??so is there some nice "monadic / anti-monadic factorization" of geometric morphisms??? ... ??and so forth ... ??...

???but ... ??what about something about stuff about ... "n-stage monadic functor" and so forth .... ???(??hmm, annoying terminology here... "binary operation" vs "secondary operation" ... ???...) .... ???something about ... "non-idempotent (??or something???) factorization system" ??? .... and so forth .... ???....

talking to alex this morning ... ran into idea that in tag doctrine as presently construed, maybe sums of dualizable objects aren't dualizable ... ??and so forth ...

??which raises a bunch of questions ... like what about the alleged "toric quasicoherent sheaves over the projective line"; what objects are dualizable there ??? .... and so forth ... ???....

??and if the projective line ag theory has more than one tag theory precursor, then what if anything singles one out as special ?? ... ???and so forth ... ????.....

??hmm, well, seems clear that in the toric quasicoherent sheaves over the projective line, 1+1 (for example) _isn't_ dualizable .... ???is that correct?? ...

??so what about something about ... ???"mutliplication toric variety of a toric variety" ??? ... and so forth ... ????.... ??whether it lives nicely at "toric stack" / "tag theory" level ... and so forth ... ???

???something about ... moduli stack of gl(1)-torsors ... "tensor product" (or whatever you call it) of such things ... ??does this qualify as a "toric product", or something ?? ... well, i think that we ought to be able to get some idea about this by thinking about the case of projective n-space, for example ... ???

??hmm, something about ... ???"toric model" of toric dimensional theory as model where _all_ quantities become invertible ... ??and toric model acting on arbitrary model in hopefully obvious "dimension-wise" way ... ????and so forth ...

????what about something about ... ???thinking of a toric model of a toric dimensional theory as a sort of "renormalization process" ??? or something ....

??hmm, to what extent does that actually make sense ????.....

??well, let's at least pretend for a moment that it makes sense ... ??but it still somewhat bugs me the way that ... ??each quantity is getting its own renormalization factor, as opposed to each dimension .... ???or something ??? ....

??hmmm, but maybe this will make sense if we think about it ... ??something about ... orbit stack wrt the torus action ... and so forth .... ????.....

??well, i'm still having some confusion between dimensions and quantities here ... ???and maybe other stuff too ... "units" and so forth ... ???....

hmmm, i'm suspecting now that maybe ... ??that "double interpretation" stuff maybe did involve some sort of "dimension/quantity confusion" ??? ... ???or something?? ... ??this really seems like it must be correct ?? ... ??something about ... given a homomorphism from a commutative monoid to an abelian group, having the domain also be a group as setting up potential for extra level slip ... ??and so forth ??? ??so then what _about_ "action of toric model on arbitrary model" idea in case of "generalized toric variety" or something ?? ... ...that bit involving 3-place chain complex ... and so forth .... ????....



???what about something about situations where people say stuff like "this is measured in these units and that is measured in those units" even though this and that are quantities living in the same dimension??? ... and so forth .... ????....

???what _about_ something about ... ??? "homomorphism" _constraint_ on these "renormalization processes" ; ??what's the meaning of that ???....

???so what about ... ???extent to which idea of "action of toric model on arbitrary model" extends from toric dimensional doctrine to tag doctrine ??? .... ????and so forth ???? .... ....hmmmm ..... ????something about bit about "keep only those models where for every dimension a scale is set" ... that is, "non-stacky models" or something ... ???something about in any toric model, any "rationally occupied" (??or something) dimension has a scale set for it ... ??and so forth ... ??? ???but in any case this still seems very ad hoc ... ???....

???what about something about ... ???possibility that in tag context (or something ...), concept of "toric model" becomes more clearly not really a model but more of a "process" of some sort ???? ????or something ?? ... ???hmmm, so what _about_ this ???... ???something about ... ???assign to each morphism an invertible number ... ???and so forth ??? ... ??or something ??? .... ????_does_ this make any sense??? ... not really sure ... ???something about "determinant" or something ????? ....

??maybe that (...) isn't quite the right idea ... ???... ??something about how a matrix gets changed when its entries get changed ... ???or something ... ????...
??what _about_ something about "gauge transformation" here??? ... something about "conjugating by diagonal matrix" ... ??but something about dimension/quantity confusion here ... ??? ... and so forth .... ????....

??so what _about_ "toric structure on quasicoherent sheaf" ???.... ??and so forth ... ????....

??so what _about_ trying to understand concept of "toric model" in way intrinsic to tag theory of toric quasicoherent sheaves over a projective toric variety??? ... and so forth ...

??so what about something about ... ??a toric structure on a variety (or something ...) as... ???a certain sort of "system of renormaliztion processes" ???and so forth ... ????....

???what about something about whether the (...) torus of a toric variety is more like a group or more like a torsor ... ??or something ... ??in its most obvious concrete manifestation, or something ??... ...and so forth ...

??some confusion here?? ... ???morphism of 2-place chain complexes ... ???and so forth ??? ... ???hmmm, something about ... ??ideal class group and so forth ??? ... ????? .... ???something about ... "weak morphism" here ... ??what does that correspond to in 2-place chain complex context ??? ... ???something about derived category or something??? .... but exactly how ??? ... and so forth ... ??...

??argument for torus being group: ???something about zero map between chain complexes?? ...??or something ??...

??argument for torus being torsor: ???....

??hmmm, in/appropriateness of terminology "toric model" (or something...) as hanging in balance here ... ??...

??something about kan extension as (left ... or something ...) adjoint to "restricted yoneda embedding" ( = "spectrum" ...) ...

???something about "single object" case, or perhaps something about "monogenic" (??in some sense?? ... and so forth ...) case ... ??... and so forth ...

??something about ... "schizophrenic object" ... "isbell duality" / "isbell conjugation" /"isbell envelope" ... "chu construction" .... and so forth ... ??something about "semantics / structure adjunction" ... ??? ...

??so maybe part of todd's point was to consider the "fixed point category" of a geometric morphism ... ??so what _is_ this like ??

todd points out that lots of nice monads on the category of sets have no fixed point objects ...

??so let x be a small-ly cocomplete category and x1 an object in x; then consider the "globalization/localization adjunction" arising from this; thus globalization assigns to a set s the object given as the s-fold sum of copies of x1, and localization assigns to an object x2 the set [x1,x2] ... ??then consider the "fixed point objects" here ...

??for example x = _set_^op and x1 = 1 ... ??? ... ???...

??hmmm... ??or what about x1=2 ??.... ??relationship to "single classical universe model theory" ?? ... and so forth ... ??...

??something about poset-enrichment here ?? ... and so forth ... ??...

??something about ... other ideas besides "fixed point objects" idea here ... ???....

??idea that term "localization" here _is_ weird because suggests idea of... ??restriction process opposite (or something ... ?????? ...) to kan extension ... ??... i mean "opposite to" in other way, i guess ???... ????... ??is this related to the usual confusion about in what sense "kan extension" "extends" ??? ... ???...

try to remember to go to colloquium talk wednesday 4pm ... !! ...

?actually seems tricky to wake up in time for this ...

??so what about some sort of "homotopy theory" / "simplicial" interpretation of globalization/localization adjunction ?? ... ??or something ?? ...

??something about semi-mysterious reference to derived functors as "original" (??or something?) example of kan extension in mac lane's "categories for the working mathematician" ... ??did we ever figure anything interesting out about what that was about ??? ...

??something about ... ???cohomology (of something ... ??...) as something about homming ... ???... and so forth ... ????.....

??so what about something about ... ??for example, model of tag theory "monoid" over commutative monoid x as ... ??something about central extension (or something ... ??something about monoid vs group here ... ???and so forth ...) of/by x ... ???and so forth .... ????....

???what about something about compact prop doctrine here, and so forth ??? ....

??so what about idea of ... applying globalization/localization adjunction idea in _lots_ of cases ... ??including relatively nice simple case such as the dimensional doctrine ... ???and so forth ??? ....

