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 ...
Tuesday, November 30, 2010
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...
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 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"...
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... ??? ...???...
??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...
??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
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.
???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 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... ??...
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 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... ??...
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???
??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??
"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 ... ??...)
??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... ??...
??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?? ...
???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...
??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...
??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?? ...
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 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... ??...
??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 ... ???....
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
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...
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 ... ???....
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) ?? ....
??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... ??...
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
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??
??_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... ??
??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...
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... ??...
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.
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->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...
. 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...
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...
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 ... ??....
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