approach.

??perhaps "kernel of cokernel" deserves to be called "image", and "cokernel of kernel" "coimage" ?? ... or something like that...

??so _is_ there some adjointness relationship between kernel and cokernel here??

hmm, i seem to be getting now that cokernel really is just the left adjoint of kernel, and that this is an even nicer way of expressing the compatibility relation... well, nicer in certain ways at least... for example as a precursor to the "stable" world where homotopy cokernel is the inverse of homotopy kernel...

??feels like the adjointness has something to do with "snake lemma" ??? or something??

well, wait a minute... how would the adjointness imply the "commutativity"??? hmmm...

i was also going to mention that the fact that the distributivity natural transformation here seems invertible (is that correct??) makes the analogy to a braiding a bit closer... ??is that in fact showing up here in some way now??

some confusion here... adjoint endofunctors don't necessarily "commute" with each other, right??

[x X y]^y vs [x^y] X y ...??...

inverse ones do, though... ??? ...???...

Monday, November 29, 2010

in an earlier attempt of mine to understand things, some decades ago actually, i had the idea that homological algebra (or something...) wasn't so much about "measuring failure of exactness" as "measuring the discrepancy between kernels and homotopy kernels" ... or something like that...

??so from the viewpoint that i'm trying to develop now, it seems interesting to take a look back at how much sense that earlier idea makes now...

Sunday, November 28, 2010

stable ... "semi-stable" .... ???pun here on "stable" os??? ??"stable" vs "absolute" os???... .... asf os... todd... ???maybe also sa street asf ... ???sa... "kan lifting" os, asf os????.....
i was thinking about the "(weighted) diagram scheme" approach to (enriched) colimits vs the "presheaf" approach... and it seemed to me that the contrast here is somewhat analogous to the contrast between "x-valued random variable" and "probability measure on x"... or something like that... here i'm thinking of an "x-valued random variable" as something like a space y (analog of the diagram scheme) equipped with both a probability measure and a map to x...

???hmm, maybe this analogy really is closer than i'd realized... the "weight" on the diagram scheme being precisely the analog of the probability measure on the domain of the random variable...

(originally i was going to specialize to the case where the colimits are set-based and so the domain of the random variable has some sort of canonical probability measure such as the equi-probability measure on a finite set, but then the obvious connection between "weight" and "measure" struck me and it seemed clear that the analogy goes deeper.)

maybe make a bit of a dictionary here...

diagram = measurable function
weighted diagram = random variable
diagram scheme = measurable space
weighted diagram scheme = probability measure space
weight on diagram scheme = probability measure on measurable space
colimit of weighted diagram = expectation of random variable

??maybe this analogy could help me understand certain ideas about "calculus of co-ends" (or something like that) ... i think i remember todd for example doing co-end calculations in some graphical way apparently strongly resembling the ordinary calculus of integrals, complete with long italianate s "integral signs"... maybe i didn't get it not only because of not getting co-ends but also because of not getting ordinary integrals... at least, i remember an early stage in my math education where i was happy with stuff like category theory and point set topology but loathed anything with an integral sign... i still loathe integral signs because my opinion on newton vs leibniz is approximately the reverse of the conventional one; i think that leibniz had a deeper understanding of what was going on but that his notation was extremely bad... anyway, i'm now half-way imagining that the main point of the "graphical calculus of co-ends" is to signify a weight on a diagram scheme by a little "ds" thing, exactly the aspect of leibniz's notation that i find the most obscurantist and annoying... (except that if that were the case then shouldn't it be called "calculus of weighted colimits" instead of "calculus of co-ends" which should have to do with some kind of "non-commutative integration" (involving traces of operators) if i'm not too badly confused??)

anyway, i was originally going to try to relate the analogy here to the semi-philosophical question as to whether "x-valued random variable" is just an awkward conceptual substitute for "probability measure on x", on the grounds that the main thing that you do with an x-valued random variable is to push forward the probability measure on its domain to x, and so why bother with the random variable in the first place when instead you could have just dealt directly with a probability measure on x? eliminate the middleman ... (middleman here = domain of x-valued random variable) ...

i remember rota taking the side that the random variable approach is the clearly superior approach because of stuff like how it psychologically promotes thinking about the correlation between a parallel pair of (for example real-valued) random variables in terms of the commutative algebra that they generate. but besides finding rota enjoyable to read i've gotten used to him presenting indefensible pronouncements as the outcome of settled arguments so i generally don't find his pronouncements very definitive.

anyway, as usual it might be interesting to work the analogy here both ways, trying to transport insight in both directions.

Saturday, November 27, 2010

so what about the relationship between "absolute limits" (or something... and so forth...) and this business about adjoints (or something...) of bi-modules?? ... ??...
?what about something about "arrow-flipping of quivers" and "flag-flipping of globular structures"??? or something??...
so given a dg algebroid s and a dg s-module w and a dg s-opmodule w*, with "weakly homming from w into a dg s-module x" naturally equivalent (??in a nice way?? or something??) to "weakly tensoring x with w*" ... ??what are we really (...) saying here? are we saying that w and w* are adjoint 1-cells in some 2-category?? or something??

well, let's see... if tensoring with w* is just like homming from w, then w tensored with w* had better have a special "point" corresponding to the identity morphism of w ... or something... ??so we're thinking of w as a dg bi-module from the unit dg algebroid to s, and of w* as a dg bi-module from s back to the unit dg algebroid... and the special point that we're talking about is a dg bi-module morphism from the unit dg endo-bi-module of the unit dg algebroid to the composite endo-bi-module w # w* ... ??or something... and this morphism should have the property that... ??what??...

hmm, also, should we have that the unit dg endo-bi-module of s, hommed from w as a dg s-module, gives the dg s-opmodule w*?? or something like that?? ... i feel like i'm doing this very unsystematically... is there some more straightforward way of assembling the facts here, seeing various parallels ... ??or something...

??maybe i should try asking todd about some of this stuff... ??...
so what about "grothendieck's six operations" ?? ... or something... ??...
so consider the walking epi finitely cocomplete algebroid... which by gabriel-ulmer duality we can describe pretty explicitly...

and then consider chain complexes of such things... ??

but we have to work out the details of what we really mean by this... i mean, we want to create something like a derived category (or more saliently its (infinity,1)-category precursor, or something like that...), but we don't actually have an abelian category to start with so we probably have to tread lightly...

but assuming that we can actually figure out what we want here, then...

??the next thing that we want to try here is to consider homotopy limits and homotopy colimits in this context ... or something... ???...

see whether they "behave as expected" or something like that...

??which might actually come down to checking whether the "derived category" that we're getting is essentially just the derived category of some certain actual abelian category?? or something??

so far there does seem to be some danger here of cheating out of confusion... or something... ??...
??so given a dg algebroid s and a dg s-module w, we want a dg s-opmodule w* st weakly homming w into another dg s-module x is equivalent to weakly tensoring x with w* ... ??or something...

