hecke operator as correspondence between modular curves...

fourier coefficients (or something) of modular form as hecke eigenvalues.... ??....

2d galois reopresentation ... automorphic representation of gl(2) .... or something... ??....

so let's consider the natural transformation from the set-valued functor "square roots of -1" on the category of characteristic p finite fields to the terminal functor; and let's linearize the sets and then take the kernel of the natural transformation... or something like that...

so for example for p=2 we get... ??the zero functor??

while for p=3 we get... hmmm... we're taking the kernel of a non-surjective map here... ??maybe we should be using some sort of homotopy-kernel instead??

so we have some vague idea how a "non-linear galois representation" "manifests itself" over a semi-simple commutative ring... but what about the case of a linear one (or whatever the morally correct concept should be)? ....???....

carchedi

on somewhat short notice i may have a chance to ask dave carchedi some questions tomorrow, if i manage to avoid sleeping through the opportunity. i should try to figure out what questions to ask...

??something about the relationship between toposes and stacks on site _locale_ with probably some obvious (and/or perhaps slightly less obvious) grothendieck topology... ??maybe something about how this relates to "syntax/semantics adjointness for geometric doctrine", and so forth...

??something about this bit about "moduli stack of foliations" or something... and so forth or something...

??something about "higher-dimensional toposes" ... groupoid-based vs category-based... and so forth...

??something about philosophy that "focusing on stacks (as opposed to objects of groupoid-enriched categories...) is a bit silly, in a categorified version of the same way that focusing on sheaves (instead of objects of categories) is" .... or something like that... ??in connection with this, something about the idea of "obtaining grothendieck topology from knowledge of homotopy colimits in a groupoid-enriched category into which the putative site (ordinary) category embeds" ... or something...

hmm, one semi-obvious (in retrospect at least) pointthat carchedi made is that what they call the "topological stacks" is pretty much ess just the grothendieck toposes, and more special than the arbitrary stacks on the hopefully obvious site here.... ???or something like that... actually, sorry, from talking to carchedi again just now i see where i screwed up here again... might try to straighten this out at some point...

??maybe i should try asking them more specifically about what happens if we try to define a grothendieck topology on the category of ("finitary"?) affine schemes (or something like that) by using homotopy-colimits in the groupoid-enriched category of symmetric monoidal finitely cocomplete algebroids... or something like that...

??this idea that there's a tendency for stacks which are well-described by some kind of nice sheaves over them to coincide with those arising from groupoid objects (or something like that) seems somewhat unexpected (to me, but it might be just because i haven't been paying attention). at least we seem to be seeing something like that in the "topological" case; do we also see it in the "algebraico-geometric" case?

serre subcategories vs something else

recently i've been noticing how there are plenty of straightforward examples of finitely cocomplete algebroids that aren't abelian categories, including also symmetric monoidal examples. it now occurs to me that this might mean that (in connection with the "doctrine" ideas that i've been trying to develop) i should switch from techniques involving serre subcategories to a somewhat different flavor of technique...

let's consider for example the category of abelian groups, and what are the full subcategories such that the inclusion has a left adjoint...

we can for example take any homomorphism of abelian groups and force it to become epi. by gabriel-ulmer duality the corresponding full subcategory should consist of those abelian groups such that ...???

let me try again... let's take a sketch of a colimits theory, by which i mean for now that we have a category c and some cocones in c that we're going to declare to be colimit cocones. so the syntactic category of the theory is going to be...

sorry, i'm still undecided about how to proceed here...

inertia subgroup and frobenius conjugacy class

trying to understand various ideas connected with this stuff...

suppose that we've got a gXh-set x....

and an elt x1 in x ....

and consider the "stabilizer in g of the h-orbit of x1" ... os....

??so... this stabilizer acts on the transitive h-set given by this h-orbit ... os....

and the aut gp of that h-orbit is the normalizer in h of the stabilizer in h of x1, modulo that stabilizer .... ???os???....

??but we're mainly interested in the case where h is abelian, so what seemed slightly interesting there mostly just degenerates??? os???.... meanwhile... ??what happens to make this "frobenius" stuff work??? os??? .....

langlands

stuff that baez has been telling me about.... ??os... asf os... ???

???sa.... "eisenstein series" .... ???sa as related to mellin transform of hasse-weil zeta fn of "affine k-space + 1" ... ???os??? .... asf os... ???......

???but wa sa automorphic form ct automorphic rep here????? os???? ..... asf os... ????.....

??bump p 4... ???says will develop functional equation for dirichlet character... os... ??sounds like should explain sa automorphic rep corresponding to 1d galois rep ???? os???? ..... asf os.... ???......

??so wa this general zeta / general theta stuff here???? os???..... asf os... ????.....

