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

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

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

??see recent idle thoughts about "abelian (infinity,1)-category" ... ??...

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

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

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

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

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

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

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

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


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

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


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

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

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

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

i also wish i could remember certain ideas about how multiple meanings of "cartan matrix" are secretly related... i'm trying to remember the name of that author... ??benson??... "representations and cohomology" ?? ...

so is there a nice "algebraic stack" (or something...) of "representations of quiver q" or something like that?? ...one reason that i'm thinking about this at the moment is that we're maybe seeing some sort of "weil conjectures" stuff going on in connection with quiver representations, and i'm trying to understand in what context this should be placed... perhaps a fairly standard algebraic geometry context if there's a certain nice algebraic stack here... or something like that...

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

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

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

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

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

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

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

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

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

hmm, i had the hom table looking like this:

. a b c ab bc abc

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

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

. a b c ab bc abc

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

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

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

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

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

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

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

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


a -> 0 -> 0

c -> bc -> ab

0 -> 0 -> 0

0 -> 0 -> 0

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

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

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

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

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

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

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

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

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


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

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

??and what about other quivers here??.... ??or something...

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

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

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

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

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

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

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




d->de->e

f->ef->e

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

d->def->ef

f->def->de

?????????

. a b c ab bc abc

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

. d e f de ef def

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

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

let's try:

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

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

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

a c
b abc
ab bc

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

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

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

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



irreducible quiver rep = simple root??

indecomposable quiver rep = positive root??

???......

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

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

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

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

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

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

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

a := 0->1

b := 1->1

c := 1->0

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

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

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

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

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

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

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

0->1->1

1->1->0

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

??apparently it's supposed to be decomposable, since the representation algebroid is supposed to be homology dimension one or "hereditary" ... i should think about this...

??so w_a_ sa maybe bumping into langlands duality ?? .... os... sa "encoding dynkin diagram as extra dot coxeter diagram" .... sa the extra dot as relating to sa... affine reflection taking zero "root" to some other root .... ???os??? ..... asf os.... ????.... "affine weyl gp" .... ???os??? .... asf os... ??w_a_ the clay-and-toothpick model on my desk??????? asf os.... ?????..... the color scheme .... ?????asf os????...... ????sa... ????long root os???...... ???????sa..... ???????root ct weight or root ct coxeter vertex... ???os????? ....... asf os.... ???????.....

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

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


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

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

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

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

hmmm... i found this:

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

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

??...also sa:

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

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

???...

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



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

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

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

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


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


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

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

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

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

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

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

so what about this vague idea of a generalization of differential calculus replacing ideal power filtrations by more general filtrations?? ... and so forth ... "renormalization" ...

so consider the free symmetric monoidal finitely cocomplete algebroid on one object x equipped with an inverse for its symmetric square (??or perhaps also with the braiding for the symmetric square being trivial... or something...) ? ...

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

quiver reps .... ??special such related to torsor??...

derived category ... ??...

schur functor..... group cohomology of symmetric group ... ???.... asf os...

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

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

6 6*8 6*57

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

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

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

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

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

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

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

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

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

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

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

g <-< s ->> q

??...

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

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

now what are the ideals in this subalgebra?

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

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

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

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

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

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

...but still...

??2-color "necklaces" ...


a,b

aa,ab,bb

aaa,aab,abb,bbb

aaaa,aaab,aabb,abab,abbb,bbbb

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

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

??just "primitive" such??

a,b

ab

aab,abb

aaab,aabb,abbb

aaaab,aaabb,aabab,aabbb,ababb,abbbb

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

3-color...

a,b,c

ab,ac,bc

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

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

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

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

some questions for gunnarsen...

1 age range of students ...

2 computer programming skills of students?? ...

3 mathematical background of students in general ...

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

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

??topos of canonical sheaves on a finitely complete small category as classifying topos for "exact presheaves" on it?? or something??

so consider the derived k-algebroid of the abelian k-algebroid of fd vector spaces over a field k...

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

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

??try next the case of reps of the walking arrow quiver ... ??also the walking loop quiver?? ...

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

maybe i should go back and take another look at ram's paper on "the connection between representations of quivers and perverse sheaves" (or something like that) ... i found it somewhat annoying (though interesting) the last time that i looked at it, but that was quite a while ago and it seems possible that my perspective might have changed enough to change my opinion about that...

so.... ??direct sums of exact functors betweeen abelian categories are exact ... but kernels and cokernels of natural transformations between them generally aren't ... ??...

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

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

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

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

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

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

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

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

so what about something about irreps of borel as always tangent spaces of... something or other ... ???...

so consider the operad for "lie algebras where all of the tetralinear operations are trivial"... so the underlying schur functor has terms corresponding to the trivial irrep of 1!, the sign rep of 2!, and the 2d irrep of 3! ... or something like that...

so the free algebra of this operad on a 2d vector space v is... the direct sum of v and its exterior square and... ??another copy og v? ...

and the "holomorph" of this lie group should be contained in g2?? in a hopefully obvious way?? ... visible in the root system... ??

??some confusion here about... holomorph... ??... automorphisms from gl(2), vs inner automorphisms, vs translations... left vs right ... ??....

consider also the b2 case here... heisenberg alg of a 2d vector space...




nothing, element, frame

1 1 1
1 2 3
1 3 6

"categorified gram-schmidt"

(so what about "categorified iwasawa decomposition" or something?? ... and so forth...)

1 -1 1
0 1 -2
0 0 1

1 -2 1
-2 4 -2
1 -2 1

x(x-1)(x-2)/6 + x(x-1)/2 + x

... ??...