??and we want to try to understand a couple of alleged motivating examples...

in particular... take s to be the "walking morphism" dg algebroid, and take w to be the "walking kernel element" dg s-module ... ??or something?? ??speaking so far in terms of _"strict"_ universal properties... i think... and we also think that the walking kernel element dg s-module is, from the weak viewpoint, actually the walking homotopy kernel element .... ??or something... but anyway, we're hoping (we think...) that w* here is ... ???a certain shift of the dg s-opmodule that in some (hopefully somewhat obvious, though i don't quite see it yet) sense represents "homotopy cokernel" ... ??or something???
so what about certain situations where something is close to involutory but with a slight "phase factor" so it's really order 4... ?? or something??...

"fourier transform" or something...

??maybe something about transforming between homotopy limits and homotopy colimits (in spectrum-enriched context...), or something??

??any relationship here?? or something??

??something about mod 2 steenrod algebra??
??so the (??co-)weighted limit of a set-valued functor f on s wrt a (??co-)weight w is essentially just hom_[_set_^s](w,f) ... ??...

??and then when f is x-valued the above holds "yoneda-wise" ??...

??and if we take x = _set_^op then we should get the hopefully pretty familiar concept of "weighted co-limit" ??... or something?? ???where a "weight" is (??now...) a contravariant set-valued functor on the diagram scheme ... ??or something??

??so let's try for example the case where the diagram scheme s looks like "m:d->c"... so a diagram f can be thought of as a "family of sets", with f(c) being the family and the fibers of f(m) being the sets in it... and take the co-weight w to be a "singleton family of doubletons"...

so the co-weighted limit of a family of sets in this case is ess just "the sum of the squares" (of the fibers)... or something...

??whereas the weighted _co-_limit of a family of sets wrt the same w interpreted now as a weight rather than co-weight (i'm being sloppy here about the way that the diagram scheme s is self-opposite here...) is ... ??...something like "one point for each occupied set and two points for each unoccupied one" ??? ... that seems a bit awkward... ??does it really work, yoneda-wise?? well, i tried working it out in my _real_ notebook... the handwritten one, which at the moment is windows journal file notebook358 on my tablet/laptop... and it seemed to work out reasonably well, though it didn't completely get rid of the awkwardness... i tried using 2 as my "point representer" in _set_^op, and it worked out to something like... "given a family d of sets, if i paint a subfamily of them blue and another subfamily of them green, such that the sum of the blue subfamily is the same as the sum of the green subfamily (as a subset of the sum of the whole family), then it's equivalent to selecting a subset of [one point for each occupied set in the family, and two (blue and green) for each unoccupied one]" (or something... ??in this story it's ok if something is painted both blue and green?? ...) ... ??because unoccupied sets don't contribute to the union of the family, so blue and green can go their separate ways in that case... there really should be some less awkward intuition about this though... ??maybe some intuitive approach towards "tensor products" that i've forgotten or never learned... or something...

i suppose that you could say something like "take the weighted sum of the underlying discrete diagram, and consider it as the vertexes of a graph where the edges ride the morphisms of the diagram scheme... (??or something?? ??something about "riding in both directions"... once covariantly (for the diagram) and once contravariantly (for the weight)... ??or something??...)... and then take the components of that graph" ... or something like that... ??...

perhaps that _is_ an intuition about set-valued tensor products that i've forgotten or never quite learned... ??...

??anyway, i guess that the primary lesson that i'm supposed to be learning from this at the moment is just about thinking of (co-)weighted limits as hom-objects (or something... ??so co-weight and diagram have same variance, so you can hom co-weight into diagram... ??or something...) and weighted colimits as tensor products (or something... so weight and diagram have opposite variance... like "contracting upper index with lower" or something ... einstein convention or something...)...

(??by the way, what about "hom : parallel :: tensor product : serial" here?? or something?? ... and so forth ... ??...)
??so what _about_ concept of "dualizing complex"??? ... or something...???

??wpa on "verdier duality" discussing "poincare duality" as special case ...???...

??hmm, the "dualizing complex" that they're talking about here is fixed for verdier daulity... ??the specialization that they're talking about seems to involve a specific "pairing" ... ??or something... ??...

Friday, November 26, 2010

??what about relationship between "(de-)categorified gram-schmidt matrix" and "euler form" ?? ...and so forth... ??some confusion here...

???something about resolution (and homology) vs filtration (and associated graded) ... ??...

??something about bilinear forms vs operators... ??bilinear forms arising from operators in combination with symmetric (or something...) bilinear forms ... ???something about "adjointness" ... ??

??something about "schur's lemma" vs non-semi-simple case ... ??? ...

so what _about_ filtration of indecomposable projective quiver representations by irreps, and resolution vice versa?? ...

notes for meeting with alex this evening

??maybe try reviewing how to derive kaleidoscope mirror reflection matrix from cartan matrix, and then try to categorify that somehow??

??also try to understand some ideas that alex mentioned coming from thind?? or something??

??something about root-string interpretation of cartan matrix ... ???sa bilinear form between roots and co-roots, os????.... asf os...

??something about "big picture" stuff?? ... doctrines... derived category ... "duality" ... distributivity.... htpy limits ct htpy colimits... flat ... triangulated ... ???or something.... and so forth...
the vague idea that i have at the moment is that a parameterized or "generalized" model of an algebraico-geometric theory is "flat" (in the sense of the inverse image functor preserving kernels) iff the map of moduli stacks is a "fibration" ... or something like that... and that "everything is homotopy-flat" (in the sense of the inverse image functor preserving homotopy-kernels) is essentially the same idea as "every map is a fibration, up to fibrant replacement" ... ??or something... ???....

??so what about the relationship between enriching the algebraico-geometric doctrine in such a way as to constrain the ag morphisms to be fibrations (or something...) and enriching the geometric doctrine in such a way as to constrain the geometric morphisms to be local homeomorphisms or open continuous morphisms or something?? ... and so forth... ??is there some kock-zoeberlein phenomenon here?? or something??...

??hmm, not clear to me yet how similar flatness of an ag morphism is to "fibrationness" in some sense .... ???...

??so what about "_faithfully_ flat" here?? .... hmmm...

Thursday, November 25, 2010

so we're hoping for something like... ??given a differential graded algebroid s and a differential graded s-module w, obtain (??perhaps subject to some sort of finitariness or other sort of niceness condition on s and w?? or something??) a differential graded s-opmodule w* st "weak homming from w* is equivalent to weak tensoring with w" ... ??or something like that...

hmm, not quite sure how well that parses yet... ??...

??so let's try to work out the alleged motivating examples that we think we know...

(s,w) is supposed to be the "weighted diagram scheme" for homotopy cokernel... or something like that... so the dg algebroid s should have two objects d and c, and be free on one degree zero cycle m from d to c... ??and the s-module w should be the strict weight for cokernel and the weak weight for homotopy cokernel ... ??or something like that?? ...
??so what about stuff about flat modules and the "abelian diaconescu's theorem" (and so forth) vs stuff about... ???how in "the derived world" (and so forth...) the relationship between homotopy limits and homotopy colimits is especially simple... or something... ??? ...

...doctrines.... ??...
so consider for example the (infinity,1)-category of chain complexes of projective (or something...) short exact sequences (or something...) of vector spaces ... ??or something... ??something about whether the "abelian" (or something...) relationship between homotopy limits and homotopy colimits holds here... ???or something... and so forth...