??so what _is_ going on here?? sa "theta fn as modular form" os??? w_a_ sa different variables.... ??and different dimensions of qfs?? ... ??os??? ..... asf os... ????...... ???.....

p 9.... 1.14 and 1.15 ..... ???try to understand???.......

???also sa 1.16 (os... asf os...) and sa "ramification at -1" ??? os... asf os... ???.....

???1.14 as saying sa... ??yes just use dirichlet coefficients as maclaurin coefficients??? os??? or sa maclaurin ct fourier os????..... asf os ... ???.....

???no wait it's very different from that??? ???sa "theta" .... quadratic... ???so what's going on here??? doesn't it seem like there's just 2 completely weirdly different patterns for 1d galois reps ct 2d ???? os????? ...... hmmmmm....... ????.....

????wa sa "g ct s" here??? os??? ..... gl(1) ct sl(2) .... ????os??? .... asf os... ???.....

???1.15 as saying sa automorphicness under something like reciprocal ... ????os????..... asf os... ???....

??so wa sa langlands correspondence for galois rep comeing from a number field?? ??os???....
asf os... ???sa getting confused about sa... ???old idea that ... ???langlands reciprocity program was in part somewhat of a retreat in some sense from original "non-abelian class field theory" program .... ???os??? ..... asf os... ???.... ... ???sa... lossiness from number field to galois rep, ct from galois rep to l-fn ... ???os??? ... asf os... ???.....

??w_a_ sa old bit about "how to name number fields" os.... ????in connection with sa... ???contentfulness of "galois correspondence" ... ??os??? .... asf os... ???sa... ???bunch of vaguely similar issues .... ???os??? ... ???sa subgps ct gp el;ts ... ???os??? .... asf os... ???sa the picture .... ???... os.... asf os... ???...... ???Sa "artin reciprocity" ... ???sa line bundles .... os... asf os... ???......

??what's the relationship between baez's "zeta structure type of a z-set" and ... ???sa.... "structure of being a separable f_p-alg ew point of scheme x over it" ???? os??? ..... asf os....
???how much have we been through this before??? os??? ... ???wa sa non-invertible morphisms between "z-sets over y" ?? .... os... asf os... ???.....

??so w_a_ strategy "figure out what ag th it is that the things that have zeta fns (namely, ... ???....) are comm monoids in the smfcca of coherent sheaves over, and then develop concept of l-function of just plain coherent sheaf over it" ??? .... ???os???? .... ????maybe something strange here??? os???.....

ralph kopperman

sometimes i think that the only sane and decent person that i've ever met in this business is ralph kopperman. i met him only rather briefly but i should perhaps tell about it sometime.

a2 schubert varieties

i've thought about the schubert varieties of a2 a bit before, in connection with invariant contact distributions on flag varieties. the a2 flag variety can be thought of as the projective cotangent bundle of the projective plane, which is close to the conceptually most natural sort of contact manifold; you can picture its points as infinitesimal little slats of a venetian blind, with the contact distribution corresponding to the blind being closed rather than open. (it's odd though that the non-orientability of the projective plane prevents the blind from providing much shade.) the big bruhat cell corresponds to the 3-dimensional heisenberg lie group which can also be thought of as the "jet bundle" of first-order taylor serieses of functions on the line, and which embeds into the projective cotangent bundle of the projective plane in a fairly obvious way. a first-order non-linear time-independent partial differential equation on the line lives as a vector field on this jet bundle, and if a wave "breaks" in finite time due to the non-linearity of the wave equation then its evolution may be continued outside the image of the embedding.

now however i want to re-examine the a2 schubert varieties from the viewpoint of my recent focus on singularities of schubert varieties. it might turn out that these particular schubert varieties lack singularities but that could still be interesting by way of contrast.

(on the other hand might relationships between singularities and "caustics" be lurking here somewhere? this seems like a long shot, offhand.)

singularities of schubert varieties

i want to try to describe here some of the motivation behind my recent interest in singularities of schubert varieties.

??kind of "schubert singularity" i'd been imagining ... ???maybe all screwed up now??? os?? ... not sure....

schubert varieties over finite fields... ???what they're like??? ??sa "singularity" os, asf os... in light of this ... ??os...

???sa stuff about "kazhdan-lusztig" os???... asf os... ???.....

coxeter ct dynkin info ... ???os... asf os....

i think that bor and montgomery say that one way to think about the relationship between homogeneous distributions and graded nilpotent lie algebras is that there's some sort of "symbol algebra" of a distribution... this vaguely suggests to me the idea of taking the graded nilpotent lie algebras corresponding to bruhat cells and finding some meaningful way of "deforming them into filtered non-nilpotent lie algebras" .... or something like that.

another stray thought, maybe related: an occasional interesting source of projective varieties is projective varieties of tangent cones of singularities. does this mean that whenever you have a projective variety you should look around for a singularity of which it might be the projective variety of the tangent cone?

nilpotent radical vs killing-orthogonal complement

i'm beginning to suspect that one aspect of the relationship between nilpotent radicals and killing-orthogonal complements of parabolic subalgebras is that there's some nice geometric interpretation of the way in which the subalgebra can be expressed as a semi-direct product of the infinitesimal automorphism lie algebra of the "residual geometry" acting on a killing-orthogonal complement (or nilpotent radical).

walking anti-shibboleth

the other day, probably via some more recent links, i bumped into this, where tom leinster described the idea of categorified universal properties as a category theory "shibboleth", something like an unconsciously acquired exclusionary trait; and where there was also some discussion of the category-theoretic usage of "walking" semi-popularized by john baez and myself.

