notes from discussion with john huerta this afternoon

??something about long root subalgebra of g2.. as a2 ... with so(3) living inside as "antisymmetric" part, as usual for a2 ... ???something about intersecting quincunx parabolic in 1 dimension, thus suggesting that this is the so(3) corresponding to "rotating the whole rolling ball system in its ambient euclidean space" (or something ...) ... ??aka "diagonal so(3)"; see below...???...

??how _does_ this relate to [relationship between incidence geometry of g2 and that of its long root a2 ?? ... and so forth ...] ?? ...

bor and montgomery ... so(3) X so(3) ... "morphism of pointed homogeneous spaces" or something, and so forth ... ???what _about_ something about ... ???how did it go ??? ... ??something about "beck-chevalley" and so forth here?? ... ???something about induced representations and so forth ... ????or something ????? ....

something about "diagonal so(3)" ... ... both balls rotating together... ??or perhaps "exactly opposite" or something ... ???then also "modified / generalized diagonal" preserving favorite geodesic ... ???something about "macroscopic vs microscopic approach to incidence geometry ... ???in maximal compact picture, or something ???? ....

???something about... ???when maximal compact subgroup of split real form (or something ...) corresponds to "antisymmetric part of whole thing" ?? .... and so forth ... ???....

????hmmm, so what _about_ possibility of general idea of "looking at incidence geometry in maximal compact picture", as here (...) ???? .... ???... ????relationship to what i'd actually wanted to talk about today, about "extent to which incidence geometry survives in arbitrary real form" ??? ... ???and so forth??? ...) ... ???...


???something about "one freeze-frame away" schubert variety for g2 ... ??something about "cubic cone" ... ???over genus zero projective curve ... ???something about the line bundle of the projective embedding here... ???something about riemann-roch and so forth ... numerological approach ... ??.. and so forth ... ???....

so what about "maximal compact picture" approach to "2d schubert variety for 2-dot dynkin diagram, and projective tangent cone of its basepoint singularity ..." and so forth ??? ... ???in general, and also in g2 particular case??? .... ???....

??so ... ??todd seems to be more or less suggesting to invent analog of "grothendieck topology" and/or of "lawvere-tierney topology" for tag doctrine ...

hmm... or i might be over-interpreting a bit here... maybe in general there's no really nice analog of these...

??what makes looking at what happens to the subterminal objects sufficient in certain cases to see what happens to the general objects?? ... hmmm...

??what about something about certain factorization systems here??? ...

there's various other possibly relevant doctrines here... for example just plain cocomplete categories ... ??also maybe some sort of categories with topos _and_ tag structure?? .... ??...

??something about reflective subcategories and gabriel-ulmer duality and so forth ... ??...

??cocontinuous monad on a presheaf category = ... ???

??algebra for it = ... ???

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

so consider inclusion (2,1)-functor from (2,1)-cat of t-modeled toposes with morphisms the "strong triangles" to (2,1)-cat of same objects but with morphisms the lax triangles ... ??or something?? ... ??then what about some sort of (2,1)-adjoint here?? .... ???....

??so... for m a commutative monoid, a commutative monoid wrt "eckmann-hilton-day convolution" of m-sets is precisely a commutative monoid under m ... ???or something...

??and ... ???is also precisely ... ???an m-set such that its slice category is commutative monoidal "in a nice way" ... ???or something ??...

??so is it true that ... ??quasicoherent sheaves over a toric variety are essentially just abelian group objects in its topos of "toric quasicoherent sheaves" ?? ... ??or something ??...

??something about k-module objects in a (cocomplete?? ??or something?? ...) category vs "k-linearization" of it ... ??? ....and so forth ...

??maybe restrict to just a topos for now, or something??? ... left- _and/or_ right-universal property of k-module objects in topos t wrt cocontinuous (??or something??) k-linear functors ... ?? and so forth ???....

??so what about relationship between lawvere's ideas about "topos correposnding to concept of space rather than to particular space" (and so forth) and something about "canonical grothendieck topology" ?? ... and so forth ...

??(2,1)-functor "category of models" from toposes to categories ... ??? no wait, actually i want the "underlying category" (2,1)-functor... ??and then is "canonical sheaves" some sort of adjoint to this?? ....??but then what about also some adjoint to "category of models" ?? ... ???and what about ways in which either and/or both of these (...) might be related to "structure / semantics adjunction" ideas ???.....

??what about something about ... ???relationship of various things here to "interpretation of doctrines" ?? ... and so forth ??

so let's try to consider various "real forms" of simple complex lie algebras ...