Wednesday, November 24, 2010

notes for next meeting with alex

??possible relationship between kaleidoscope mirror reflection associated with particular dot d in dynkin diagram, and "flipping the direction of all the arrows touching d" or something... ??possible "pull/push" ideas here?? ... underlying/free ... ??? or something??...

??guess about isomorphisms between underlying dynkin diagrams of dynkin quivers, and problems with it... ??...

??stuff about various concepts of "cartan matrix" (??as discussed by benson for example??... ???something about "euler form" or something like that??) and relationships between them and de/categorification ideas, and quiver representations, and basis (??or something??) of irreducibles vs basis of indecomposable projectives, and "projective cover" (or something) and so forth...

(what were those buzzwords (?buzznames?) associated with basis of irreducibles vs basis of indecomposable projectives (??or something?? ... and so forth...) in "category o"?? ... there's "bgg resolution" and "jantzen filtration" and so forth, but isn't there some other buzzname or buzznames that i'm thinking of?? ... hmm, i think that i'm thinking of some kind of "reciprocity", but what _was_ the buzzname attached to it?? ??maybe "bgg reciprocity" or something like that?? also, just what _is_ "reciprocity" supposed to mean here?? ??did it have soemthing to do with transforming between more than just two basises?? or something?? something about getting the same transformation matrix for two different changes of basis or something??... ??any relationship to situations where "reciprocity" means some sort of "adjointness"?? ...still unsure about which traditional sorts of "reciprocity" get included there... ??lawvere suspected all of them?? ... or something...)

?"derived morita context" as chain complex of bimodules, and "core of a t-structure on a triangulated category" and so forth... ??...

Tuesday, November 23, 2010

i got philip hackney to try to fill me in on the last couple of meetings of the rational homotopy theory seminar that i missed...

he told me an interesting sufficient condition for a rational space to be "formal" (in a certain hopefully obvious sense...??...), that its cohomology be concentrated in even degree and be a "complete intersection" graded commutative algebra (or something like that...) ...

i don't have a good feeling yet for what this "complete intersection" condition means, "geometrically" or otherwise...

but one thing that seems interesting here already is that (if i didn't misunderstand what he was trying to tell me...) this is a condition on the cohomology... not sure yet what to make of the idea of a formalness condition expressed in terms of the cohomology alone ... ???....

Saturday, November 20, 2010

?? a blank slate has more automorphisms than a marked one... ??...

(??how does this relate to woozles and bread crumbs and universal unwrappings and so forth?? hmm, maybe it does fit together...)
what about the derived category of an abelian category as "filtered" (and/or "graded" ?? .... or something....) in certain way?? ... something about postnikov fibration ... ??"space of spaces" and so forth???... ... spectral sequence ... ????and so forth???....
gunnarsen mentioned to me an idea (that he thought might have been hilbert's) about understanding the theory of factorization in a "rng" such as the even integers as being analogous in certain ways to the theory of the ideal class group of a dedekind domain...

i started thinking about this analogy a bit, and it seems somewhat interesting...

might relate to some things that i've thought about before (like kummer's chemistry analogy), but not sure yet...
??so as preparation for trying to understand weighted homotopy limits and weighted homotopy colimits of spectrums (or something like that...), let me try to remind myself about certain aspects of weighted limits and weighted colimits of vector spaces (if i ever actually understood those aspects...)...

let's consider weighted limits and colimits where the k-enriched "diagram scheme" is the unit k-enriched category (aka "the walking object") ... ??so the weight is a k-object w and the diagram is a c-object x with c some k-enriched category... and the weighted limit is something like "x raised to the power of the external exponent w" or something like that... k-enriched right-universal property of x^w as... ??...

y -> x^w
----------
w -> [y,x]

??or something??

??is there some nice relationship between "spanier-whitehead duality" (or something...) and "verdier duality" ??? ....

hmm... googling on these two together seems to give a lot of hits connected with someone named "roy joshua" ...

hmmm... checking notebook357 from when simon willerton was here... and some wikipedia articles... some things look promising (??some idea from simon about "derived category as motivated by duality" or something... ??...) but also some confusion between various types of "duality" ... serre ... verdier ... poincare ... and so forth ... ???....

Friday, November 19, 2010

somewhere recently i saw mention of something like "cohomological functors" in the context of triangulated categories, or something like that... probably wpa on "triangulated category"... ??might these have something to do with "bimodules between abelian (infinity,1)-categories" or something?? ... something about (infinity,1)-functors with one adjoint and thus (??) also the other... ??in certain "stable" context ... ???.....

??hmm, is the apparent fact that verdier invented triangulated categories perhaps a good sign in terms of trying to relate "verdier duality" to a sort of "equivalence" between homotopy limits and homotopy colimits in "the stable case" ( =?= the case of an "abelian (infinity,1)-category", or something like that) ?? ....

Thursday, November 18, 2010

??the relationship between homotopy limits and homotopy colimits of spectrums that we're imagining... that they're somehow almost the same thing in fair generality, and that preservation of one amounts to preservation of the other... ??or something like that??... trying to imagine this as some sort of more perfect form of the relationship between finite products and finite coproducts in an abelian-group-enriched category... (??is there any nice way to break up the progression towards the more perfect form into stages??) ...??compare to relationship between limits and colimits of truth values?? ??something??? (??level slip??...) ...??where preservation of one is very different from preservation of the other, despite some sort of close relationship... ??....

Friday, November 12, 2010

so does a half-exact functor between abelian categories nicely induce a htpy-exact fr between the derived (infinity,1)-categories?? or something?? ?and is this what "restoration of exactness" is really about?? or something??

is the (infinity,1)-category of chain complexes of quiver representations of .->. ess the "walking morphism" wrt some hopefully obvious doctrine?? ??or something??...

??given a symmetric monoidal bi-complete (infinity,1)-category (or something...??maybe "stable" or something?? ... ??also something about "negatives of morphisms" or something??...), and given an "environment" of the same type, consider the hom simplicially-enriched groupoid .... ??or something??...

let's consider an example... "[z/2,-]" as a half-exact endofunctor on the abelian category of fp abelian groups... ??...

Wednesday, November 10, 2010

so given a young diagram d, consider...

"walking d-line object" ...

"walking d-invertible object" ...

"walking d-trivial object" ...

??... and so forth ... or something... ??...

Saturday, November 6, 2010

??so what happens if you try something like taking the "triangle" (or something...) groupoid-tri-span of a triangulated category and trying to interpret its degrouidification as a lie bracket operation?? ... and so forth...

??_is_ there some tendency for "semi-simple" and/or "frobenius" property in connection with "triangle" (vs something about nilpotence in connection with ordinary twisted sum?? ... ??? or something???

i think that i'm making a lot of progress on understanding "reflection functors" but i might not have much time to write about it in the next week or so...

??does "reflection functor" live over f_q??? ... ???...

what about trying to relate "reflection functor" to relationship of quiver reps to flags and "springer flags" and so forth ... (or something...) ??? ... ??...