(it's probably possible to find online a bit of my version of the history of "walking" if you look very hard, but perhaps i should give more of it at some point.)

in the discussion leinster did suggest that the shibbolethic exclusivity of the idea is unfortunate, and mike shulman did mention the centralness of the idea to the "doctrines of algebraic geometry" program. nevertheless there's something ironic (or worse) here to me: that an idea seen from within one culture as a shibboleth of it should in disguised form be a crucial shibboleth of another culture, and the two cultures (in this case category theory and algebraic geometry) go on miscommunicating with each other in such close quarters.

(i'm not planning to get into whether this is actually a common pattern with shibboleths.)

i should try to find out where it was that i first ran across an algebraic geometer saying something like "projective n-space is the classifying space for line bundles generated by n+1 sections", and to remember the process by which i gradually realized that what they were really saying was that "in the world of symmetric monoidal finitely cocomplete algebroids, the category of coherent sheaves over projective n-space is the walking example of a line object embedded in the direct sum of n+1 copies of the unit object"; unifying the mundane characterization of projective n-space as "the space of lines through the origin in affine [n+1]-space" with the more sophisticated characterization in terms of line bundles via the category-theoretic (and/or 2-category-theoretic) analysis of "variation" in terms of "variable elements" aka "generalized elements" (and/or "variable objects" aka "generalized objects").

i did at one point make somewhat of an effort to explain the "doctrines of algebraic geometry" philosophy to mike shulman, but i felt that i didn't get the real point of it across. i felt that he thought i was just trying to explain the idea of categorified universal properties, and moreover doing a bad job of it by belaboring only very simple examples in comparison to the much more sophisticated ones that he's used to; whereas what i was actually trying to explain was that if category-theorists will condescend to think about the very simple examples of categorified universal properties that algebraic geometers have used to build the edifice of algebraic geometry, they'll discover and understand the spectacular things that algebraic geometers have accomplished with such simple examples. (and if they think about how to extend the work of algebraic geometers using more sophisticated examples then they may discover something new and interesting.)

up until fairly recently i didn't have a clue as to what algebraic geometers were really doing; namely, exploiting categorified universal properties in amazingly powerful ways. once i stumbled upon that realization, the aspects of algebraic geometry that had previously seemed mysterious to me became practically transparent, to the point where shulman (quite falsely i'm sure) accused me of having more of a background in algebraic geometry than he did.

one of the alternate titles of the "doctrines of algebraic geometry" program is "algebraic geometry for category theorists". category theory sometimes shows up in algebraic geometry in conscious, mundane, complicated ways, but if category-theorists can instead tune in to to the unconscious, spectacular, simple ways then they're uniquely positioned to appreciate what the algebraic geometers are secretly really doing. this is what i think i failed to get across to shulman.

b2 geometry as "hoop geometry"

since i'm using the lie group "b2" as a test case for understanding singularity vs non-singularity of schubert varieties, i'll try to describe here a certain mental picture that i have of the b2 incidence geometry (in the split real case) as what i call "hoop geometry".

among the various names of various forms of b2 are "sp(4)" and "so(2,3)". "sp(4)" means the automorphism group of a 4-dimensional symplectic vector space while "so(2,3)" means the automorphism group of a pseudo-euclidean vector space of signature (2,3). one of the things that the symplectic picture is good for is for understanding the invariant distribution on one of the grassmanians as the natural "contact distribution" on the projective space of a symplectic space, while one of the things that the pseudo-euclidean picture is good for is for the "hoop geometry" idea, which i'll try to explain.

perhaps the most famous pseudo-euclidean geometry is so(1,3), the lorentzian geometry of special relativity. this is the geometry of light beams, which we can think of as linear isometries from euclidean 1-dimensional "time" to euclidean 3-dimensional "space". (light moves in straight lines and always at a certain speed aka "the speed of light".) so(2,3) works the same way except that "time" is now a 2-dimensional euclidean space instead of 1-dimensional; thus now we have the geometry of "2-dimensional light beams" aka linear isometries from 2-dimensional euclidean time to 3-dimensional euclidean space. as with so(1,3) we have symmetries of time alone and of space alone but also "boosts" that mix time and space together.