??hmm, so what _about_ this ?? ... ???something about .... ???interaction with doctrine interpretations ... ??and so forth ...

??something about ... ??doctrine of "toric dimensional theories where all of the quantities are invertible" ??? .... ???and so forth ??? .... ??however i'm not really thinking of this in a "toric" context ... ???.... ??really just 2-stage chain complex, or something ??? ... ???....

??what _about_ something about... ??certain cases we might be imagining here of ... ??applying globalization, followed by applying _right adjoint_ part of doctrine interpretation ?? ... ???and so forth ... ???....

??something about ... "globalization of ideal class group" ... and so forth ... ???....

???something about ... ??deciding that i may have been getting two vaguely similar ideas mixed up with each other:

1 tag doctrine as arguably un-2-topos-like ...

2 dimensional doctrine as arguably un-2-topos-like ...

??or something ???...

??so do these work against each other, or what?? ... and so forth ... ???.... ... ??"level slip" or something ??? .... and so forth ...

(for todd)

just some afterthoughts from the discussion this morning ...

so one of the things that we've been talking about is this general phenomenon of a right-adjoint "spectrum" process that yields a pre-stack from an object of some sort, and a kan extension process left-adjoint to this that tries to re-assemble an object of that same sort from a pre-stack ...

and sometimes i've been calling this left-adjoint "reassembly" process a "globalization" process, for hopefully evident reasons ...

but it occurs to me now that maybe we should similarly call the right-adjoint spectrum process a "localization" process ... that is, maybe it really does make good conceptual sense to think of "localization" and "globalization" as adjoints to each other in this context ... not that the word "localization" is completely untaken and free to be given a new meaning, but... not even clear that some of its existing meanings aren't special cases of this proposed meaning ...

but in any case, we've also been talking about considering the category (or higher category) of "fixed points" of the globalization/localization adjunction here ...

and i just wanted to mention the idea that perhaps one of the ways in which the concept of "stack" (as opposed to the concept of mere "pre-stack") emerges here is precisely from the condition on a pre-stack of being a fixed point of this adjunction ...

that is, we can ask what is the least restrictive "sheaf condition" (or in this context really a "stack condition" ...) aka "grothendieck topology" for which all those pre-stacks which qualify as fixed-points under the adjunction are in fact stacks ...

and if the site category that we started with is "topos-like" (mainly in the sense of having "distributivity" aka "exactness" properties similar to those holding in toposes) then it may be that the sheaves for this topology (as a special case of the stacks) are not so far different from the original site objects, and also not so far different from the pre-sheaves that are fixed-points under the adjunction ...

whereas if, as seems to happen in the "toric" case, the original site category is rather un-topos-like, then it may be that the sheaves for this topology are considerably more general than the original site objects, and than those pre-sheaves that are fixed-points under the adjunction ...

i guess that one sort of "consistency check" that i should probably apply to my attempted reasoning here is as follows: is it true that the "sheafification" and/or "stackification" functors here are of the correct "handedness" (that is, left- vs right-adjoint) as would be expected for a process forcing a pre-stack to become a fixed-point of the globalization/localization adjunction ???

not sure yet just how much sense all of that makes ... ??...

??hmmm, i guess tha there's some possibility (that maybe we even already alluded to) that maybe processes of _both_ handednesses might exist here ... ???....

??so wa barr-beck (or something ...) distributivity decompositions of yoneda monad (or something ... and so forth ...) and something about ... "filteredly cocomplete analog of connected projective" ??.... and so forth ...

??examples... ??...

??connected ...

??projective ...

??"compact" or something ??? ....

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

??something about "relative cocompletions" of various kinds, vs ... ??other ideas, or something ???...

??something about ... ??first adjoining connected colimits, then discrete ?? ... and so forth ... ??....

??so what about for example... ??freely adjoining filtered colimits to _finset_ ... ??vs relatively (??to finite colimits, or something ?? ...) freely adjoining all small colimits to it ... ??? .... and so forth ... ???...

??so what about relationship between ["quasicoherent vs non-quasicoherent sheaf" as to do with strong vs lax gloabalization ... ??and so forth ...] and ["big vs small topos" as to do with strong vs lax triangle ... ??and so forth ...] ?? ... ??? and so forth ....

notes for next discussion with alex

?? tag doctrine ...

?? "globalization" approach to "quasicoherent sheaf" .... toric and non-toric case ... ???...

fix this ... rewrite somewhat more intelligibly ... ??...


toric stuff here .... ??? ....


??sa "ab=cd" example ... ??as maybe pretty easy to sort out... os... ??various ways of visualizing and /or otherwise grokking ... ?? ... os ... asf os... ??which somehow made me think about sa ... :

stratification... example of ... ??? ??leading to something about ... :

singularity...

??then sa stack...

??orbit stack of the torus acting on the toric variety .... ??and so forth ...

??something about ... from "toric" vs "non-toric" viewpoint??? os?? asf os....

??also sa that "double interpretation" stuff i was trying to remember recently ...

??also sa yet more "toric toposes" to add to catalog ... ??os... asf os...

asf ...


"torsic variety"

??something about g-variety with dense g-torsor ... ??os ... asf os... ???...


conflict ... ???....

??something about ... ??at the moment i seem to be getting a conflict between two ways of trying to understand toric variety accidental topos as geometric theory ... :

1 ??sa... graded actions as forming slice topos ... ???sa "frame at Z level" =?= grade ... ???os, asf os... ???

2 ???sa "analog of diaconescu's theorem for filteredly cocomplete categories in place of just plain categories" ... ???no hint here of anchoring grade ... ??os ??? ... asf os... ???.....

??hmm, ok, i think that this is beginning to fit together better now... ??seems to have been confusion... for example something about two extremes, toric variety vs its torus ... and so forth ....

??on the other hand, now i'm getting even more bugged by this "filteredly cocomplete analog of projective" (or something ...) business ...

??so what _about_ "toric ideal class group" (or something ...) and whether it's ever non-trivial ... ???and so forth ... ???

???what about whether ag morphism involving "associated graded vector space" (and so forth ... or something... ??...) qualifies as tag morphism ??? .... and so forth ... ??if yes then what this might mean ... ??? ....

??so what about ... ??extent to which relationship between what i'd call "toric small zariski topos" and "toric big zariski topos" (??but how clear of an idea do we really have as to what this latter thing should be?? ... ??some daughter of the mother topos, but which one ?? ...) fits some pattern of "small vs big" suggested (or something ...) by certain other person or persons ... ??...

??so what about something about ... ??forgetful geometric morphism from _set_^[_finset_^x] to _set_^x ??? ... something about ... "explicit" understanding of coverings / axioms forming grothendieck topology here ... ?? nice axiomatization of "flat" co-presheaf... or something ... and so forth ... ??...

??so what about analog of diaconescu's theorem for filteredly cocomplete category in place of plain category ?? ... ... and so forth ... ???something about how this relates to ... ??taking "accidental topos of toric variety" seriously as geometric theory ... and so forth ... ???something about classical model as cone of fan, sort of ... different sort of from "toric small zariski ..." ... ??... ...relationship here ... ??also something about relationship of this (...) to vague ideas about "presheaf / co-presheaf confusion" that i've been mentioning recently ... and so forth ... ??also something about comparison of this approach to "graded action" / "slice topos" approach to understanding the geometric theory here ... ??... and so forth ...

(for todd)

suppose that x is a "filteredly cocomplete" small category, and consider the grothendieck topology on x where for any filtered colimit diagram in x, the filtered colimit is covered by ... ??

??then is there some nice way of recovering x as a nice "intrinsically defined full subcategory of the sheaf topos here?? ...

??idea that... ??this question seems like a special case of a much larger family of questions that i ought to be able to answer but perhaps don't quite know how to ...