??so consider the (?...) "hall representation" of a hall (?lie??...) algebra, by which i mean the representation obtained by considering short exact sequences where base space and total space are both flat ... ??or something... ??... ??does this make sense, and is it just some obvious sub-representation of the ... ??enveloping algebra rep, or something???? (should go back to a_n case (??or something??) to orient myself here... hmm, i think some stuff here is screwed up; try to straighten it out...) ... hmm, so what about also short exact sequences here where both base and fiber are flat??... ???something "frobenius" (or something...) going on here??.... ??what about derived category and "triangle" here, and/or something about "associativity between (or something...) multiplication, action, and killing forms" ?? ...

what about this business about "fp flat = fp projective" or something???.... ??something about us maybe almost noticing this in quiver representation context?? ... (??hmmm, what about something about "direct sum k-theory vs exact sequence k-theory" here?? or something??? ??something about direct sum k-theory of fp projectives vs exact sequence k-theory of more general objects ... ???or something??...) ... ??relationship to "constructive" aspect of flat?? ...??something about fp in module sense ct in algebra sense here... ??or something... ??...

??to what extent does concept of "flat object" (???vs "flat module" or "flat functor" or something...) make sense here??

Wednesday, November 3, 2010

??sa aesthetic differences os...

??sa "putting stuff in standard form" and/or "taking advantage of knowing that some reliable algorithm applies to a given situation"... me vs lockhart ...

??sa the overlooked feature as "blemish" os... ???... ??sa that particular
case where lockhart's overlooked feature actually seems to go _against_ his dislike (according to me) of applying an algorithm / "turning the crank" ...

??sa "my history with the problem"...

??sa the other article that i think i remember...

????sa discussions with huerta... ??os???... about "insight (??os) ct rigor" os... asf os... ??

Tuesday, November 2, 2010

one of my teaching ideas involves a game that's supposed to teach students about "group theory" (which to me is so all-pervasive in mathematics and in life and the universe in general as to encompass a whole lot of other stuff)...

i'm thinking about this now because of my planned upcoming trip to visit laurens gunnarsen... i'm planning on mainly learning rather than teaching on this particular trip, but since it's all about teaching i'm thinking a lot about teaching ideas in general...

like most of my teaching ideas this is one that i've had almost no chance to put into practice, so it's in very rough form... i think that i have many different variations of the game in my mind, and i'm not sure which would work better for various purposes... it would probably take a lot of experimenting with actual students to figure that out...

i probably don't have a good name for that game yet, but for now i'll call it "the masquerade game"... and the basic idea is pretty simple: to put some of the players in a situation where they have limited information (or limited means of acquiring it) about the "true identity" of some of the other players (or in some variations, of game tokens of some sort), to try to get them to realize how their state of knowledge or "perceptual power" can be measured as a permutation group which controls and shapes their experience within the game; how invariance or covariance wrt the group corresponds to the "observability" or "realness" of a concept from the viewpoint of the player whose perceptual power is being measured.

of course some of the variations of the game are more heavy-handed than others in terms of how explicitly various aspects of group theory are forced upon the players...

it occurs to me at the moment that there might be variations where a change in perceptual power over time more or less explicitly brings out the idea of "symmetry-breaking" ... which is pretty much always implicitly there anyway, though...

perhaps big julie's "dice with no spots" would make a nice silly illustrative example... ??is that from one of the original stories?

??in a somewhat related vein, there's also the "game" where the player is given a pair of n-variable rational functions, one at least as symmetric as the other, and tries to express the more symmetric one as a rational combination of the less symmetric one with completely symmetric ones... for example, express "x" as a rational combination of "x^2" and symmetric rational functions of x and y...

Monday, November 1, 2010

notes for discussion with baez this evening

derived equivalences between categories of quiver representations...

1 ??something about "functoriality paradox" or something? ...

2 ??big picture motivations ... various... ??maybe not get into these too much tonight... quivers and perverse sheaves... ??or something...

3 numerology...

4 weyl group vs artin-brieskorn-coxeter braid group... something about q = -1 and so forth... something about "verdier duality" and related stuff that willerton tried to explain to me...

5 "derived morita context" and so forth... decategorification thereof...

6 "abelian (infinity,1)-category" ... htpy kernel and htpy cokernel as inverses...
??possible improvement on "triangulated category" concept?... ...??model cat approach... ??...
hmm... john huerta asked for my opinion on "lockhart's lament"... i told him that i'd try to read it...

i have a couple of reactions... not sure how many of them i'll get around to mentioning here...

at one point lockhart sets up as a bad example a presumably somewhat traditional proof that an angle "inscribed in a semi-circle" is a right angle, and following that as a contrasting good example a proof of the same fact, more or less found by one of lockhart's seventh grade students... so of course before i read the good proof i decide to think the problem over myself, in part to see whether i could guess what the alleged good proof would be...

as it turns out i guessed correctly. in fact i had a bit of an aha moment while thinking about it, and correctly suspected that i was on the track of lockhart's preferred proof. roughly the "good" proof is as follows: complete the semi-circle to a full circle by reflection through the center of the circle; now you've got a parallelogram inscribed in a circle, and it's intuitively obvious (in a way that can pretty easily be fleshed out into a proof) that such a parallelogram is a rectangle.

one of my excuses for mentioning thia now is that for various reasons i'm interested in trying to reconstruct the genesis of my aha moment...

this is probably a digression from the main issues that john huerta was interested in my opinion on... i hope that i get back to that, but no promises...

i have somewhat of a history with this particular problem, and i should describe some of that history if i get around to it... my aha moment of a few minutes ago is i'm pretty sure a new addition to that history...

let me also mention that some time ago i read another article somewhere which lockhart's lament reminds me of... i'm wondering now whether that was lockhart too...

in reconstructing my thoughts it helps that i have here the actual literal back of an envelope on which i sketched a few ideas...

i think that first i sketched my thoughts about the "bad" proof... pretty vigorously disagreeing that it's bad, rather that lockhart just presented it unfairly... which he subsequently pretty much admits is what he did, though i hadn't read that admission at that point... anyway more about this later, if i get around to it...

fishing around for other ideas, i sketched a few other pictures...

in one picture, instead of drawing a line segment from the apex of the angle to the center of the circle (as in the "bad" proof), i tried "dropping an altitude" from that apex to the boundary diameter of the semi-circle. i had some vague ideas about this... vaguely to do with connecting the original angle-inscribed-in-semicircle picture to the traditional "angle at the center of the unit circle" picture of trigonometry... which probably requires introducing another circle of twice the radius ...

in another picture i drew inside the triangle-in-semicircle the half-scale similar triangle gotten by connecting the midpoints of the sides... not sure whether there was any interesting idea there...

i returned to the idea of making the triangle-in-semicircle into a traditional "trigonometric" right-triangle inside the unit circle (again of twice the radius of the original circle)... and when i started fleshing out this picture, the parallelogram began to automatically emerge somehow, which led to the aha moment... but it's still somewhat mysterious to me as to how this happened... whether it was actually a sensible working out of my vague unit-circle idea, or just a random visual coincidence that i happened to draw that parallelogram...

not sure how worth it it would be to pursue this further...

about the "bad" proof... roughly, my idea of the good presentation of it is that you draw a line segment from the apex of the angle to the center of the circle, dividing the original triangle into two manifestly isosceles triangles; then you write down the obvious system of equations where the variables are all the angles of all three of the triangles; then you solve that system of equations (somewhat gingerly because of the clockish "cyclical" aspect of the angular variables; perhaps treating them very "geometrically" if you want to disguise the noticeably "algebraic" nature of the proof). nobody could care too much about the details of this system of equations, which presumably feeds lockhart's rhetoric about how ugly the proof is. but to me that's the nature of this problem: you identify the tool that's used to solve it, namely the particular system of angular equations, and then actually solving the system is just a foreordained anticlimax. i guess that i'm thinking of it as an example of "reducing a problem to an already solved problem", namely the solution of systems of equations involving the addition of angles.