but now picture the unit sphere in 2-dimensional time as a 1-dimensional "hoop" which fits around an equator of the 2-dimensional unit sphere of 3-dimensional space, sort of like the rings of saturn, thought of as a hula hoop around its equatorial waist.

the grassmanian of total linear isometries from time to space is now just the configuration space of these hoop placements. the other b2 grassmanian, of rank 1 partial linear isometries ("partial light beams"), can be pictured as "lines" of configurations where the hoop swivels around an antipodal pair of nails driven through both the hoop and the sphere; or alternatively as the nail placements themselves, around which the hoop can freely swivel.

now let's try using this mental picture to visualize the 2-dimensional schubert varieties in the grassmanians, and check whether they seem singular or non-singular as predicted.

the 2-dimensional schubert variety in what i'm now thinking of as the "point" grassmanian is the hoop placements that can be reached from an initial such placement by a single swivel. it seems visually clear that this is a non-singular variety ...

well, actually it wasn't as visually clear as i thought at first. nevertheless, the situation seems visualizable enough that after mulling it over for a day or so i think i can see that the real points of the variety form a projective plane, at least in the naive topological sense, which suggests that algebraically it's in fact probably just a projective plane.

we can also revert to the symplectic picture, where the point grassmanian is now the 3d projective space of the 4d symplectic vector space. so the 2d schubert variety in the point grassmanian is just a projective plane in a projective 3-space, quite vanilla it seems.

but now let's try to visualize the 2-dimensional schubert variety in the "line" grassmanian...

actually, first i'm going to dispense with visualization... this schubert variety is the space of rank 1 partial linear isometries from 2d euclidean space ("time") to 3d euclidean space ("space") that are compatible with a given one of them. we can map from this schubert variety to the projective space of the 3d space by taking the image of the partial isometry. this map is 2-to-1 except over the image of the given one, where it's only 1-to-1. is this enough to establish that the variety is singular in a naive topological sense?

does it make sense to talk about the "tangent cone" of a point of a cocommutative coalgebra? have we ever thought about that sort of question before?

schubert varieties of b2

as suggested earlier, i want to use the 2-dimensional schubert varieties in the grassmanians of b2 to test some ideas about when schubert varieties are non-singular.

we can think of these grassmanians as the spaces of partial linear isometries of ranks 1 and 2, respectively, from euclidean 2-space to euclidean 3-space. (thus the rank 2 ones are actually total linear isometries.) these are 3-dimensional projective varieties.

the 2-dimensional schubert variety in the grassmanian of rank 1 plis consists of all those that are compatible with a fixed one of them, in the sense that both are restrictions of some tli.

the 2-dimensional schubert variety in the grassmanian of tlis consists of all those whose equalizer with a fixed one of them is at least rank 1.

my naive prediction seems to be that the first of these schubert varieties will be non-singular and the second will be singular, because there's an invariant 2-dimensional distribution on the first grassmanian but not on the second one.

let me try to argue that the first half of the prediction is correct and that in fact the first schubert variety is a projective plane, isomorphic to the projective space of the 3-dimensional target space. given a 1-dimensional linear subspace v of the target space, there's a unique pli from the source space with image v such that .... hmmm, this doesn't seem to be working.... ??

worse yet, the other half of the prediction seems to be failing too. naturally i suspect that i've got something backwards somewhere, but i've been looking for such a mistake and haven't found it yet.

i suppose i should consider the possibility that the problem involves using my intuition about real-algebraic geometry in a situation where maybe complex-algebraic would be more appropriate. i doubt that that's the problem though.

ok, wait, i think that i found the problem, and that i've almost got this straightened out now. i seem to have made a pretty silly mistake, but i should take a bit of time to make sure that i've really got it straightened out now.

ok, so the mistake is that i got it backwards when i said:

my naive prediction seems to be that the first of these schubert varieties will be non-singular and the second will be singular, because there's an invariant 2-dimensional distribution on the first grassmanian but not on the second one.

instead it should have been:

the first of these schubert varieties will be singular and the second will be non-singular, because there's an invariant 2-dimensional distribution on the second grassmanian but not on the first one.

in trying to straighten out which one has the invariant 2-dimensional distribution, i found it helpful to remind myself about the other way of thinking about b2, associating it with the geometry of a 4-dimensional symplectic vector space instead of that of a (2+3)-dimensional pseudo-euclidean vector space. then the first grassmanian is the "lagrangian grassmanian" of 2-dimensional isotropic subspaces while the second grassmanian is that of 1-dimensional isotropic subspaces. this latter is just the projective space of the symplectic vector space, which carries a natural contact distribution which is the invariant 2-dimensional distribution that we're supposed to be thinking about here.

if we really do have this straightened out now then i'd like to go further and try to understand the nature of the basepoint singularity of the 2-dimensional schubert variety in the first grassmanian in greater detail, for example what the tangent cone is like; i'm still confused about some aspects of that.

rupert's question

coincidentally, someone named rupert posted a question to the newsgroup sci.math yesterday or so:

Suppose that G is an almost simple algebraic group. What would the
unipotent radical of a parabolic subgroup look like, in general?