??something about where this question is coming form ... "torpos" and so forth ....

notes for next discussion with todd

??something about tag theory of, for example, commutative monoids ... ??how close it comes to being a "torpos" ... what it's model stack is like, and the toric quasicoherent sheaves over it's model stack.... and so forth ... ?? ... ??something about stackiness in toric context ?? ... and so forth...

?? "toric globalization" and so forth ...

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

???something about filtered colimits and so forth ... "the filtered colimit topology" or something ... ???something about "intrinsically flat object in topos" ... and so forth ... free vs non-free filteredly cocomplete categories and relationship to affine vs non-affine toric varieties ... and so forth ...

??catalog of toposes associated to toric variety ... and so forth ... ??...

??something about localicness here ... and so forth ...

??something about "frankenstein doctrine" ... ??and so forth ... ???...

?? catalog of toposes associated to a toric variety ...??or something ....

??all the not-especially-toric ones ...

?? toric mother topos and its daughters ...

"toric small zariski" ... (or something...) ???relationship to mother topos and so forth ... ??

actions of "structure" comm monoid object ... aka "toric non-quasicoherent" ???

torpos ... aka "toric quasicoherent" ...

relationship between these last two ... formal properties of inclusion, and so forth ... ??maybe partially special to toric case, partially not ??...

...other relationships here ... ???...


??perhaps also something about ... ??graded actions ... ??though perhaps not quite fully intrinsic... ??well, though you could try making it intrinsic by "saturation" or something ... ??not quite sure to what extent that works ... ??....

?? ... "torpos" ... = topos st among the "flat" (???or something???) objects there's one special "torus" one... with the others having it as "localizations" ... ???or something?? ... and so forth ... ??consider affine case here, for example ??? ...

??something about... lots of symmetric monoidal (??and/or pro-monoidal ... ??) structures on a topos, in general ... ??but maybe torposness really is just a property of a topos ...

??so what about something about ... ??"intrinsic concept of flat" here?? ... and so forth ... ??again, some presheaf / co-presheaf confusion here ??? ... ?? formula / model ... ???and so forth ... ???something about ... ??"flatness in which input" ... and so forth ... ??.... diaconescu tricks ... and so forth .... ?????? .....

???what about something about "flatness of localization" (and so forth ...) and
addition / multiplication confusion here (...) ??? ....



??sa "object-pair classifier" over topos ... ??...

??sa object-tuple classifiers as forming ... ???operadal category, os ??? asf os... ??sa... relationship to [relationship between pro-monoidal cat and quadratic operad... asf ...] ... ???....

??so what _about_ "filteredly cocontinuous presheaves" ?? ... and so forth ... ???...

??hmm, how about in terms of grothendieck topology ?? ... and so forth ... ??...

?? "the filtered colimit topology" ... ???or something?? ...

??so what about this example that we have... involving projective line as toric variety ... ??...

3-object site category ... ??explicit description of hom-sets and so forth here ????...

Nx0 NxZ 0x0
0 NxN 0
0 ZxN N

??or something... ??? ..fix that ... ??....

??hmmm, maybe it's more like ...

N+ N+ 0
0 1 0
0 N- N-

???or something ???...


??something about ... ??still peculiar that we're sort of using "accidental topos" ... ??... and so forth ... ????....

??something about "accidental site" ... ??... ??with semi-monoidal structure ... ??and "quasi-pro-unital" or something ... ???... ????.....


??what about something about "eckmann-hilton" and "property vs structure" (and so forth...) here ??...

??so what _about_ "localization as filtered colimit" ?? ... or something ... ??...

??ag theory coming from symmetric monoidal algebroid ... vs from commutative algebroid ... ??and so forth ...

??something about dimensional theories and stackiness here ... ?? .. and so forth ... ???...

??geometric interpretation of filtered colimit here (...) .... ???.... ???seems peculiar ?? ... ...hmmm....

??what about something about ... colimit of endomorphism, vs colimit of it's "unwrapping" ??? .... ???and so forth ... ???..... ...hmmm...

??so what about something about ... ???single-environment completeness theorem vs multi-environment .... or something ... ??did we have some sort of other jargon about this?? ???something about "duality" or something ?? .... ... ???.... ??anyway, some thing about some vague memory ... ??generalized gabriel-ulmer duality or something?? ... ??something about ... ??various kinds of limits and/or colimits ... ??sometimes centralizing each other, or something ?? ... ??something about geometric theories maybe fitting into this somehow?? ... ??though seems weird because geometric doctrine seems to need multi-environment model theory ... ??or something ... ???....

??so what about taking nice small examples of toric quasicoherent sheaves over projective line (for example ...) and trying to see what their underlying ordinary quasicoherent sheaves are like ... skyscraper sheaves or whatever ... ??...

??so what about something about simplicial toric quasicoherent sheaf ?? ... and so forth ... ??something about "toric case of derived category of quasicoherent sheaves" and so forth ... ???....

???something about "resolution" here ??... and so forth ... ???... ??injective resolution ??? ... ???....

??todd tried explaining to me a bit about where "newton polygon" idea comes from ... newton allegedly developed some sort of method for giving solutions with puiseux series coefficients to algebraic equations with laurent series coefficients ... ??or something ???...

??so then _is_ there something here about case of "taking nth root" as "toric" or something ?? ... and so forth ... ??...

??so what about ... ?? toric projective line ... ??maybe "toric small zariski topos" here ends up being just presheaves on walking pushout data ??? ??or something ??... ??that is, something about "family of set-pairs" or something ... ??with structure sheaf of commutative monoids being ... ?? Z-indexed family of pairs ... first coordinate singleton if 0 <= n and nothington otherwise; second coordinate singleton if n <= 0 and nothington otherwise ... ??or something ?? ... ??which sounds vaguely suggestive in various ways ... ????something about "formal inverse" and / or "riemann-roch" ... ??or something, and so forth ... ???...

??then seems somewhat clear what sheaf of actions over structure sheaf amounts to here ... ??and seems like ... "sheaf : quasicoherent sheaf :: lax pullback : strong
pullback" here ....?? or something ?? .... ??so does that at all match our other intuitions (and so forth ...) about what non-quasicoherent sheaves are like ??? .... ??hmmm, perhaps it sort of does??? ???because it seems like you could have
extra local sections over the "overlap", for example, or over one of the big pieces, which don't contribute to any global sections ... ??or something ?? ... ??though on the other hand, might need a bit more work to match other vague idea about ... "strong pullback, but of the wrong diagram ..." ... that is, "at the affine level is where it goes wrong" ... or something ... ??is there maybe something here about ... ??systematically re-expressing lax colimit as strong colimit of different diagram ??? .... ???or something ?? ... and so forth ... ?? ???hmm, vaguely reminds me of something about ... ??"homotopy colimit as certain sort of weighted colimit" and so forth .... ???.... ??or is something backwards here, or something ?? ...

??what about that other (...) intuition here, something about "localness wrt base space vs wrt total space" ?? ... or something ...

??so what about something about "ab=cd" as affine toric variety ??? ... or something ... and so forth ... ???

??hmm, so what about the torus of this toric variety?? ... and so forth ... hmmm ...

??so what about something about "frankenstein doctrine" ?? ... ??semi-monoidal ... every object is idempotent ... filteredly cocomplete ... ??anything else ??? ...

...hmmm...

something about taking the idempotent objects... from an ag theory or something, i mean... ???or just the "good" ones?? ??or something??? ???better than flat ??? ...??or what ???...

??we _did_ sort of think about stuff a bit like this before... in connection with comparing invertible to idempotent, and so forth ... ??maybe trying to find common generalization, or something ?? ... and so forth ... ??but didn't have these "filtered colimit" ideas back then ?? ... ??or something??? ...

??something about ... ??"categorified semi-lattice of basic affine opens" ... ??or something ?? ... and so forth ...

(??what _about_ something about "site as sort of like categorified basis" or something, and something about ... basis of affine opens for small zariski ... and so forth ... ??...)