(one good thing about this "bad" proof is how it generalizes so straightforwardly to the case of an angle inscribed in some fraction of a circle other than one half; does the "good" proof generalize in any interesting way to that case??)

from this viewpoint you can think of the parallelogram proof as really just supplementing the two isosceles triangles with two more, obtained as their mirror reflections (through the center), giving a larger system of angular equations but with the solution more intuitively transparent...

lockhart seems to like proofs that turn on suddenly (as in "aha!") seeing some (pretty much literally) overlooked part of the picture; for example the complementary semi-circle in the case of the "good" proof, or the center of the circle in the case of the good presentation of the "bad" proof...


for some reason i feel like mentioning that this morning i was trying to recall einstein's semi-famous proof of the pythagorean theorem, and was annoyed that i couldn't remember it even though i could remember little bits of it... i actually gave up and looked it up... roughly it's like this:

first, notice that although it's often stated as a theorem about squares, you could equivalently use almost anything else in place of squares, say triangles, or elephants:

"the area of the elephant on the hypotenuse is equal to the sum of the areas of the elephants on the other two sides".

(surface area if your elephants are 3d.)

since it's indifferent what we use, let's use right triangles similar to the original right triangle, with the one erected on the hypotenuse being the original right triangle itself (thus pointing "inward" rather than "outward"). then the theorem is obvious.

(it would probably help a lot if i'd ever gotten around to working out some easy way to post pictures here...)

more explicitly: when the figure erected on the hypotenuse is taken to be the original triangle itself, then not merely do the areas of the smaller figures erected on the other two sides add up to its area, but moreover those smaller figures themselves add up to it itself (with only negligible lower-dimensional boundary overlap).

in some sense this is the ultimate "dissection proof" because no steps need to be taken to enact the dissection; the figure on the hypotenuse comes automatically pre-dissected into the two smaller figures.
a->b->c

. a b c
a 1
b 1 1 1
c 1

??or something... ???... ??may have omitted some signs? ...
a->b->c->d vs e<-f->g->h ... ??


. a b c d ab bc cd abc bcd abcd
a 0 0 0 0
b 1 0 0 0
c 1 0 1 0
d 1 0 1 1
ab 0 0 0 0 0 0
bc 1 0 0 0 0
cd 1 0 1 0
abc 0 0 0 0 0 0
bcd 1 0 0 0
abcd 0 0 0 0

??something about trying to visualize this (or just part of it...) in terms of edges of 4-simplex... ??of course go back and look at for example 3-simplex and 2-simplex cases...
extended a_n quiver rep... ??sa eigenvalue around the loop, os??


simon... ??sa... verdier duality in "singular" case???? ??as requiring derived cat rather than original cat?? os... .... os.... ???hmmm... sa... "singular" stuff and perverse sheaf ... ??sa singularity of alg variety (os...) ct of coherent sheaf over it ... ???os??? hmmm... sa skyscraper sheaf at non-singular point ... ??? ???ct skyscraper sheaf at singular point... os... asf os... sa schubert-bruhat stratification... ??os... asf os... ???... ...sa perverse sheaf and quiver rep...

simon... ????..... not sure now... ???... ??perhaps i was thinking about "quiver varieties" and supposed "tame" case of extended a_n ... ??or something... mixing together "verdier duality" stuff with quiver rep stuff in certain ways... ??or something...

baez... fan... kaleidoscope... stack... ???..... toric geometry...
russell paradox vs burali-forti paradox... ??possibility of some sort of "duality" or something?? ... ???something about product vs sum?? .... or something... ???...
consider mutliplication of real numbers as about "areas of rectangles" vs as about "iteration of magnifications" ... or something... ??what's the relationship here?? ...vaguely reminds me of stuff about... ??blocks to intuitive understanding of fundamental theorem of calculus .... ??... and so forth ... ??....

Sunday, October 31, 2010

??chain complex of morphisms of vector spaces vs morphism of chain complexes of vector spaces...

??chain complex of morphisms of abelian groups vs morphism of chain complexes of abelian groups...

??something about "spectral sequence" ... chain complex... filtration ... ??...

??see recent idle thoughts about "abelian (infinity,1)-category" ... ??...
ab_[j+1] = e_j
bc_[j+1]= def_j
a_j = d_j
c_[j+1] = f_j
b_[j+1] = de_j
abc[j+1] = ef_j

still not sure that that's actually correct, but let's go ahead and try decategorifying it (i guess pretending that it really was categorified originally) into a 3-by-3 matrix of laurent polynomials... or something...

hmm, does this really make any sense?? hmm... 3 vs 6 ... simple roots vs positive... ??...

. a b c
d 1 1/q
e 1/q
f 1/q

??maybe specialize q=-1 for some reason??

. a b c
d 1 -1
e -1
f -1

i have no idea what i'm doing here but it sort of seems to be holding together for some reason...

??this matrix is involutory?? ... ???...


??so what about the possibility that it's in the weyl group somehow, and that with the q's it's in the artin-brieskorn-coxeter braid group? ...or something...

again, i wish that i could remember some of the stuff that simon willerton tried to explain to me about "verdier duality" and so forth... i have the feeling that some aspects of that might be relevant here... ??.... ??something about... ??for certain purposes, cohomology as effectively z/2-graded... ???or something like that??


. -2 1 0
. 1 -2 1
. 0 1 -2

a |-> a + -2a = -a
b |-> b + a = a + b
c |-> c

. -1 1 0
. 0 1 0
. 0 0 1

hmm... that does look similar to the "reflection functor" matrix above... more precisely it seems to be its negative... or something... ??so what's going on here??

i also wish i could remember certain ideas about how multiple meanings of "cartan matrix" are secretly related... i'm trying to remember the name of that author... ??benson??... "representations and cohomology" ?? ...
so is there a nice "algebraic stack" (or something...) of "representations of quiver q" or something like that?? ...one reason that i'm thinking about this at the moment is that we're maybe seeing some sort of "weil conjectures" stuff going on in connection with quiver representations, and i'm trying to understand in what context this should be placed... perhaps a fairly standard algebraic geometry context if there's a certain nice algebraic stack here... or something like that...

let's take for example q = a->b->c ...

let's try to formulate this as a tensor-products-and-finite-colimits theory...

well, so there's the free finitely cocomplete symmetric monoidal algebroid on the quiver algebroid... which is something we've probably thought about before... and we're thinking about it again, maybe from a slightly new viewpoint... or maybe it's really the same viewpoint and i just didn't recognize it...