(this qualifies as a coincidence because this past week or so i've been thinking about such unipotent radicals, even though it didn't occur to me that that's what they are until after i read rupert's question- i'd been thinking about them from a different point of view.

i guess that i'll try to compose an answer to the question here. obviously it's not ready for prime time yet and probably has a fair amount of mistakes, though, so i don't know whether i'll get around to posting it to the newsgroup.)

can you give some idea of what kind of answer you're fishing for? i can try to give you my current take on it, based mostly on stuff that i've been thinking about in the past week or so, but i could imagine some pretty different sorts of answers as well.

(i'll probably be somewhat careless here about the distinction between lie groups and lie algebras, in part because a lot of what i want to talk about involves root systems. for example i'll probably be talking more directly about "nilpotent radicals of parabolic subalgebras" than about "unipotent radicals of parabolic subgroups".)

first of all, notice that as the parabolic subalgebra gets bigger it gets "closer to being semi-simple", so its nilpotent radical actually gets smaller. (for example the biggest parabolic subalgebra is the simple lie algebra itself.) another nilpotent subalgebra that gets smaller as the parabolic subalgebra p gets bigger is the killing-orthogonal complement of p, which in fact is not only isomorphic to the nilpotent radical of p, but is itself the nilpotent radical of the isomorphic-but-"opposite" parabolic subalgebra to p.

thus instead of talking about the nilpotent radical of p, i can equivalently talk about the killing-orthogonal complement of p, and i'll do that because that's the way that i've been thinking about these subalgebras, as killing-orthogonal complements of parabolic subalgebras rather than as unipotent radicals of them.

the killing-orthogonal complement of p is naturally identified with the tangent space of the partial flag variety associated with p. and my basic answer to your question is this: the structure of the killing-orthogonal complement of p as a multi-graded nilpotent lie algebra tells you about the invariant distributions (in the differential geometric sense) on the partial flag variety, and conversely, the nature of those invariant distributions tells you the structure of the multi-graded nilpotent lie algebra.

(the relationship between graded nilpotent lie algebras and distributions is discussed in for example:

N. Tanaka, On the differential systems, graded Lie algebras and pseudo-groups, J. Math. Kyoto Univ. , v. 10, 1-82, 1970.

actually i'm not sure how useful a reference this is, though.)

for example, when the partial flag variety is a hermitian symmetric space, its geometry is "isotropic" and so there are no invariant distributions; thus this is the case where the killing-orthogonal complement of p is not just a nilpotent lie algebra but an abelian one.

the coefficient sequence of the l-series of a nice 1d galois representation is very straightforwardly periodic... but then when you start multiplying them together... ??.... even though .... ????......

discussion with huerta today

so huerta and i are trying to carry out this plan that we have to find the correspondence between the null subalgebra geometry of the split imaginary octonions and the geodesic rolling geometry of a ball rolling on another one of 3 times the radius...

somewhat as expected, we're agonizing at certain points about certain factors of 2 and so forth...

we're trying to decipher bor and montgomery's "theorem 1", and we're still finding certain aspects of it a bit tricky...

discussion with baez today

john and i were trying to straighten out some ideas about l-functions... in the course of the discussion though we got pretty confused at various points and began to realize how much work it might take us to get un-confused...

"(m,n)-category" terminology

alex and i have run into a situation where the terminology "(m,n)-category" seems to suggest a wrong pattern... i have a feeling that this is related to some sort of disagreement and/or miscommunication between me (on the one hand) and toby bartels and mike shulman (on the other hand)...

i once heard michael johnson talk about "posets with all pullbacks", and i helpfully pointed out to him that it's silly to talk about pullbacks in posets because a pullback in a poset is just a product...

(as it turned out i was the one being a bit silly...)

it might be interesting to categorify this phenomenon now in a certain way.... perhaps there's some concept of "category with homotopy-limits of all diagrams on 2-connected diagram schemes" (or something like that) which could be misconstrued as silly in an analogous way... ?and which might even be relevant to some things that we've been thinking about recently?

in connection with stuff about "doctrines" and so forth...

what about constraining the size of a model, vs constraining the size of a "sketch" of a model?? or something like that... this contrast seems to be showing up here... ????...

decategorified gabriel-ulmer duality and infra-sketching of meet-semilattices

alex and i have been talking a bit about decategorifying gabriel-ulmer duality instead of categorifying it, in part as a way of trying to ground what we're doing to some extent. and it seems to be somewhat interesting and useful to think about this stuff.