??what about something about ... ???categorified "lattice-ordered abelian group" here ???? .... ????or something ??? ....


??something about ... ???"semipro-monoidal" or something ????? .... ???maybe ... "pro-unital semi-monoidal ..." ... ??or something ??? .... ??exept maybe ... something about ... ??"filteredly cocomplete analog of pro" ??? ... ??? or something ??? .... ?????....

??so what about something about doctrine combining absolute and filtered colimits?? ... and so forth ... ??? ... ??non-toric case ... ???...

??so what _about_ something about ... ??"associativity for pro-monoidal category", or something, and ... ??trying to get something like a "quadratic operad" out of this ?? ... and so forth ...

??so what _about_ doctrine of symmetric monoidal filteredly cocomplete (??k-linear ... or omit this for the toric case ... ??? ...) categories ??? ... and so forth ...

??what about something about... ??given an ag theory, taking the theory of this new doctrine given by something about ... ??just the stuff generated from the unit object by filtered colimits, or something ??? ... ???is this part of an adjunction / interpretation of doctrines ??? ... ???... and so forth ... ???

???something about ... ???stuff here about ... "local vs twisted" ... ??and so forth ... something about relationship to doctrine of dimensional theories ... and so forth ... ???...

??what _about_ something about ... ??relationship between colimits of formulas and ... ???(weak ... ??...) geometric colimits of theories?? ... ?? ... or something ... ??perhaps sort of showing up here... or something ... ??...

??still confused by "change of doctrine", as for example here ???...

??found myself saying something about "same moduli space, but with more / less structure" (... or something....) ... ??but then tried refining it to "same moduli stack ..." ... ???which apparently just made things worse?? ??or something ... ???but then tried "same model groupoid ...", which might be helping ...

??so maybe it was just a big old level slip that was bugging us ... (categorfied ... ??...) structure on a model groupoid, vs (not categorified ...) structure concept inherent in given groupoid ... ??? ??or something ...

??so what _about case where it's not just same model groupoid, but same model category ... ??and so forth ... ??...

??so what _about_ trying to understadn lots of examples of extra structure on model groupoids here ... toric ... dimensional ... ??something about this "filteredly cocomplete ..." stuff ... and so forth ... ??? ....

??so what about something about ... postnikov-like factorization (and so forth...) of certain (2,1)-fmnctors here ?? ... and so forth ...

??so what _about_ projective n-space as theory of doctrine of symmetric monoidal filteredly cocomplete categories?? ??or something .... ??hmm, so what about something about "intended environment" here ?? ... and so forth .... ???...

??confusion??? ???something about ... ??maybe unit object is _missing_ here ??? .... ???or something ??? .... ???.... ??something about "semi-monoidal ..." ... ?? ... ??...

???relationship to something about ... ??unit object being non-projective?? ... ??or _something_ ?? ???what about "projective : presheaf :: ?? : filteredly cocontinuous presheaf" here ?? ... or something ... ??? .... hmmmm .... ????.... ??what _about_ relationship to "flat", or _something_ ???? .... and so forth .... ????

???hmm, what _about_ still model/formula confusion here ??? ... something about diaconescu's theorem and flat _co_-presheaf here ??? ... and so forth ... ???...

??what _about_ something about ... ??finite top space ... ??something about "minimal neighborhood of a point" ... and so forth ... something about variance confusion ... vs closure of a point ... ??...

???something about ... ??model/formula confusion and something about ... ??torsor of commutative monoid ... ???and something about tensor product of such things ... ???and so forth ... ???.....

??filteredly cocontinuous presheaves on a symmetric monoidal filteredly cocomplete category .... ????....

??something about ... when underlying filteredly cocomplete category (hmmm ...?? ??something about toposes here ????....) is free filteredly cocomplete ... (??something about special case of on a single-object ... ??or something ?? ...) ... ??? ...

???something about filteredly cocontinuous bimodules ... ??and so forth ... ??

??something about "non-toric" case ... ???.... ??...

???category where an object is an open subscheme of scheme x ... ??and a morphism from y to z is ... ?? ... ???...

???anything "kz"-ish going on here ??? ??? or something ??...

??sheaf and/or pre-sheaf of commutative monoids ...

??something about getting from a commutative monoid its filteredly cocompletion ... ??or something ... ??something about in contravariant way, or something ????....

????.....

pre-sheaf of actions over a pre-sheaf of ocmmutative monoids ... ??_is_ this just a pre-sheaf over some category somewhat straightforwardly obtained from the pre-sheaf of ocmmutative monoids ???? ..... and so forth .... ????....

??what about something about "toric quasicoherence" here ???? .... and so forth ... ???....

???lots of confusion ??? ....

??what about idea that "the crucial aspect of quasicoherence is in the affine case" (or something...) in this (...) toric case ??? .... ??and so forth ... ...hmmm....

??something about ... toric small zariski topos... and ... ... ??something about .... geometric (weak ...) colimits (2,1)-category where object is "filteredly cocomplete category equipped with filteredly cocontinuous presheaf of commutative monoids over it" ... ...??or something ... and so forth ...

??something about "toric voodoo"... ???something about "generic point of toric (sub-)variety ... ??or something ... and so forth ...

??bunch of ideas about filtered colimits ....

???is this a "tame" (??something about "small" ... ??...) monad on _cat_ ?? or something??? ... and so forth ...

?? weak colimits of categories with filtered colimits .... ??as related to weak colimits of toric schemes / stacks or something??? ... and so forth ...

??also something about geometric filtered colimits of geometric theories and of tag theories...


??also algebroid / "ordinary" case ??? ... or something.... and so forth... ...something about ag theories ... ??...


???something about ... eckmann-hilton... ??stretched in some peculiar way ???? or something???

??any variance confusion here ???? .... ???or something ??? .... hmmmm .... commutative monoid .... ????.... ...hmmmm.... ????....

???something about "accidental" vs "deliberate" here??? .. and so forth ... ???danger that accidental works better than deliberate here ?? ... ??or something ?? ...

??symm mon cat with filtered colimits... and so forth ... ??something about modules and graded modules and so forth here .... ??? ...

??something about... "free filteredly cocomplete" =?= "affine" ????? or something???.... and so forth... ????.....

??something about gabriel-ulmer duality ... ??some sort of "strange loop" here, maybe?? ... something about filtered colimits ... as involved in foundations of gabriel-ulmer duality ... (??or maybe just some special case or something???...) ... ???but then also something about ... model category of topos ... ??... tending not to have too many other colimits ... ??or something ??? ... and so forth ... ??...

??every groupoid has all filtered colimits ?? ...

so what about something about... ????tag vs ag and ... ??various kinds/levels of base change ... ?? ... and so forth ...

??something about base change for theory vs for doctrine ... ??or something?? ... and so forth ... ???linkage of some sort here??? ... hmmmm.... ??relationship to for example linkage between comm ring and its module ag theory ?? ...

?? ... formula ... theory ... doctrine ... ???? or something??? and so forth ...

??something about formula vs model here ??? ...

something about ... [ some commutative monoid vs mutliplicative monoid of some commutative ring ] vs [tag doctrine vs ag doctrine] ... ??and so forth ...

??something about doctrine of k-ag theories and / or m-tag theories, and so forth ... ??doctrine of toposes nicely (??) "over" given elementary topos ... ??and so forth ...

??something about module ag _environment_ of commutative ring ... and so forth ... ???...

??so what about something about ... taking any "de-glueing" of a "scheme" (or something...) into affine pieces and interpreting it as giving a ringed (or whatever ... monoided in the toric case ...) topos ... ???and then applying to that ringed topos hakim/tierney "spectrum" idea ... ???or something ... and so forth ...