??for example let's consider so(5) ... ???

so for example we have so(5), so(4,1), so(3,2) as real forms ... ??...

first let's consider sl(2) ... with sl(2) and su(2) as real forms .... ????....

sl(2) = rrr ??

su(2) = ???something about "pauli matrixes" ???...

01
10

0 -i
i 0

1 0
0 -1

??not at all clear that this is helping yet ... ??...

well, wait... that last one looks like it's in the cartan or something ... ???

hmm, but maybe this is suggesting that... ??the reason that macroscopic and microscopic incidence geometry doesn't survive in arbitrary real forms has to do with... ??relationship between a root and it's negative .... ???or something ...

??for derek...

??something about non-split signatures... particularly "physical" penrose diagram case...

??something about "cartanian dual of lightlike curve in 2+1 conformal spacetime" and legendrian and "generalized legendrian" submanifods ... ??and so forth...

??what about something about "pfaffian system of diff eqs" here ??? ... or something... and so forth ... ???something about the "non-trivial tangent cone" case ... ???or something ... and so forth ... ???....

??something about dimension of coadjoint grassmanian ... ???and so forth ... ??something about long root subalgebra???....

???what _about_ something about idea of... "trying to get incidence geometry and invariant distributions and so forth to work essentially the same way for all real forms, just by sharing the same root diagram" ? ... ??? .... or something ... ???...

notes for next discussion with todd

??try to concentrate on "doctrines of toric algebraic geometry" ...

??what about something about?? ... idea that "canonical grothendieck topology" idea here... something about "best topos approximation" or something... ??might be peculiar here because of nature of doctrine ... ??or something ???....

??something about "boilerplate" idea ... ??...

??something about whether "k-linearization" process connects the two doctrine ladders in the intended (or something...) way?? ... ???....

??something about "viable sub-lizard" approach to grothendieck (and/or lawvere-tierney...) topology ... ???....






??puzzle about "minimalistic syntax" ??? ??save for end?? ... ??or something ???....

??so... "toric quasicoherent sheaves over P^n" allegedly form certain topos ... ??and we think that we sort of know what it's the classifying topos for ... but we're mainly interested in it as theory of poorer doctrine, of course... ??so... model of it wrt poorer doctrine in topos t as .... ???or something???

??or something about ... ???model of it wrt poorer doctrine in actions of commutative monoid m in topos t ... ??or something ... ??? ??something about special case m = 1 ... ??... ???maybe very degenerate??? ....or something ... ???...

???given model of it wrt geometric doctrine in topos t, forgetfully get model of it wrt poorer doctrine in t ... ???...

??so consider a slice topos of the object classifier... say over object x ... ??then... ???this is the classifying topos for ... ??algebras of the free "substitution" monoid on x ... ??or something?? ???what about generalizing this somehow to non-free such monoids ?? ... ??or something?? ....

no wait, that's not correct... try some examples ...

x = "the object square", for example ... ???so a model should be... ???an object equipped with ... ???a pair of points??

x = "the object to its own power" ...??"object equipped with endomorphism" ???/ ???or something????

??compare this to some other hopefully straightforward interpretation of "theory of object equipped with endomorphism" ... ??....

??actually, now this (...) whole idea is seeming wrong ... ??simply because of exponentiation not really being part of geometric doctrine ?? ... ??and i sort of almost knew that already or should have... ??but i think that i was influenced by this alleged "minimalistic syntax" idea... about which i'm now a bit puzzled... how can you get "interesting structure" using just (??...) "adding a generic point of a given object" (or something...) ???.....

???for example, how to get classifiyng topos for "dynamical system" using the minimalistic syntax?? ...??maybe ask todd?? ...


??something about ... ??burroni monad ... as _not_ an example of a substitution monoid in object classifier over topos of directed graphs ... ??... ??or something ???...

??so what about "karoubi envelope of an operad" ??? ... or something ...

??suppose we have a morita equivalence (??of operads?? ... or something ??...) e : x -> y ... ??and also a morphism m : y -> z .... ???then ... ??is there a nice concept here of... ??"the x-analog of z" ??? or something?? ....

hmmm ... ???some sort of "distributivity between morita equivalences and morphisms" ?????? or something ???? .....

???what about something about ... ???expressing a morita equivalence as a span of morphisms .... ????or something??? .... ???span of "morita morphisms" ??? ...

a,b : x <- s -> y span of morita morphisms ... ????....

??what about "weak pushout of bm along a" ... ???or something ...