??explicit description of the syntactic algebroid?? ...

Saturday, October 30, 2010

so what about the idea of "abelian (infinity,1)-category"? seems fairly clear how it ought to go... htpy kernel and htpy cokernel being inverse to each other or something... so probably has already been developed...

??so what about "derived category of representations of quiver q" vs "representations of quiver q (or something) into the (infinity,1)-category of chain complexes of abelian groups"?? ... and so forth ... ???.... ??also something about representations of q into the abelian (infinity,1)-category of spectrums... something about equivalence of spectrum-enriched (infinity,1)-categories ... ??and so forth...

what about model category approach to "abelian (infinity,1)-category" ?? ...
??what about "abelian (infinity,1)-cat" as possible improvement on concept of "triangulated category" (_without_ t-structure ... ?? ...) ? ...

??what _about_ grothendieck's "derivator" concept? ... os... ??....

??about the idea of "improvement" here... if the emphasis is on how a triangulated category can carry many t-structures, then i think that "abelian (infinity,1)-category" might qualify as an improved version of the concept... ?? ... whereas if the emphasis is on how a triangulated category is a mere category rather than higher-dimensional such, then of course "abelian (infinity,1)-category probably doesn't qualify... ??... ??or something...

Friday, October 29, 2010

hmm, i had the hom table looking like this:

. a b c ab bc abc

a 0 0 0
b 1 0 0
c 1 0 1
ab 0 0 0
bc 1 0 0
abc 0 0 0

but now i think that that might be off... ??maybe it should be more like:

. a b c ab bc abc

a 0 0 0
b 1 0 0
c 1 0 1
ab 0 0 0 0
bc 1 0 0
abc 0 0 0

??? or something like that?? what about the homotopy kernel and/or cokernel
of bc->ab, though??

i'd been playing around with the idea that the hom table has some relationship to the mirror quandle of the kaleidoscope, but maybe this correction (?) makes that seem less likely?

let's see, let's try working out the long exact homology sequence of bc->ab, or something like that...

hmm, now that i think about it, in the d<-e->f case we noticed that for example there seems to be a morphism from def to e with decomposable kernel... so what happens if we try transporting this along one of the alleged derived equivalences that we have to the a->b->c case? and how did we manage to not notice that hole in the d<-e->f hom table, if that's what happened?

hmm, so under our first stab at an alleged such derived equivalence, def->e in fact corresponds to bc->ab ...

so anyway, back to the long exact homology sequence...

by the way, it seems possible that our mathematica program trying to find the derived auto-equivalences of the standardly flipped a_n quiver is at least checking for a necessary condition for such an e
.quivalence, namely that the special triangles where all 3 corners are indecomposable are preserved... ??... ??hmm, not sure whether this might give the mirror quandle idea a better chance... ??...


a -> 0 -> 0

c -> bc -> ab

0 -> 0 -> 0

0 -> 0 -> 0

??is that the way it works?? ??simply something about... given a morphism in the original abelian category, you can interpret it as a 2-place chain complex... and by another one of these suspicious coincidences, we spent some time (more or less successfully) trying to convince ourself that in fact the 2-place complex bc->ab is decomposable in the derived category...

??hmm, is there something here about how "long exact homology sequences" work (the "period 3" aspect... or something...) that's pointing (?again?) towards the idea that "in the derived world the opposition (or something) between kernel and cokernel gets resolved" ??? or something??...

??hmm, so what _about_ "restoration of exactness" vs "preservation of homotopy co/kernels" (=?= "homotopy-exactness"??) here????? or something??? ....

??hmm, so if hom(ab,bc) is completely trivial (including the higher aspects) then... ??we're down to just two derived auto-equivalences for a->b->c, which would match the automorphisms of the "undirected quiver"... but we still don't seem to have the a2 case working that way, unfortunately... ??...

hmm... ??the only potential derived auto-equivalences that my mathematica program seems to be finding are ones that relate pretty directly somehow to the cyclic symmetry of the extended dynkin diagram .... ??or something like that...

??so what about "syntactic description" (??in some sense...) of these (...) derived equivalences? ...???... ??something about "derived correspondence"?? ??something like "derived morita equivalence" or something??

??"derived morita context" gets about 10 google hits...

hmm... the potential derived auto-equivalences for a->b->c that we've found... parameterized by z/4... but the one corresponding to 2 would exchange [b,abc] and [abc,b] ... or something... hmm, maybe that's not so bad... both perhaps trivial... ???....

some stuff here is vaguely reminding me of joyal's (?) "inner vs outer horns" ...
could be just a phantom...


??so... ??what about decatgorifying these derived morita contexts into invertible matrixes over the laurent polynomials? ... or something... hmmm... ??...

Wednesday, October 27, 2010

??so what about the multiplicative groups of the quiver algebras of all the different "flippings" of the a_n quiver, as subgroups of gl(n+1)? ??or something...

??and what about other quivers here??.... ??or something...
l(b,0) -> l(ab,0) -> l(a,0) -> l(b,1)

l(ab,0) -> l(a,0) -> l(b,1) -> l(ab,1)

l(a,0) -> l(b,1) -> l(ab,1) -> l(a,1)

????????????

l(c,0) -> l(bc,0) -> l(b,0) -> l(c,1)

l(c,0) -> l(abc,0) -> l(ab,0) -> l(c,1)

l(bc,0) -> l(abc,0) -> l(a,0) -> l(bc,1)




d->de->e

f->ef->e

df->def->e ??df decomposable???

d->def->ef

f->def->de

?????????

. a b c ab bc abc

a 0 0 0
b 1 0 0
c 1 0 1
ab 0 0 0
bc 1 0 0
abc 0 0 0

. d e f de ef def

d 0 1 1
e 0 0 0
f 1 0 1
de 0 0 0
ef 0 0 0
def 0 0 0

??so it looks like ab and bc are going to have to match up with e and def (?not nec resp?) ...

let's try:

ab_[j+1] = e_j
bc_[j+1]= def_j
a_j = d_j
c_[j+1] = f_j
b_[j+1] = de_j
abc[j+1] = ef_j

that one seems to work ... ??but let's try to make an exhaustive list here....

a = f
b = ef
c = d
ab = e
bc = def
abc = de
--
a = ef
b = d
c = de
ab = def
bc = e
abc = f
--
a = de
b = f
c = ef
ab = def
bc = e
abc = d
--

a c
b abc
ab bc

a b bc c abc ab ????...

a bc b abc ab abc ???....

(ab bc) (a abc) (b c) ?????

???seems rigid????? or something??....



irreducible quiver rep = simple root??

indecomposable quiver rep = positive root??

???......

Sunday, October 24, 2010

??... ??"recognizing a triangulated category as a derived category" ... ??"core of a t-structure" ... ??in situations where a derived category is "shared" (??carries multiple t-structures or something?? ??also something about extent to which a t-structure breaks symmetry... that is is really extra structure...??... ??something about distinct t-structures with the same core, or something??...) to what extent can that sharing typically or generally be lifted to the chain complex (chain-complex-enriched) categories?? or something??...