??also... take such de-glueing... in toric case ... say for projective line ... and get commutatively monoided pre-sheaf topos ... ??or something ??? ... and then take the topos of actions of that commutative monoid object ... ???and then find in there "toric quasicoherent sheaves" as alleged full subcategory... (?? not sure what sort of formal properties the inclusion functor should have here... ?? or something ... and so forth ...) ... ??and consider whether this leads to way of interpreting the toric quasicoherent sheaves as forming a pre-sheaf topos ... ??or perhaps at least gives some hints as to whether or not this is possible ... ???.... or something ... and so forth ... ????....

??hmm, so what about something about ... ???(almost ... something about karoubi envelope ...) recovering the site category from a presheaf topos as the connected projectives, or something... ??also something about algebroid version of this ... ??relationship to something about "enough projectives" and "cohomological approach to non-/affineness" and so forth ...

(??did i mention recently something about ... ??"enough projectives" (or something) as something measured by "cohomology" (or something...) vs as some sort of precondition for getting it (...) to work well ... ?? ... or something...)

??something about ... "best presheaf topos approximation to given topos" ... ??or even to given category or something ??...

though there's approximation from above vs below, and maybe the one that's sort of working nice here isn't the "useful" one ??? ... ??or something ??? ....

notes for next discussion with todd

??is it (...) a presheaf topos??? ... and so forth ....

?? algorithms for converting between "graded" form and "glued" form ... and so forth ...

??something about ... ???lots of sort-of related toposes...


???lots of other stuff .... ????....

???something about cartesian bicategory and _topos_ (or something...) as non-example ... or something ... something about relevance to stuff above ... ??...

??something about all of this "filteredly cocomplete ..." stuff... and so forth ...
doctrine ... ???something about how this relates to that "cartesian bicategory" stuff that i just mentioned above ... or something ... and so forth ...


??lots of stuff which would be interesting and fun to talk about but perhaps shouldn't because it's not on the main track at the moment...

for example ...

"tiny" (or something... and so forth ...) object in presheaf topos (or something ... and so forth) ... relationship to variants of barycentric subdivision and "twisted morphism category" and so forth ... exponentiable, powerable, tensorable, and so forth ...

??something about forcing and so forth ...

following up on this ...

??so what about something about ... ???isbell envelope of full subcategory of _top space_ containing just 1 and sierpinski space ?? ... ???does this parse?? ... ??even if so, doesn't it also suggest some sort of alternative approach ... ??starting with just a bimodule, or something ...??

??what about something about simply (??...) some sort of comma object and/or "twisted comma object" here, or something ??? .... and so forth ... ??...

??what about variants ... ??in place of sierpinski space, use ... ???discrete 2 ?? ... ??"affine line" ... ???and so forth ?? ...

??so what about something about ... ?? "diaconescu's theorem" for ag theory and/or tag theory given by day convolution ??? ... or something ... ???

??hmm, first wrt actual monoidal structure; then wrt pro-monoidal structure ??? ... ??or something ???... ... hmmm ... ???...

??hmm, actual monoidal structure case as maybe "the trivial case" here ??? ... ??or something ?? ....

so what about "destructiveness" of "quasicoherence" adjunction in toric case ??? ... or something ... ?? compared to "ordinary" case??? ... ??maybe moreso, or is it telling us that the destructiveness is pretty great in the "ordinary" case as well??

??and what about ... ??possible ways of avoiding so much destructiveness here ... ??or something??? ... and so forth ... ??for example what about categories vs groupoids here??? ....

??so what _about_ "coherent cohomology" (or something... and so forth ...) in toric case ?? ...

??are we suggesting that ... ???being a _pre-sheaf_ topos is a kind of analog of being an _affine_ scheme ??? .... ???or something ?? ... ??hmmm, but as part of _what_ analogy or analogies?? ... ??and as not part of what other analogies ... ??...

??something about "accidental" here ... ??? ??accidental analogy, or something ??? ...

we think we had one main analogy where the topos analog of "non-affine" was missing, right?? ... ??or ... ??was that before we started thinking about ... ??something about idea of ... ringed topos as analogous to framed topos instead of to topos ... ???or something ?? ...

??meanwhile we seem to be developing some other analogy and/or analogies here ... "variety as analogous to toric variety" or something, and so forth ... ??including "accidental" variant, or something ... ???....

???with this latter alleged analogy, what about something about ... ??maybe analog of "cohomological interpretation of non-affine", or something ??? ....

??maybe this is on the right track, and we (??fortunately?? ??or something ??...) don't want the "accidental" variant here, because we want something like ... ??"the unit object to be non-projective" ... ???or something ???...

??well, still some confusion here ... ??maybe actually not being a pre-sheaf topos ... does involve the "accidental" stuff ... ??or something ??... and so forth ....

??so... consider this "localization" left exact left adjoint from N-sets to Z-sets ... ??so i guess that part of the point is that it's _not_ a right adjoint ... ??or something ?? ...

??something about ... N-Z-biaction ... ???not having adjoint, though sort of coming close ?? ... ???or something ??? ....

???something about ... ???flat vs ... ??projective or something??? ... here ... ???or what ????....

??hmmm, so what _about_ something "finitary vs infinitary flatness" (or something ...??...) here ??? .... and so forth ... ???....

??... ask todd about some of this ... ??something about "cartesian bicategory" (or something...) and so forth ... ???....

??so do we have any indication as to whether the "toric quasicoherent sheaves" over the projective line form a presheaf topos?? ... ask todd about this ...

??so what about something about "abelian diaconescu's theorem" for ringoid object in topos ?? ... and so forth ... ???....

??model of topos given by site as ... ???flat co-presheaf, satisfying extra property ... relating to covers in grothendieck topology ... ???something about "believing that the covers actually cover" ... ??or something ??? ...

??what about something about interval object which is not only a flat co-presheaf but also "connected" ??? or something ... and so forth ... ???something about doctrine incorporating something about exponentials?? ??or something?? .. and so forth ...

??hmm, is ".->.<-." an interval object in _simplicial set_ ????... ???is the "concatenation" of interval objects an interval object ??? .... ???? .... ??and so forth ... ???... 2 3 4 5 6 .... 4 6 8 10 12 ... 3 5 7 9 11 ... ??... ??even if this does work (still not clear to me...), staring at the interval squared makes it seem like this is giving something other than barycentric subdivision ... ???or something ?? ... ??what about something about edges of simplex with "rectilinear backbone with all steps the same size" as having edges of lengths sqrt(1),sqrt(2),...,sqrt(n) ?? ... ???... ??hmm, so maybe this really is an interval object, and maybe the left exact geometric realization process here is a sort of "modified version of barycentric subdivision where the barycenter of a simplex is pushed over to the barycenter of its longest edge" ??? ... ??or something ???... ??what about something about .... ??as maybe implicit in this, some way of encoding subset (??or something??) of {backbone edges} as total order on {vertexes} ...??and/or of binning the total orders into the subsets, or something ?? ???hmm, maybe something about "only keeping track of comparisons between immediate neighbors wrt the original total order" ??? ... or something ... ???sounds vaguely familiar for some reason?? ??what about whether this alleged modified barycentric subdivision is "good enough for purposes to which ordinary barycentric subdivision is put" ... ??whatever they are ... ?? ??or something ... ???.... ??what about if you do this to the nerve of a category?? ???is it then again the nerve of a category, and if so then which one ???.... ???hmm, so _is_ this just the twisted arrow category of the original category ?? ???or something ???... ??what about something about extra structure on twisted arrow category allowing you to recover the original category from it ?? .. and so forth ... ???... ??maybe _pair_ of factorization systems ?? ...???or something ???... ??hmm, so what about something about ... ???untwisted arrow category as corresponding to _another_ kind of "subdivision" ??? ??or something ?? ... and so forth ... ???.... ??something about dependence on "orientation" here ...??or something ???... ??something about sort of (again, somewhat orientation-dependent, or something ...??...) "ternary subdivision" ... ??flatly cosimplicial [simplicial set] ".->.->.->." ... ???as corresponding to something about.... ???internal homming from ".->.->." ???? or something ????...... ??????.... ???some sort of "poincare duality" here????? ???or something???? ....

