?? notes for next discussion with todd ...

?? trying to settle whether attempted definition of "big zariski topos ..." works in case of affine scheme ... or something ...

wish list ?? ...


??follow up on relationship between ringed topos and pre-stack over category of ("finitary" ?? ...) affine schemes ...

??something about structure / semantics adjointness here ?? ... or something... and so forth ...

???stuff about quasitoposes??? ... and so forth ... ????... ??maybe something about "dense sub-topos" ?? ??or something??? ... and so forth ... ???....

so let x be an algebroid equipped with a symmetric monoidal structure on its (contravariant) module cocomplete algebroid ...

??then consider the "concrete operad" of the unit object ... ???or something ???....

??does this (or something like it?? ...) maybe remember a lot about the symmetric monoidal structure??? ??or something ???....

?? so consider the ag theory of _N_-graded modules of polynomials in d+1 grade 1 variables ... ??and so forth ... ??as theory of _what_ ??... ??relationship to something about ... ???representations of algebraic monoid gl(n) as opposed to algebraic group gl(n) ??? ... and so forth ... ???...

???something about "object with invertibility property" as same as "object with invertibility structure" ?? ... ??on the other hand there do seem to be some weird sorts of "near-invertibility" or something ... and so forth ....

??what about the small and/or big zariski toposes of the ag theory of a g-torsor, for example for g finite ???.... and so forth ... ???

???zariski locale as coherent ??? meaning corresponding to distributive lattice??? .. ??_what_ distributive lattice???? .... ???shouldn't be just the ideals ?????? or something???? and so forth ....


ag theory over z/2 .... commutative algebra objects... and / or cocommutative algebra objects ... and so forth ....

wish list of topics to discuss with todd...

??including small puzzles that might be bugging me??? ...

??also vague dreams and so forth ... ???...


galois shapeshifters ...

etale fundamental group and so forth ... via ag theory ... ??...

"galois stack" ...

"motive" and so forth ...

forcing ...

"renormalization" ... and so forth ...

"normal bundle to substack" and so forth .... ???....

derived categories ... "dg ag theory" or something, and so forth ... rosenberg and so forth ... ??interpretation of doctrines??? "homotopy-flatness" and so forth ... "restoration of exactness" ... ??...

reflection functors ...

"triangulated category" ... or better moral euqivalent ... and so forth ...

perverse sheaves ...

??something about "core" ... ??... something about different abelian categories sharing same derived category, and so forth ...

"abelian diaconescu's theorem" ...

free symmetric monoidal cocomplete algebroid on a cocomplete algebroid ... and so forth ...

???representations of algebraic monoid gl(n) as opposed to algebraic group gl(n) ??? ... and so forth ... ???...

"sesquicoherent sheaves" and so forth ... and crackpot matrixes and so forth ...


??baez-kim ... ???



moduli stacks of curves and so forth ...

???something about "shimura stack" ... or something ... ??...

??"langlands ..." ... ??...

hllbert scheme...

string theory ...

???ag theory over Z/2 and something about p = np and so forth ... ???....

"deformation cohomology" ...

rational homotopy theory ...

singularities ... ???....

toric algebraic-geometric doctrine ...

toric dimensional doctrine ...

so consider the presheaf topos of the walking arrow category as a ringed topos, via the identity arrow of some nice commutative ring k, perhaps a field ... and consider the (alleged...) dimensional category of invertible modules in this ringed topos ...

??essentially just one object ?? ... ??endomorphism ring of that object ??? ... ??or something???? .... ????

??something about ... "upper triangular matrix together with lower triangular matrix ... " ???maybe somethng about them being mates of each other or something??? hmm, but that doesn't constrain things too much ... ??? ...

?????....

hmmm....

?? ... "pointwise" tensor product ...

??so what _is_ the unit module here ??? ??maybe i mean as a module of 2X2 triangular matrixes?? ???or something ???? ...???....

??so what about "strong triangle" universal property of ringed topos defined by "lax triangle" universal property?? ... and so forth ... ???....

?? also vice versa ... ???....

d doctrine...

m : t -> e .... in d ... ???os...