??visualizing (or something... and so forth...) tangent space at point of apartment variety as rep of cartan... specific examples... b2... and so forth... ??sa "tame vs non-tame orientations" or something... and so forth or something...

??left-universal property of algebroid of reps of given quiver, wrt various doctrines... involving or omitting tensor product, kernels, cokernels ... and so forth... ??more generally, general case of abelian gp-valued or vector space-valued pre-sheaves (??or sheaves wrt a grothendieck topology?? ... and so forth... or something...) on a cat... ??"generalizations / analogs of "diaconescu's theorem"...

??modified version of "tangent space at a point" for a stratified space?? or something?? something about normal bundles... ??as seems to be going on with flag varieties??... ??any relationship to all that other stuff people do with this sort of thing... perverse sheaves and "intersection homology" and so forth ... ??...

??map from positive roots to weyl group?? seems to work in a_n case... can understand tangent spaces of schubert varieties that way ... ???....

hmm, so the obvious map (a quandle homomorphism or something?) from the positive roots to the weyl group (in fact slightly different from the map that i was thinking of at first, in the a_n case) is (or at least seems to be from casual inspection of some examples) "order-preserving" but not "fully" so... which is perhaps "good" in some sense, in that the order on the roots contains more delicate (dynkin as opposed to coxeter...) information than the order on the weyl group does...

john huerta and i started reading a paper by bryant referenced in the paper by bor and montgomery... pretty interesting, talks about a lot of stuff that i've been trying to work out here...

so given a total order on the interval [1,n] of natural numbers, consider the collection of all subintervals for which the order restricted to them is completely backwards of the usual order...

i guess only bother with subintervals with distinct endpoints...

for example n=3:

1 123 []
2 213 [12]
2 132 [23]
3 231 [12]
3 312 [23]
4 321 [13 12 23]

ok that's not quite doing what i wanted it to... but maybe i see a way to modify it so that it will ... ??...

so the b_n root poset...

(1,1,0,0)
=== (1,0,1,0)
(0,1,1,0) == (1,0,0,1)
=== (0,1,0,1) == (1,0,0,0)
(0,0,1,1) == (0,1,0,0) == (1,0,0,-1)
=== (0,0,1,0) == (0,1,0,-1) == (1,0,-1,0)
(0,0,0,1) == (0,0,1,-1) == (0,1,-1,0) == (1,-1,0,0)

is there a nice way to identify this with some sort of "triangular self-adjoint matrixes" (or something...)? ... imitating the a_n case, as baez suggested...

??hmm, so suppose that we compare the "matrix picture" of an a-series bruhat cell to the "matrix picture" of its zariski tangent space ... ???or something...

so suppose that k is a field, x a k-algebroid, and that for any "tame multi-additive k-functorial operation" on the k-algebroid of k-vector spaces, there's a corresponding such operation on x... preserving composition and so forth... ??...

?"flat" as "preserving limits of virtual objects", sort of?? hmm,but is a composite of flat functors flat?? ...

also, what about "flat" as certain kind of co/limit of representables?? ... or something... ??...

so let's try comparing flat actions of the additive monoid of natural numbers to flat modules of its monoid ring...

so by analogy with my understanding of the role of the concept of "flat presheaf" in diaconescu's theorem, i'm guessing that a "flat module" of a k-algebroid r should be... what?? ...

we're trying to understand the left-universal property of the k-algebroid of fp r-modules, as an abelian k-algebroid... we start out by understanding its left-universal property as a finitely co-complete k-algebroid... which is that it's the "walking r^op-module" (or perhaps "walking r^op-shaped diagram"...) ... so an exact k-functor from the k-algebroid of fp r-modules to an abelian k-algebroid x is given by kan-extending an r^op-shaped diagram in x which has the special property that the kan extension (?which is the process of "tensoring" an fp r-module with the r^op-shaped diagram...) preserves kernels.

??a flat k-functor r->x is the restriction along the yoneda embedding r->_r^op-module_ of an exact k-functor _r^op-module_->x ... ??...

is there a modified version of the concept of "natural number object" in a topos which would have the same relationship to iterating the "partial map classifier" functor as the usual version has to iterating the functor "coproduct with 1"? i imagine that someone has looked at this... i vaguely wonder whether it might relate somehow to "alternate topologies on the natural numbers"... not sure how much sense this actually makes...

the "walking mono" finitely complete k-algebroid is, by (?some version of) gabriel-ulmer duality, the opposite of the k-algebroid of monos between fd vector spaces over the field k. there are two indecomposable objects 0>->k and k>->k.

now what are the fp op-modules of this k-algebroid? i guess that you can think of them as modules of the endomorphism algebroid of the direct sum of the two indecomposable objects. so what is this endomorphism algebroid like? is it the algebroid of 2-by-2 triangular matrixes?? ...

i guess that that sounds correct... ??so are we claiming that the k-algebroid of fd reps of the a2 quiver is the "walking short exact sequence" abelian k-algebroid? if so then this seems somewhat suggestive... except that i'm not really sure what it's suggesting yet...

let's try working this out a bit more carefully...

the 2 representable modules of the 2-object algebroid should correspond to the quiver reps 1>->1 and 0>->1 ...

so what are the indecomposable a2 reps? do they correspond to just the positive roots? i haven't thought about this stuff in a while...

so is it really true that every object in the "walking short exact sequence" abelian k-algebroid is a finite direct sum of the three indecomposable objects "the subobject", "the total object", and "the quotient object" ??

so if this is on the right track then in what ways can/should it be generailzed??

i'm vaguely wondering whether there's something vaguely like "diaconescu's theorem" lurking here. i don't really know the history of diaconescu's theorem but i always had the vague feeling that it was developed as an analog of something else...