??what about something about "segal category" here, or something ??? .... ... and so forth ... ???....

??... hmm, so what _about_ something about "flatly cosimplicial [simplicial set]" here?? ...?? or something??? ... ??somehow i didn't even think about this aspect of it??? ????something about ... ???geometric morphism from topos of interval objects to topos of interval objects ... ??something about ... ??relationship to some vague sort of "poincare duality" (or something...) that i thought i vaguely perceived showing up here .... ???or something ????....

??interval object in _simplicial set_ ... ??corresponding to geometric endomorphism of topos of interval objects ... ???or something ...

??i forget, does "interval object" include something about the extreme points being distinct ??? ... ??or something ?? ...

???internal homming from "." ... as identity geometric endomorphism ... ???...

???internal homming from ".->." as geometric endomorphism corresponding to interval object ".->.->." ... ???...

???so _is_ "." an interval object, and if so then what geometric endomorphism does it correspond to ??? ....

???internal homming from walking 2-simplex as geometric endomorphism corresponding to interval object ".->.->.->." ?????.... ??or something ????

???internal homming from representable presheaf as ... ????left exact left adjoint ??????? ..... ???left exact because of "connected projectiveness" of representable ??? (something about splitting idempotent here ... ??...) ... ???left adjoint because right adjoint is .... ???hmm, or did i get it backwards here ??? ... ??or something??? ... ???is there something here about "essential geometric endomorphism" ?? ???or something ???....

???hmmm, is there something here about "tiny object" or something ??? ... and so forth ... joyal ... differential forms ... ??or something ??? ....


??hmmm, so far we're saying what the interval objects are that correspond to these geometric endomorphisms, but we're not really saying much yet about what the geometric endomorphisms themselves really are ... as constructions on models of the topos ... ?? or something ... and so forth ... ???....


??is (left exact left adjoint part of ...) [geometric endomorphism corresponding to interval object ".->.<-." ] equivalent to internal homming from something ??? .... ??? .... ??or something ??? ....

??something about ...???maybe some "poincare duality" stuff here as wrapped up with something about ... ???using left exact left adjoint f evaluated at walking 1-simplex as shorthand for f ... ???or something ?? ... and so forth ... ?? ...

???are representable presheaves always "internally" projective ???? ??? or something??? ... and so forth ... ???....

??not at all clear to me how to use arbitrary "compact model" (??or something??...) to get geometric endomorphism ... i mean at intuitive "construction on models of topos" level ... ???or something ??? ....


??so what _about_ something about concatenation of interval objects??? ??as non-symmetric tensor product, or something ?? ... and what _about_ whether there's a unit object for it??? ... that is, again something about whether "interval object" is automatically (...) "strict" ... ??or something ... ?? ...

??and what _about_ relationship of some stuff here to something about ... ???whole bunch of batanin / joyal / berger / [and so forth] stuff ??? ??? or something ... ???....

???hmmm... ???so let's try evaluating "internal homming from walking 2-simplex" (for example...) at walking 1-simplex ... ???or something ... ???hmmm, well, maybe right _there_'s your poincare duality; something about "(??internal?? ... ??...) homming into the walking 1-simplex" ...

2-simplexes in walking 1-simplex ... ??... ??something about 3-simplex vs backbone of 3-simplex here ???? .... ???or something ???? ....

??we still seem to sort of have an argument saying that "self-model of topos" is roughly _equivalent_ (??or something??) to model of topos ...??or do we?? ... but in any case, this sounds insane, right?? ??"in both directions", or something ??? ... ??so where's the resolution of the confusion??? ... ???something about "projective vs internally projective" or something ???? .... or what ??? ... and so forth ... ???...

??hmm, a lot of this could probably be straightened out fairly easily ... ??...

??so what _about_ possibility of relationship (??or something... ??maybe with level slip in between, or something ... ??...) between "internally projective" and "exponentiable" or "powerable" or "copowerable" or something ... ?? ...

??maybe walking j-simplex and it's "backbone" (or something ...) are _both_ interval objects ... ??? ??or something ???...

??something about "generalized (in certain direction... still monomial but allowing coefficients... or something ... ??...) toric variety" and 3-place chain complex and so forth ... ???something about getting around to discussing with todd ...

??something about nilpotence and "infinitesimalness" in this context ??? or something ?? ....

??or wait, maybe you can already talk about nilpotence (or something ...) in the toric context without needing the generalization ... ???... ??still seems slightly peculiar though?? ... ??something about "0 as aspect of addition" ??? ... ??or something ... ??vs as a "scalar" ??? ...?? or something ??? ... and so forth ... ???...

todd (??yesterday or so?? ...) mentioned something about schur functors and so forth ... possible relationships to various (mostly algebraic-geometry-related ... ??...) stuff that we've been talking about recently ... ... i'd really like to get around to talking about this stuff...

??hmm, so what about something about relationship between [toric maximal atlas of projective space (??and/or of more general grassmanian or partial flag variety ... or something ...) as having interpretation in terms of something about (??overlapping/superimposed??) bruhat classifications and so forth] and [??something about "kaleidoscope as fan" and so forth ...] ?? ... and so forth ...

??maybe level slip here, or something??? something about situations where some partial flag manifold _is_ a toric variety for some (perhaps good ... ??...) reason, vs situations where some sort of toric variety is nicely systematically associated to a partial flag variety ... or something, and so forth ... ???.... ??probably have run into this level slip before ??? ... and so forth ... ??...

something about ... the way isbell conjugation as being related to chu construction (and so forth ...) probably makes more sense to me so far than it being related to "spectrum" (or something ... ??...) ideas ... ???hmm, except that in the middle of saying that maybe i sort of see how that second relationship might relate to that first relationship ... ???something about ... ???"duality between geometry and algebra" ... ???"duality between point and function" ... ?? "duality between model and formula" ... ??... and so forth ... ??... ??"satisfaction relation" ... "galois connection" ... ???... ???...

??but anyway ... ??i was going to say something like ... ???let's take some nice more or less familiar "chu construction" situation ... (?? i was thinking for example of concept of "topological space", though now i'm not sure exactly how "pure" an example of such a situation that is... but maybe that's just as well ... or better, or something ... ??...) ... and then ... ??try to think of it in "isbell envelope" terms ... or something ... ???that is, something like ... ??asking what is the thing (... ??...) that we seem to be suggesting the category of topological spaces (or something at least vaguely like that ... or something ... is the "isbell envelope" of??? ... and so forth ... ???...

??maybe also currently more relevant (...??...) examples ... ???something about ... ?"algebraic geometry" ... and so forth ... categorified and/or decategorified ... and so forth ... ???... soemthing about "galois theory" ...

???what about something about non-idempotentness of "isbell envelope" ??? ... and so forth ... ???....

??something about ... "factorization of hom bifunctor" idea ... ??something about "slice and co-slice" ... something about factorization as "interpolation" ... and so forth .... ????....

??todd mentions something about ... ???... some functor (??something about "inclusion of accidental topos of smaller affine toric open into larger" ... or something ...) being a geometric morphism of toposes because of something about some action of some commutative monoid being flat, or something ... ???which makes me wonder about something about "flatness of localization" and maybe vaguely related stuff ... ??and so forth ...

??something about ... diaconescu's theorem as about model of certain geometric theory ... ??something about construeing that model as a geometric morphism ... ??so is that part of what todd was saying, and/or does it fit with what todd was saying ?? ...

?? given flat action x of commutative monoid m ... ??construe this as model of theory of "m-torsor" ... or something ... ???... ??so something about geometric morphism from geometrically terminal topos to

???hmm, something about... ???model here as thing that you tensor with to perform geometric morphism .... ??hmm, something about "building blocks co-presheaf" and bit about simplicial sets as classifying topos for "interval object" or something... and so forth ... something about "flat building-blocks co-presheaf" ... ?? and so forth ... ??_is_ this really all fitting together ??? ...

