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... ??...