??sa ... ??? "t-modeled d-theory as giving rise to stack over ... " ... ???os??? ...

???secretly d = geometric doctrine, t = theory of commutative ring ... ??? ....

??and so forth or something?? ...

??so what about how general theme of "non-degenerate inner product produces canonical flag generic to given flag" (and so forth.... ???relationship of flag variety to conjugacy class and/or coadjoint orbit, and so forth ... ???) relates to ... ???something about "algebraic geometry issues" ?? ... or something... ??? ???something about whether some version of this idea works in an "ag" context ??... ??something about that bit about how... ??"gram-schmidt fails in complex case ..." ... ??or something??? ... i'm probably misremembering how that went; i should have sufficient notes written down somewhere to check up on it ...

??so what about some concept of "local module of local ring" ?? .... ??not finding any relevant discussion so far... ??seems to be something in noncommutative case but doesn't seem relevant ...

??so what about "geometric" case?? ... and possible relationship to "quasicoherent module" or something ... ???....

??so are all locally ringed toposes "small zariski toposes" ? .... ??... or something ... ???...

??so what about ... ???taking the "total variety of a coherent sheaf" seriously ?? ... and so forth ... ???something about "schanuel variety" and so forth ??? ....

??so for example, let's consider the symmetric k[x]-algebra of a k[x]-module m ...

??consider a vector space v with a linear operator f on it ... ???...

???sa ringed topos ct "stack" ... ????os... ???asf os...

??sa possible misunderstanding with carchesi ?? os??? ... asf os... ????....

???so what about something about ... ???framed topos (or something... ??...) and stack over site of locales ... ???or something ... and so forth ... ???...

notes for next discussion with todd

a lot of scattered notes lying around... might not manage to incorporate everything here...

??something about ... ??extent to which "locally (k- ?? ...??or something ?? ...) presentable (n,1)-category" works same / as nicely for n = 2 and/or infinity and so forth as for 1 ... ??? ....

?? sa doctrine interpretation and (2,1)-sketch...

??something about pun on "local presentation" and "locally presentable" ??? ....

??examples of doctrine interpretations...

finitely cocomplete k-linear to abelian ... ??sa "abelian diaconescu theorem" ...

dimensional theory to ag theory...

finite products to finite limits...

finite limits to geometric ...

??and so forth ...

??bit about quasicoherent as sort of analgous to "atomic" (or something ...) object in commutative monoid ??? .... and so forth ... ???....

???sa ringed topos ct "stack" ... ????os... ???asf os...

??sa possible misunderstanding with carchesi ?? os??? ... asf os... ????....



??????sa.... ??? (2,1)-full-and-faithfulness (or otherwiase... asf os...) of certain passage or passages from ag theory to ringed topos ... ???os??? ... asf os... ????

finitely cocomplete category of "finite injections" ... ??as "walking epi" ??

finitely cocomplete category of "finite maps" .... ???as "walking map" ... ??...

???theory interpretations between these ???....

"epi as special case of map" ... ???... ???mapping a finite map to its image inclusion here (...) as preserving finite colimits ... ??but re-including the finite injections among the finite maps as _not_ preserving finite colimits?? ??something about cokernel where "two codomain elements are identified together " ... ??as not preserved... ???because of whether domain elements going to them are also identified together... ???or something?? ....

??so what about the possibility that... ???there's a nice doctrine interpretation from the ag doctrine to the "ringed g" doctrine, but/and that ... ??because the module category of a ring object in a topos (os... ???) is abelian (os...) ... ??the monad unit isn't invertible ??? or something???... ??hmm, do we know _any_ instances where it's invertible, and if so then what are they like???.... hmmm... ??relationship to various old confusions ... ???or something??? .... something about what we mean by "same theory ..." ... "... in different doctrine ..." ... ??? and so forth ... ???...

maybe ask/tell todd about some of the quasitopos ideas here ... ???....

??so what about whether the non-quasicoherent sheaves in the small zariski topos give (??in some maybe fairly obvious way??? or something??) similar examples in the big zariski topos?? ... ??and so forth ... ???...

???hmm, so ... ??what _about_ idea that we should expect monad unit here to be non-invertible???....