??what about triangulated categories vs some sort of categories enriched over the derived category of the category of abelian groups, or something like that?? ... and so forth...

Saturday, October 23, 2010

??so... think of a tame (not trying to be technically accurate here...) quiver as a nilpotent matrix x and exponentiate 1+x .... and apply this to the underlying vector of a rep of the quiver... ??then the explosion counit is surjective?... ??and the kernel is exp(x)(v)-v ... ??then we keep iterating or something?? hmmm....

Friday, October 22, 2010

so consider the indecomposable objects in the derived category of reps of the a2 quiver...

??then is there some obvious involution corresponding to reversing the direction of the arrow?? hmmm...

can we think of an abelian category of some restricted homological dimension (or something) as enriched over some category of (doubly) bounded chain complexes? or something? ...

vague feeling that various things that simon told me about might be relevant here...

a := 0->1

b := 1->1

c := 1->0

free resolution of c = c<-b<-a ...??... [l(c,0),l(a,1)] = 1 [l( ... -> c0 -> a1 -> b1 -> c1 -> ...

... <- a1 <- ??sa "triangulated category" ... ???.... ??peculiar multiple-level interpretations of "cartan matrix" ...???... ??perhaps this example is somewhat claustrophobic ... ??... try a->b->c vs a<-b->c ... ???....

d:=1->0->0 e:=0->1->0 f:=0->0->1 g:=1->1->0 h:=0->1->1 j:=1->1->1

d':=1->1->1 e':=0->1->1 f':=f g':=d' ?? h':=0->1->1 j':=d' ????.....

?? g':=1->2->2 h':=0->1->2 j':=1->2->3

d<-d'<-h'<-f' e<-e'<-f' f<-f' g<-g'<-h' ????? not very systematic ... ??... h<-h'<-f' j<-j'<-h' ?? h<-h' j<-j' ??maybe we should try some hopefully obvious way of matching up distinguished triangles in the .->.->. case with such in the .<-.->. case? ??suggesting in particular that we _are_ trying to get an equivalence of categories but of triangulated categories... ?? ... or something...

??also... consider the complex of reps of .->.->. as follows:

0->1->1

1->1->0

with vertical maps hopefully pretty obvious... ??is this an indecomposable complex or something?? ... ??are there a whole lot more indecomposable complexes to worry about here, or is this the only "unexpected" one?

??apparently it's supposed to be decomposable, since the representation algebroid is supposed to be homology dimension one or "hereditary" ... i should think about this...
??so w_a_ sa maybe bumping into langlands duality ?? .... os... sa "encoding dynkin diagram as extra dot coxeter diagram" .... sa the extra dot as relating to sa... affine reflection taking zero "root" to some other root .... ???os??? ..... asf os.... ????.... "affine weyl gp" .... ???os??? .... asf os... ??w_a_ the clay-and-toothpick model on my desk??????? asf os.... ?????..... the color scheme .... ?????asf os????...... ????sa... ????long root os???...... ???????sa..... ???????root ct weight or root ct coxeter vertex... ???os????? ....... asf os.... ???????.....

hmmm..... ??? sa extra coxeter vertex "at middle"????? os?????...... asf os... ????.....

???wa sa whether "langlands dual" lives at level including gl(1) (??? or something... sa "abelian" langlands reciprocity.... ???asf os??...) and/or even "the additive group" ??? os??? .... asf os... ???


??finding interesting stuff in old paper notebook #287 from october of 2002...
stuff about g2 and octonions and rolling ball (something about on projective plane...), and getting nice basis for tangent space of partial flag manifold by something about "hemispheres that the figure (?= coxeter vertex?) lies in", and so forth... more relevant to point of moment though, is... ??some stuff about affine weyl groups and so forth.... ??idea that "the blacks are the roots" ...???or something like that?? ...that seems pretty helpful/crucial or something,,,

??some "grand overlay" stuff here??? .... ???...

hmm, there really _is_ a pretty grand overlay picture here... in margin there's a quincunx and a crown ... don't see explicit "roll, spin, skid" yet, though...

??hmm, so w_a_ sa "langlands duality" and "root ct weight" and sa kac-moody?? ... os, asf os... ???... ??transpose of cartan matrix?? ... asf .... ??? ???hmm, so w_a_ sa "co-root" os, asf os??...

hmmm... i found this:

"???so... ??is it maybe the case that ... ??the extended coxeter complex "for" (?...) a loop group g-tilde has black dots forming a lattice which looks like the root lattice for the _dynkin dual_ of g?... ?...o_s_? ??..."

...with a "??hopefully not?..." in the margin... ???also a "still not quite clear" ... ????....

??...also sa:

"??so wa sa a prescription for adding the extra dot (os...) that goes something like this:

?? "find the (...) parabolic that centralizes a short (??hmm...) root, and attach the extra dot so that it's touching just the dots out of that parabolic...." ... ?os..."

???...

??hmm, so wa langlands duality and short/long reversal here???? os... asf os... ?????....



???hmmmm.... ????sa voronoi cells of root lattice os??? as... ??what???....
??some nice thing with kaleidoscope symmetry, but _what_?? and is there some "langlands duality" here, os??? ...asf os...

???maybe sa "coxeter supertile associated to extra dot" ???? os???....

wait a minute, i still think that there's some voronoi cell thing going on here, but maybe it's not quite the root lattice... staring at some of these pictures in #287... ??maybe in the _good_ picture it _is_ the root lattice?? os??? ... g2 case ... os ... ???...

hmm, perhaps more salient than "voronoi cell" here is ... ??well, some sort of fundamental domain for the "affine kaleidoscope" group acting on the root lattice space... ??something about some canonical way of getting a nice fundamental domain or something??


????wa sa "systematically discretizable polytope" vs "tile polytope" here???? os???.... asf os... ???...


hmmm... reference to pressley & segal p16-17 ... ??...

???sa "co-root lattice and weight lattice as dual" ???? os???....

??hmm, sa qa326 h83, p2... manin... "moduli space of curves of arbitrary genus : virasoro gp :: partial flag variety : semi-simple lie group" ... ????os??? ...
and me asking what in the world that's supposed to mean ... ??...

??so... the coadjoint partial flag variety tends to be "very partial", so the corresponding parabolic subalgebra tends to be close to maximal... ??which maybe fits with the idea that the extra dot tends to touch not that many other dots?? ... or something...

??so _is_ it clear whether "langlands duality" is really showing up here (?..) ??
??maybe it's _not_ ?? because... ??... hmmm... ??seems like it shouldn't be that hard to check ... ??? ...unless the killing form maybe causes confusion here... ??...

??hmm, so wa sa semi-direct product of kaleidoscope group acting on "heisenberg lattice" of root system here??? or something...