???something about "model" and "realization" ... ???something about "geometric realization of simplicial set" (for example) as ... how "formula" (= simplicial set ...) is realized in model (= flat building-block co-presheaf) ...

??"geometric realization process" associated to flat building-blocks co-presheaf as "the realization" (of "the theory" ... ??...) more or less synonymous with "the model" associated with the flat building-blocks co-presheaf ...

??so something about walking 1-simplex simplicial set as "main formula" of the theory... "generator" ... ???something about "carrier" ... ???or something... ??? ... and so forth ... "the interval object" itself ... ???any other simplicial set as built up by limits and colimits from (mostly ... ??...) just walking 1-simplex... some fuzziness here, but ... ??...

??so what about something interactions between various doctrines here... ??... diaconescu's theorem in doctrine of geometric theories... but with perhaps even more fundamental (??or something??...) analog in doctrine of abelian categories ... ??but with doctrine of ag theories perhaps more relvant ... ??but then with doctrine of tag theories perhaps even more relevant ... ???and so forth ??? ...

??something about ... ??toric small zariski topos (or something) vs accidental topos ... ??? something about ... ??maybe "arising from same glueing scheme but using different "building blocks co-presheaf"" ??? ... ??or something ???... and so forth ...

??what _about_ something about "point over walking idempotent commutative monoid" here?? ....

??so are we suggesting now some sort of fairly sensible inclusion (or something ... hmmm ... well, _something_ ... some sort of system of functors or something ... maybe a span or something ... ???....) of accidental topos of toric variety into its "toric small zariski topos" ?? ... ??or something ?? ... ??and how does this relate to semi-recent ideas of maybe vaguely similar flavor ... ??and so forth ... ???....

??so what _about_ something about "isbell duality" and "chu construction" and maybe also "double-sided yoneda embedding" (or something ... ???did we run into something sounding (...) vaguely like that just yesterday or so??? ...) and so forth going on here ?? ... ??something about vague ideas from long time ago but also more recently ... ???...

?? :

09-3 - Mount Allison University :: Sackville, New Brunswick Canada
Jul 28, 2009 ... The Isbell envelope being strikingly reminiscent of the Chu construction, it is natural to ask how they're related, which I'll offer an ...
www.mta.ca/~cat-dist/archive/2009/09-3 - Cached

something about ... ??recurring discussions with baez about something about ... confusion and/or culture clashes (??especially math vs physics ?? ...) about "covariant vs contravariant" .... (??something about "daylight savings time" and "active vs passive" and "heisenberg vs schroedinger" and so forth ...) ... ???something about extent to which we may have consciously mentioned something about "unremoveable carpet-lump ("has to go somewhere / somewhen") op in yoneda embedding" in this connection ... ??or something...

??hmm, just pretty recently i remember thinking somewhere (??perhaps couple of somewhat related contexts ... ??or something ??? ... ???something about "d-module" ... ??but then also something about some stuff that todd suggested ... ??which made me think about hopf algebras and/or some sort of generalization of them ... ???...) about something about ... ??monoidal structures on (??especially enriched, especially vector-space-enriched ?? ...) pre-sheaf categories (or something ...) in some level of generality ... ??and i started thinking about vague idea of something about ... ??"generalization of hopf algebra where co-multiplication algebra homomorphism gets generalized to some sort of tri-module (??perhaps twice covariant and once contravariant??) ..." ... ??and then i started thinking about perhaps construing the structure that seemed to be emerging here as something like an operad ... and so forth ...

??apparently not quite noticing at that time that i was apparently re-visiting vague ideas that i've had before ... ??about trying to relate day's concept of "pro-monoidal category" (as something like "context where day convolution makes sense" ...??...) to concept of "operad" ... and so forth ... ???

???so what about... ???analogy/relationship (and so forth ...) between [background topos stucture on topos with day convolution operation ... and so forth ...??...] and [??something about vague idea of "(??perhaps lie or nonassociative... and so forth ... ??...) algebra with nice canonical basis" ??? ... and so forth ... ???] ??...

?? todd seems to be suggesting something about "flatness of localization" and "diaconescu's theorem" ... in toric case ... ???something about doctrine crossover (??or something ... ??...) here ... ???...

??something about ... ???... accidental topos of toric variety ... ??something about "toric quasicoherent sheaf" ... ??taking concept of "quasicoherent" here slightly seriously ... ???... ??something about sheaf topos over "toric small zariski topos" ... equipped with "structure sheaf" commutative monoid object ... actions of this commutative monoid ... forming a topos... ???but with day convolution more relevant than cartesian product... ?? and with quasicoherence constraint ... ???? and so forth .... ???

??something about relationship to face poset of fan, or something??? .... ???...

??but then what about relationship to "_right_" way of using topos theory here ??? ...

?? ... "either the graph of a linear function from x-axis to y-axis, or vice versa... or perhaps both ..." ... ??... ???something about ... ?? "frankenstein theory" ... ???vs something about "bruhat classification" and so forth ??? .... ???something about arnold and so forth ... lagrangian submanifold as "graph of symplectomorphism from ... to ... or from ... to ... .... " ... ???or something .... ???and so forth ... ???...

??something about overlapping / superimposed bruhat classifications ... ??...

??so what about something about "toric proj construction" ?? ... ???as maybe sort of involved in ... ??certain questions here (??...) ... ??something about "toric version of serre's theorem" ... ??... and so forth ... ???....

??so(4,2) .... ????conformal compactification of flat (3,1) spacetime ... "event" corresponds to middle dot; "light ray" to both end dots together?? ... ???something about "twistor" ??? .....

??so what about something about ... ??good definition of "quasicoherent sheaf" via kan extension (and so forth...), vs conventional annoying definition ... ???trying to systematize / generalize this ??? ... ??systematic way of interpreting object in value of (??(2,1)- ??...) kan extension (or something...) as "sheaf" in some way ... ??? and so forth ...

??something about version of adjunction between [_pre-stack(_comm ring_)_ ??... or something...] and _ag theory_ using categories instead of groupoids ?? ... (2,2)-adjunction instead of (2,1)-, or something ???...

??so what _about_ "fixed points" (or something ...) of various adjunctions and/or "galois connections" here ??? ... something about including toric case ... and so forth ...

??stabilizer parabolic in so(2,3) of "event in 2+1 conformal compactification of flat space-time" ... as rectangular rather than triangular; pretty easy to remember because the translations of flat 2+1 space-time form an abelian subalgebra rather than a heisenberg one ... ??but then what _about_ the triangular one, involving the heisenberg?? ...??any similarly nice conceptual description of the semi-direct product in this case ??

??"entropy" of a probability measure, or something...

??say x has weight 1/2, and y and z each 1/4 ... then use 1 bit to record x, two to record y or z ... 1/2 * 1 + 1/4 * 2 + 1/4 *2 ....

= expected 3/2 bits per selection ... ???

vs x,y each 1/2 ...

1/2 * 1 + 1/2 * 1 ....

=1

1/2 1/2 1

1/4 1/4 1/4 1/4 2

1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8 3

????something about... "maximizing over the ways of using probability measure x as a random-bit generator, of the average amount of information needed to record... " ... ???something about tweaking by y ... ??or something ???...

??something about ... ??"proportion of randomness of x which can be filtered through y" .... ???or something ???....

??idea that most sensible thing to attribute "entropy" to is... ??a "random variable" (or something ...) ...??especially (maybe not exclusively though??) finite-codomained, or something???... ??something about "expected length of recorded results... quantifying over possible coding schemes ... " ... ??or something ... ???...

(hmm, might i be making rota's mistake here??? ????or something ??? ....)

??what about whether there might be something about ... ???getting weighted colimits _and_ weighted limits involved here ???.... something about decategorification ... exponentiation ... and so forth ... ???....