i wonder to what extent gabriel and ulmer might have explicitly thought about the decategorified version while developing their actual version. though i also wonder to what extent anyone anywhere has developed the decategorified version; i guess that it's very likely that for example some sort of "lattice theorists" have nailed it pretty far into the ground.

part of the attraction of the decategorified version is of course that some annoying "set-theoretical size" issues pretty much go away, allowing the duality (and some other things) to become more "perfect" in some ways, and allowing more definite answers (?more robust wrt change of set-theoretical foundations?) to some questions. this latter seems now to be helping to convince me that some of the tentative answers to questions that we've arrived at in the categorified case are on the right track, because they seem to resemble the more definite answers that we think we're seeing in the decategorified case.

let me try to give some sort of example here...

recently i've been suggesting that for example a doctrine whose syntactic (2,1)-category is essentially just the syntactic ordinary category of a finite limits theory can be sketched by giving a sketch of the finite limits theory, and then supplementing this by, for each generating object x in the original sketch, including in the new sketch as a homotopy-limit cone the "constant" diagram with value x on the diagram scheme given by "the cone over s^1". i probably wouldn't have stated it that way until today, though, because stating it that way is the result of alex and me thinking about the decategorified analog which goes something like this:

a finite limits theory whose syntactic category is essentially just the syntactic poset of a meet-semilattice can be sketched by giving a sketch (by which i mean here basically just a presentation) of the meet-semilattice, and then supplementing this by, for each generator x in the original sketch, including in the new sketch as a limit cone the "constant" diagram with value x on the diagram scheme given by "the cone over s^0".

and the pattern here is supposed to extend pretty straightforwardly to higher cases as well.

grothendieck topology from homotopy colimits in a (2,1)-category

i don't understand much about what people are doing with various sorts of grothendieck topologies on various categories of "schemes", but i have a vague idea connected to this that i want to explore here. i want to get a grothendieck topology on the category of affine schemes (or something like that... ?perhaps just the "finitary" ones?) by embedding this category into the (2,1)-category of symmetric monoidal finitely cocomplete algebroids, and then using the homotopy colimits in this (2,1)-category to determine (in a hopefully near-obvious way) the grothendieck topology...

invariant "higher-degree distributions" on partial flag varieties

so recently i've been talking a bit about stuff like invariant contact distributions and other sorts of distributions on partial flag manifolds, and in particular about how to "read off" such things from root systems. now however i'm beginning to think that it'd be nice to put this stuff in a somewhat larger context, where we consider not just "linear distributions" but also "higher-degree" ones...

i first started thinking about this (as far as i know) when thinking about the "rolling" distribution on the short-root grassmanian of g2. it occurred to me that since this distribution is really just a manifestation of the incidence geometry "lines" on the grassmanian, and since the axioms of the g2 incidence geometry are invariant under switching the long and short roots, the long-root grassmanian should display some similar manifestation. but there's no invariant linear distribution of the appropriate dimension (namely 2) on the long-root grassmanian to be read off from the root system, so we have to start looking farther afield for the corresponding manifestation, and my suspicion is that we're now dealing with an invariant "higher-degree distribution"...

but then will there be some reasonable way to "read off from the root system" such invariant higher-degree distributions? of course it's probably more complicated than in the linear case, but is it at least reasonable? maybe you just take the symmetric power of the relevant "weight diagram" and look at the relevant filtration... or something like that...

is a "lorentzian conformal structure" perhaps an example of a higher-degree distribution? that is, is the conformal structure equivalent to the field of light-cones? i think that it is... this is mainly from vague memories of when i thought about the cone-field on an infinite-dimensional smooth manifold of legendrian submanifolds... "lorentzian conformal structure" as equivalent to "causality structure"...

there also seems to be an opportunity to play the "mythical etymology" game here, trying to relate these "higher-degree distributions" with distributions in the "schwartzian" sense, especially in this lie-theoretic context...

so i've been thinking some more about the invariant higher-degree distributions on the g2 long-root grassmanian... i'll try to describe some of this...

all right, so let me try to describe some of my current thoughts about this...
let's consider a g2 "point" p, and then let's consider the variety given as the supremum of all the "lines" through p. so perhaps we can think of this as a schubert sub-variety of the point grassmanian... the zariski closure of a certain bruhat class (or something like that). anyway, it seems to be a non-singular variety, basically a 2-sphere (i'm thinking in terms of real algebraic geometry of course!) or projective plane or something like that, and this (the non-singularness) is related to how the relevant "invariant distribution" is linear. in contrast, if we look at the point/line dual situation, then i think that we're going to get a non-singular schubert variety, because of the lack of a corresponding invariant linear distribution. and i want to get some idea of the nature of the singularity by thinking about the invariant higher-degree distribution and trying to interpret it as the tangent cone of the basepoint (the unique element in the most special bruhat class), and so forth...