Wednesday, October 20, 2010

so what about this vague idea of a generalization of differential calculus replacing ideal power filtrations by more general filtrations?? ... and so forth ... "renormalization" ...
so consider the free symmetric monoidal finitely cocomplete algebroid on one object x equipped with an inverse for its symmetric square (??or perhaps also with the braiding for the symmetric square being trivial... or something...) ? ...

??how might problematicness of internal hom of abelian algebroids (or something...) relate to attempt to develop nice doctrine incorporating tensor product and kernels and cokernels?? ....
quiver reps .... ??special such related to torsor??...

derived category ... ??...

schur functor..... group cohomology of symmetric group ... ???.... asf os...
???something about formal/conical analogy (and so forth...) and poisson and gerstenhaber (or something) graded (in some sense or senses... ??...) operads (or something...) and their somehow similar numerologies... (was it "stirling numbers of first kind" or something??) ... ??something about maybe relating this to recent ideas about cohomology of irrep of n! (??so what _about_ developing program or calculation or something for this??) and preservation of substitution by decategorification and euler characteristic of free "homotopy lie algebra" (or something... and so forth...) ...???

Tuesday, October 19, 2010

so consider the representation theory of for example z/p over a finite field f_q ...

6 6*8 6*57

q-1 q^2-1 = (q+1)(q-1) q^3-1 = (q^2+q+1)(q-1)

1+q+...+q^[a*b] = (1+q+...+q^a)(1+q^a+...q^[a*b]) ???....

??something about "cyclotomic reciprocity" here?? or something...

Monday, October 18, 2010

??what about the idea that "stuff is structures"?? ...or something... ??...

consider a g-torsor equipped with some structure under the cartan limit... at most a pair of flags...

then consider forgetting some of the structure and some of the stuff... ??though maybe with another "cartan limit" here?? ??don't forget more than a certain amount of stuff ... ??...

(??vs don't remember more than a certain amoount of structure?? or something??)

??this as corresponding to "subquotient" ?? quotient algebra of subalgebra....

??lattice of such subquotients?? combinatorial description of it??

??relationships among a2,b2,g2,...?? here... ???....

??might also be interesting to look at subquotient lattice of finite group...

g <-< s ->> q

??...
so consider the maximal parabolic subalgebra of g2 indicated by abcfgjkp in the root system picture below:

===a===
b=c=d=e
=f=g=h=
j=k=m=n
===p===

now what are the ideals in this subalgebra?

-,bj,bfj,bcfjk,abcfgjk ???

then is the quotient algebra by the ideal bj isomorphic to the triangle-shaped maximal parabolic subalgebra of b2? if so then what does this mean? is there a nice "geometric interpretation" of this (in a certain sense that i'm vaguely imagining...)?

then also the quotient by bfj, and a maximal parabolic of a2 ... ??...

and what _about_ the relationship between ideals and invariant distributions here?? ... or something... ??...

??the relationship between the ideals of a2,b2,g2(,...???) should translate into a relationship between invariant distributions?? ??or something??...

hmm, wait a minute... certainly the cartan subalgebra has a continuum of ideals in general... ??perhaps somewhat typical for a lot of larger subalgebras as well... ??...

...but still...

Saturday, October 16, 2010

??2-color "necklaces" ...


a,b

aa,ab,bb

aaa,aab,abb,bbb

aaaa,aaab,aabb,abab,abbb,bbbb

aaaaa,aaaab,aaabb,aabab,aabbb,ababb,abbbb,bbbbb

aaaaaa,aaaaab,aaaabb,aaabab,aabaab,aaabbb,aababb,abaabb,ababab,aabbbb,ababbb,abbabb,abbbbb,bbbbbb

??just "primitive" such??

a,b

ab

aab,abb

aaab,aabb,abbb

aaaab,aaabb,aabab,aabbb,ababb,abbbb

aaaaab,aaaabb,aaabab,aaabbb,aababb,abaabb,aabbbb,ababbb,abbbbb

3-color...

a,b,c

ab,ac,bc

aab,aac,abb,abc,acb,acc,bbc,bcc

aaab,aaac,aabb,aabc,aacb,aacc,abac,abbb,abbc,abcb,abcc,acbb,acbc,accb,accc,bbbc,bbcc,bccc

Friday, October 15, 2010

i asked julie bergner some questions about rational homotopy theory today... there's a seminar going on where a bunch of us, none of whom know too much about rational homotopy theory, are trying to learn about it... so far julie has been taking a somewhat more active role than i have, and furthermore some of my questions were relating to the use of model category structures in rational homotopy theory, so julie seemed like a good person to ask...

anyway, one of the things that we realized is what the table of rational homotopy groups of rational spheres looks like...
some questions for gunnarsen...

1 age range of students ...

2 computer programming skills of students?? ...

3 mathematical background of students in general ...

Thursday, October 14, 2010

so given a small abelian ringoid x and an exact functor from x to _ab gp_, we can consider the composite x -> _ab gp_ -> _set_ ... which will still preserve limits... ??so will be a flat presheaf on (the underlying category of) x?? ...

so consider the grothendieck topology on x with coverings as follows... ??...

??topos of canonical sheaves on a finitely complete small category as classifying topos for "exact presheaves" on it?? or something??
so consider the derived k-algebroid of the abelian k-algebroid of fd vector spaces over a field k...

i guess that we should specify whether the chain complexes that we want to consider here should be "finitary" or "bounded" or something... probably yes... i'll try proceeding for the moment without worrying about this too much yet...

so it seems like the derived k-algebroid here is "semi-simple" with basis z... ??or something like that??...

??try next the case of reps of the walking arrow quiver ... ??also the walking loop quiver?? ...
so let's consider the derived algebroid of the algebroid of (say for example...) representations of the quiver a->b->c and see whether we can find inside of it a nice copy of the algebroid of representations of the quiver a->b<-c ... or something like that...

maybe i should go back and take another look at ram's paper on "the connection between representations of quivers and perverse sheaves" (or something like that) ... i found it somewhat annoying (though interesting) the last time that i looked at it, but that was quite a while ago and it seems possible that my perspective might have changed enough to change my opinion about that...
so.... ??direct sums of exact functors betweeen abelian categories are exact ... but kernels and cokernels of natural transformations between them generally aren't ... ??...

??so direct sums of flat functors from an algebroid to an abelian algebroid are flat ... ??...

??so maybe there's no particular good "tensor product" of abeian categories... ??...

i should probably ask toby bartels about some of this stuff...

so consider flat presheaves on the walking arrow quiver... vs flat reps of it... ??...

??so what _about_ topos / abelian category analogies here?? ... ??idea of "spectrum" of an abelian k-algebroid... exact functors into
the k-algebroid of k-vector spaces... (??and/or a stack of such functors into certain intended environments... ?? ...) ??something about karoubi-saturation here, but then beyond that as well ?? and so forth... ??...

?so what about the idea of "alexandroff abelian category" in analogy to concept of "alexandroff topos" (??in analogy to concept of "alexandroff locale" or something...)?? ... and so forth... ??...

??so what about the idea of "classifying topos for exact module of abelian ringoid x" ?? (or maybe introduce "base ring" here in certain way or ways ... something about ringed toposes... ringoid vs algebroid... module vs bi-module... and so forth...)

...and so forth... ??...