??so what _does_ it tend to mean in general when monad unit is invertible (or something...), and how does that relate to "same theory in different doctrine" stuff?? ... ???? ...

???and what about something about... ???getting algebra (2,1)-cat of monad involved here??? ???as yet another doctrine??? ???or something???.... ????..... hmmmm..... ??????.....

???any relationship between ... ???what's (???) going on here (??...), and ... ???something about bit about ... "full homotopy category

???something about (??loose ??...) analogy between "quasicoherent module in ringed topos" and "atom in commutative monoid" ??? ??or something?? ... ??something about "trying to recover the free generatros ... " ... ???or something ???....

???so what about something about trying to take seriously idea of ... doctrine interpretation from ag doctrine to "ringed g" doctrine as being expressed in terms of ... ??something about ... (2,1)-sketch for ag doctrine ??? .... or something... ??ask/tell todd about this ??? ....

???so what _about_ relationship between non-quasicoherent sheaves over (big and/or small ??? ...) zariski (??or something) ringed topos of commutative ring, and ... ??something about ... ??modules of category of quasicoherent sheaves ??? .... or something... and so forth ... ????....

??so what about something whether monad unit is ... ???? "(2,1) full and faithful" ?? ... ???or something??? .... and so forth ??? .... hmmmm.... ?????.....

...hmmm.... ????..... ???something about "good ag theory" here ??? ... ???or something????.... ... ????..... ??as one where the monad unit _is_ full and faithful ??? ???? or something??? ... vs ... ???? ???some vague analog here to something about "dense vs closed" ??? ??or something??? ... ?? ??or maybe closed vs dense" ?? ...??? ...

??hmm, so what about "quasicoherent sheaf" concept (or something...) from "big topos" vs from "small topos" viewpoint????? .....

??so what about ... ???(2,1)-category (or something...) of all ringed toposes st the global sections of the ring object is a given ring ... or has a homomorphism from the given ring... and so forth... ??something about various sorts of "extremal" objects in such (2,1)-categories ... ???....

??and ... ??relationships among modules objects in these (...) various ringed toposes... ???something about (non-)quasicoherence and so forth here ... ??...

??also "categorified" version ... ringed topos st the global sections of the "module category" stack is a given ag theory (or something...)... or has a homomorphism (or something...) from the given ag theory ... and so forth ... ???? .....

(for derek)

i want to try thinking outloud here a bit about the geometry of the conformal compactification of 2+1 space-time... because some of the ideas that came up during the discussion are bugging me a bit...

... something you mentioned about non-"reductiveness" of "parabolic geometries" ...

... also something about the shape of the generic region ... ???....

derek... bruhat cells... conformal .... ??something at infinity ... ???...

???zariski topos of "propositional" ag theory .... ??? os... asf os... ????....

so what about quasicoherent modules in ringed (?...) topos of z/2-sets, for example??....

??hmm, so _is_ the stalk of the sheaf of open subspaces of a tdch space x at a point pof x a boolean algebra?? ??and if so then _is_ there some sort of "formalization" (wrt "geometric" doctrine, presumably...) of the concept of boolean algebra here?? ...

also, if so, is there a nice way to describe this boolean algebra directly in terms of the "point" of the corresponding boolean algebra('s spectrum) ?? ...

seems not boolean... ?? at non-isolated point, consider germ of its complement ... ??...

(for todd)

i hope that we get a chance to talk this wednesday morning...

i have a question about boolean toposes and related stuff that's really bugging me...

(seems like things like this have been bugging me for a long time... i think there's hope i'll get it straightened out eventually though...)

did i mention that this is probably a very stupid question??? ....

the category of boolean algebras is allegedly equivalent to the opposite of the category of totally disconnected compact hausdorff (tdch for short) spaces... i think that i understand reasonably well how this works...

consider the one-point compactification of the discrete space "N" of natural numbers; this is tdch. the corresponding boolean algebra consists of the clopen subspaces, which in the tdch case is equivalent to the regularly open subspaces, i think. we can also think of them as the subsets of N which are either finite or co-finite, or the sequences of truth-values that are eventually constant.

now in the topos of sheaves over this tdch space, ... ???....