??maybe for operads the morita span apex should be allowed to be a prop ??? ... ??or something ???.... ??or maybe not necessary??? ??? or something ???...

??so... ???karoubi envelope of typed k-linear operad .... ???...

"boilerplate" ...

??"legalese" ... ???

law ... computer programming...

??something about... being told that "the zariski topos (or whatever...) is the classifying topos for local rings" as like having a lawyer tell you the boilerplate without telling you the actual relevant details ... ??or something ...

??so consider forgetful (2,1)-functor from geometric theories to "tag" theories... ???and left adjoint to this??

??vs ... ??classifying topos for models of tag theory over comm monoid m ... ??and so forth ... ??...

so what _about_ "free boolean frame on a frame" and / or "free boolean grothendieck topos on a grothendieck topos", and so forth ?? ... ??_are_ there "divergence" problems here?? ... ??or something?? ...

??so _is_ it true that ... ???the left adjoint to the forgetful (2,1)-functor from symmetric monoidal cocomplete k-linear categories to symmetric monoidal cocomplete categories takes the "toric quasicoherent sheaves" over P^n to the quasicoherent sheaves over P^n ?? ... ??or something ... ??...

so consider "the free symmetric monoidal category on one invertible object with trivial self-braiding" ... martin asks about whether the inverse object here has the same property ...???

??something about... ???taking inverse as contravariant symmetric monoidal equivalence on the invertible objects?? ...??or something?? ... ??something about mates??

hmmm... ???mate of identity morphism as identity morphism ... ??only if... you're careful to "use the same inverse" on both domain and co-domain ?? ... ??or something ???...

??what about something about ... "adjoint equivalence" here, or something... ??was that supposed to be different somehow from an ordinary equivalence??? ... sounds weird... ??maybe that issue is a level-slip away???...

??or maybe it's _not_ a level-slip away???

??inverse objects vs adjoint-inverse objects??

??so why poset of forcing _"conditions"_ ??? .... ... and so forth ... ???

hmm...

??well, there's the vague idea about how the double-negation topology "causes everything that can happen to happen", or something like that... ??"as much as can happen to happen" ... or something...

??so consider the "toric ag theory" of... ???

??well, consider the graded actions of the free Z-graded commutative monoid on n+1 generators in grade 1 ...

??which we can think of as forming a pre-sheaf topos?? ... objects of site category = integers .... morphisms = ... ???..

??but then consider the sheaves for a certain topology here ??...


??so what about something about... ??the per-sheaf topos here as a slice topos, and some conceptual interpretation of that ... ???and so forth ... ???

???something about ... ??torsor of group completion of commutative monoid ... ???something about with frame for certain associated torsor ... ???or something ???.... and so forth ... ????

??maybe reminding me of something about toric varieties here, in fact ??? ....

my experiences with john baez

i met john baez via the medium of "usenet newsgroups", particularly the newgroups "sci.math" and "sci.physics"...

he was clearly very articulate and knowledgeable... he also struck me as apt to take the "safe", "establishment" side in any dispute, or at least in any scientific or mathematical dispute... even in cases where i had good reason to disagree with the establishment side...

i remember that at some point he posted a message saying that he wouldn't mind hearing from people who had what they thought was some brilliant new theory of physics ...

(i wonder if i can find this message somewhere...)

i had ideas that i wanted to tell someone about, but i didn't think of them as constituting a "brilliant new theory of physics", exactly... it was more that i had found an amazingly simple way to understand some of the brilliant old theories of physics... a way that i thought was probably already more or less understood by everyone who really knew what they were talking about, but which for some reason seemed to be kept secret from beginning students... this is pretty much always the way it is with me; it's what i do... try to find the amazingly simple ways of understanding things that are usually kept secret from the beginning students, so that i can try to teach them... to beginning students...

so the kind of ideas that i wanted to tell someone about didn't exactly match the kind of ideas that he seemed to be looking for, but they seemed close enough... as an unemployed drop-out from a mathematics doctoral program i found it difficult to get anyone in the academic world interested in my ideas, so my standards as to what constitutes a sufficiently receptive audience were set very low...



??something about ... "peculiar early work" / "i want to be famous" ... ??or something... ???....


??something about ... ??being pretty honest about not wanting to (intensively...) work with me... at some points ... ???

[?? a "go-between", apparently, is someone who lies to you about what the other fellow said and then goes back and lies to him about what you said ...

?? butch cassidy ... ?? "the fall will probably kill you" ... ??? ....]