now i'm struggling to remember what i'm supposed to understand about "tangent cones" and so forth, though... let's see, one idea is that a "singular point" is one where the _zariski_ tangent space is too big to be the actual tangent cone; the actual tangent cone is of the "right" dimension but isn't shaped like a vector space, and the zariski tangent space is its minimal linear hull. except that i'm not sure whether some part of what i just said might have depended on the assumption that the singularity is "conical" in some sense...

hmm, an idle thought occurs to me.... if hermitian symmetric spaces are the case where the invariant distribution on the grassmanian is as non-linear as possible, then is the schubert singularity of particular interest in that case? or something like that? not sure how much sense that might actually have made... particularly since i'm having trouble seeing _any_ invariant distribution in this case, even higher-degree...

an even more idle thought: i'm wondering about the relationship between various contexts where some mathematical culture has been preoccupied with "conic" things... "conic sections" ... "conical singularities" and "cone of a projective embedding" (or something like that) ... and so forth...

anyway, i seem to be guessing that for the g2 line grassmanian, the zariski tangent space at the basepoint singularity of the 2d schubert variety is the 4d space corresponding to the invariant linear distribution visible from the root system... ??and that the further 2 degrees of constraint will come from... well actually, i seem to have no idea whatsoever where they could come from... i must have something screwed up here...

i guess that one possibility that i should consider here is that the invariant distributions that i've been calling the "visible" ones are maybe just the most photogenic such, and that i've been overlooking some wallflowers...

so is the "nilpotent radical" of a parabolic subalgebra just the "triangular dual" of the killing orthogonal complement of the subalgebra?? or something like that? ...don't know much about standard terminology here... i just mean that wrt the "triangular decomposition" of the root system these things look oppositely placed... or something like that.

how do interpretations of "nilpotent radical" that we might be getting at here relate to other ideas about them, such as "stuff forgotten by residual incidence geometry" or something about classification of irreps?

another possibly interesting example for trying to understand the singular or non-singular nature of a schubert variety is b2, again comparing the 2d schubert variety in the "point" grassmanian to that in the "line" grassmanian...

discussion with john huerta today

i think maybe i see how to build up the identification of the 1d null subalgebras of the split real octonions with the configuration space of a ball rolling on another one 3 times the radius.

my vague idea is to build it up one step at a time, each step being an added degree of genericness wrt the favorite flag. or something like that.

from an old e-mail to huerta:

here's my rough description of the false rolling behavior:

the rotation plane is always "the same color" that it would be if the
point of contact was as given, but the orientation was the standard
one and true rolling was happening.

(as a special case of this, true rolling does occur when the
orientation is the standard one, which is part of what we already
observed.)

(that e-mail is from when i wrote a mathematica animation which related the null subalgebra geometry of the split imaginary octonions to a ball rolling on a ball of _the same_ radius.)

so how do i make this more explicit??

discussion with alex today

we're trying to work out some of this stuff about homological algebra for the algebroid of short exact sequences...

trying to think of some stuff here as analog of "coherent cohomology" but for baby doctrine of finitely cocomplete algebroids...

something about coherent cohomology as derived functor of "sections" functor right adjoint to theory interpretation... ??something about measuring failure of "sections" functor to preserve colimits... ???something about recurring idea of "sheaves imitating spaces" or something like that... something about expressing variety as colimit, then interpreting this as expressing the structure sheaf as a colimit of pushforwarded structure sheaves... or something like that...

also something about vague idea that.... ???"restoration of exactness" and/or "serre duality" ... or something, and so forth or something... as relating to idea that "exactness of something fails, but secretly what's really happening is that homotopy-exactness succeeds" .... ?? though even if that philosophy is on the right track, is it still going to apply, just as easily, in this not-quite-abelian context that we're exploring??

anyway... we have a couple of theory interpretations from the walking object finitely cocomplete algebroid to the walking epi one: v |-> 0>->v->>v (which extracts the domain of an epi) and v |-> v>->v->>0 (which extracts the codomain)... and we can try to measure the failure of the right adjoints of these interpretations to preserve colimits. and the right adjoint of the "domain" interpretation seems to be "middle term of the ses", which _doesn't_ fail here. while the right adjoint of the "codomain" interpretation seems to be "first term of the ses", which _does_ seem to fail...

chris rogers pointed out fairly obvious fact that using injective resolution when "measuring failure to preserve colimits" goes along with idea that 0th derived functor being the original functor f follows from f preserving kernels... or something like that...

stuff to read

trying to somewhat organize here a list of things that i should probably make some attempt to read...

background foundational material on "doctrines" and so forth...

??makkai and pare

??makkai and reyes

??article in handbook of mathematical logic

material on langlands program ... ??...

etale ... descent ... geometric morphism ...

consider for example projective modules over z[1/3] .... or something like that ...

suppose that we have a morphism of symmetric monoidal finitely co-complete algebroids from the algebroid of representations of a finite abelian group g to the algebroid of z[1/3]-modules...

this was supposed to help us in thinking about the relationship between "galois stacks" and algebroids of coherent sheaves over a base scheme ... ??or something like that... ??and maybe it's actually working, sort of?? ...

let's consider for example z[6^[1/2]], or something like that...

no wait... let's consider some cyclotomic extension of z... well in fact we could have used z[6^[1/2]], but that example i tend to use as a "base", since it's got a nice explicit non-trivial line bundle over it... or something like that...

so perhaps try thinking about the "20" cyclotomic extension of z, for example ...
and then we have some idea of where this is ramified... "at 20", or something like that, because 20 is the discriminant, or something like that... ??though do we have to worry about ramification at archimedean primes here?? or something like that.... ???... ??maybe to avoid such trouble we should focus on some "real" sub-extension, or something like that...

hmm, in a slightly different direction... (or maybe really more or less the same direction, but pushing the idea further... to the point where it should probably really break if it's ever going to...) suppose that we consider morphisms of symmetric monoidal finitely cocomplete algebroids (and equivalences between such morphisms...) from the algebroid of representations of for example a finite abelian group to the algbebroid of vector spaces over, for example, the algebraic numbers, or the complex numbers, or something...

i'm still confused here... "vector space over the field of algebraic numbers" sounds so boring... whereas "representation of the etale fundamental group of spec(q)" (or something like that...) sounds interesting ... ???....

??am i making some level slip here?? something about endo-equivalences of the identity functor on the algebroid of modules of a commutative ring .... ???or something????...... ??what about models of the theory here???? ..... ????...... ??maybe we're dealing with theories that 'lack classical models" ?? .... ??how does this relate to some ideas that we were thinking about recently about certain "galois stacks" corresponding to "pure property theories" ?? ??or something like that???.....

chain complexes of short exact sequences

short exact sequences of chain complexes are a somewhat familiar concept; however i want to exploit that familiarity for purposes of re-interpreting them as chain complexes of short exact sequences, as an example of chain complexes in a finitely co-complete algebroid that's not an abelian category...

i'm getting mixed up here about how profinite groups and so forth work.... or something like that... something about the inverse limit of all of the finite quotient groups of a group, compared to the inverse limit of a dense (in some sense) collection of them... ??something about "congruence subgroup" here?? .... ???......

so given a functor from minimal algebraically closed fields of finite characteristic to vector spaces, we get a functor from finite fields to vector spaces by assigning to a finite field the corresponding "fixed subspace"... then we can push forward to a functor from finite sets to vector spaces... or something like that... and take the "generating function"...

this should be parallel (or something...) to: given a representation of z, we get representations of each z/n by taking the elements fixed under the action of n... so what was baez's description of the "zeta function of a z-set" as the generating function of a certain structure type? ...

so for example let's consider the functor assigning to a field the free vector space spanned by the square roots of -1 in it... and the quotient functor obtained by modding out the hopefully obvious subfunctor... or something like that... but for finite fields this splitting off goes negative ... ??or something like that??.... so the l-function here is really the generating function of a _virtual_ functor... ??or something like that??

??the progression from representation of z to representations of each z/n preserves direct sums??

(but not equalizers...???)

ok, there's a bunch of mistakes and confusion here... i'm going to try and clear some of it up...

so what about (for example...) "the green structure type of being a separable (?) algebra over the gaussian integers" ?? i think that baez already mentioned this idea once...

also what about "the structure type of being of the form x^2 + y^2" ? or something like that... ??something about how this relates to the (ordinary) structure type of being a separable algebra over the gaussian integers?? ....

doctrine of classical first-order theories?

on the one hand, i'd like to think about whether there's a doctrine (in the sense that i've been defining it recently) of "classical first-order theories" (by which i really mean something like "classical first-order theory as conceived of by category-theorists", probably a bit different from how some logicians conceive of such theories).

(for this purpose it might be a good idea to look at the books by makkai and reyes and by makkai and pare, and at that section on "doctrines" in that logic handbook...)

on the other hand, i'd also like to think about the doctrine whose syntactic (groupoid-enriched) category is the opposite of the category of finitely presented groupoids (or something like that). i mentioned this doctrine to baez today as perhaps having its theories be equivalent to something like classical first-order theories, but when we discussed it further things seemed to get a bit more complicated.

hmm, the objects of the syntactic category of a doctrine of "classical first-order theories" should be something like "finitely axiomatized classical first-order theories"...



also i'd like to think about pro-finite groupoids (or something like that) and whether these give theories of the doctrine just mentioned.

anyway, suppose that i have a "classical first-order theory" (in a certain sense which i'll try to clarify as i proceed here). i should be able to extract from it a groupoid of "formulas"...