???so then (...) what _about_ something about ... ??... ??given "alternative probability measure" (or something ... ???maybe something "riesz-ish" (??or something??? ?? ??maybe phrase "absolutely continuous" popping into my mind here ??... ??hmm, maybe i was groping for something like "radon-nikodym derivative" here?? ??? or something?? ... and so forth ...) going on here??? ...), considering entropy of "the (??...) filter yielding it (??in bayesian fashion) from the original probability measure" ?? ... or something... and so forth ... ??? something about ... ??"filter" as boolean random variable ... ??something about negation / complement filter here, and something about ... "distributing (or something ... ??"attributing" ... ??...) total entropy of question among its various answers ..." ... ???attributing entropy more fundamentally to "answered question" than to "question" ... or something ... and so forth ... ???.... ???something about ... highly restrictive filter having lowly restrictive one as negation / complement ... ???... ??hmm, so what about something about geometry (or something ...??...) of 3 probability measures in row (or something?? ...) : no-update, original, yes-update ... ?? ... and so forth ... ???...

???so what _about_ something about ... ?? "filter with continuous support but finite entropy" (??or something ... and so forth ... ???hmmm, what about something about continuous _filter_ here ??? .... or something ... and so forth ...) ?? ... ???any relationship to ... ???"conceptual foundations of quantum mechanics" (or something... and so forth ...) ?? ... ???something about "discrete spectrum vs continuous spectrum" ?? ... and so forth ... ??? ....

??ask martin about kan extension approach to "quasicoherent sheaf" ??? ... and so forth ...

notes for next (??and/or last ...) discussion with todd

??[ag theory of [quasicoherent sheaves over [a pre-stack (??or something) on site category of affine schemes]]] defined by kan extension ... "globalization of concept of module of commutative ring" ... ???something about adjunction to "spectrum of an ag theory" and so forth ... ???something about "galois connection" and "fixed point" here ... ???....

??todd suggests relationship to lawvere's ideas about "isbell conjugation" and so forth ...

???so how do toposes get involved here??? ..... and so forth .... ???.... ???something about ... ???toposes vs stacks (or something...) here .... ???something about ... ??the way at the moment "quasicoherent" seems to make some sense in "stack" context but less so in "topos" context .... ???or something ???? .... and so forth .... ???....

??really seeming now like ... ??in fact "you don't have to worry about quasicoherence in the topos context" ...??or something ... ???which perhaps we more or less already realized before .... ???...

??something about "toric case" and so forth ... ???

??something about "fan as maximal atlas" and so forth ... ???

??something about "toric quasicoherent sheaf" ... ??as ordinary quasicoherent sheaf equipped with kind of extra structure, defined in terms of extra toric structure on variety .... ???and so forth ... ???...

??todd asks about "quasicoherentization" (or something ....) and relationship to something in toric case ... ??concerning "accidental topos" ... ??? and so forth ... ???....

??something about ... toric mother topos ... and so forth .... ???....

discussion with john huerta this evening...

i mentioned something about root diagram interpretation of cartan involution and maximla compact subgroup in special case of split real form... and vague idea of "extending cartan involution to irrep" or something like that... playing around with b2 case a bit, but then trying to imitate it somewhat in g2 case ... then trying to connect it with other stuff we think we sort of know about the g2 case ... including some stuff that we've been discussing recently coming form bor and montgomery ...

??something about ... root/weight diagram interpretation of "extending cartan involution to" (or something...) 7d irrep, and/or decomposing it wrt maximal compact subgroup... trying to relate this to what we've been (for some strange reason) calling "good vs evil" ... split octonions as some version of cayley-dickson construction from quaternions, or something ...

???also something about .... canonicalness (or something...) of product decomposition of maximal compact subgroup here ...

??something about trying to match up certain "dualities" here ... and so forth ....

??"good vs evil" ...

?? "fermion rolling on projective plane" (??something about how long root subalgebra relates to (??this?? ??or something???) projective plane ... and so forth...)

?? "rotation vs revolution" ... ???something about "heliocentric vs geocentric" and "interaction picture" (so to speak...) and "gauge-fixing" and so forth ... ???...

?? "space vs time" (analogies and/or relationship to split so(n) case, and so forth...)

?? "orientation vs point of contact" ...

???something about connecting "good vs evil" with both split octonion picture (via something about decomposing imaginary split octonions wrt maximal compact subgroup...???) and with rolling ball picture (via something about decomposing imaginary split octonions wrt maximal compact subgroup, which is something like so(3) X so(3), which we want to try to pretty directly relate to rolling ball... ??via ideas about "increase of homogeneousness" ??? ... and so forth ... ???), perhaps/hopefully helping to understand "tweak map" and so forth ...

??something about "tweak map" affecting evil coordinate in way depending on good coordinate or something ... good coordinate relating to projective plane and evil to fermion rolling on it ??? .... or something ... ???...

??i'm getting confused here about stuff like ... ???how idea that [quasicoherent sheaves are sheaves with a certain special property... rather than just arbitrary module objects over the ring object ... ??/or something... and so forth ...] relates to idea that [??(2,1)-colimits of ag theories (and/or of tag theories?? ... and so forth ....) get along nicely with "pasting together" of varieties (resp toric varieties)] ... ??? .... ?????? .....

???something about ... how module ag theory of ringed topos gets along with "pasting together" of ringed toposes ... ???and so forth ... ???.....

??hmmm.... ??something about certain questions that martin was asking me about semi-recently ... ???..... ??something about geometric pullback of schemes, and how it gets along with ag theories of quasicoherent schemes ... ??or something, and so forth ... ???but something about "pasting" was getting involved as well ... ???....

??so again, let's consider nice simple example of non-quasicoherent sheaf ... ???...

??something about ... ???non-quasicoherent sheaf over _affine_ scheme ... ??? ... ??? ...

???something about quasicoherence as one of those properties that's essentially saying something about localness wrt point of base space rather than of total space ??? ... ??? .....

??as opposed to mere sheaf of modules ... ??...

??so what about a sheaf of sets st the free module on it is quasicoherent ??? .... and so forth ... ???....

???something about ... "locally [isomorphic to sheaf of modules coming from global module] ..." vs "locally [having global sections isomorphic to those of a global module] ... " .... ???.... ??the latter being pretty much vacuous?? ... and so forth ...

??something about... ??idea of definition of "quasicoherent sheaf" as designed precisely so as to make colimits of schemes correspond to (??(2,1)- ... or something ...)colimits of their quasicoherent sheaf ag theories ?? ???... or something ... ??... ?????.....

??i mean... something about ... ??in such a way as to generalize the affine case, or something ... ???? ....

??something about kan extension of (2,1)-functor "module ag theory of comm ring" along yoneda embedding ??? .... and so forth ...

???what was that question that i posted to n-lab, exactly ???.... ben-zvi gave a partial answer ... ???..... ???something about colimits of _schemes_ ... and so forth ...

??something about "toric case", and so forth ... ??

??something about... ???how this relates to "spectrum of ag theory" and so forth ... ???.... ????something about ... ???vague "galois connection" idea here ?????? .... ???or something ??? .... ???hmm, how literally ???? .... ????....

??maximal compact of pgl(4) as po(4) and of cryptomorph po(3,3) as po(3) X po(3) ... ???or something ???? .... roughly ... as seeming to fit together reasonably well ...

so is the (??) fan of a toric variety something like a saturated / maximal atlas ???....

??talking about toric varieties with todd recently... bringing back memories of ideas from, for example, around december of 2008 (i think ... hopefully can find some of this in various notes from around then ...) ... ??something about toposes and so forth here... also something about ... "double interpretation ... " ... ??how did that go ?? ... ???something about stacks or something ??? .... ???.....

??something about .... ???kaleidoscope as fan, and so forth ... ????.....

??maybe the main (?...) "double interpretation" that i was trying to remember here is... ??something about ... "toric points" ... vs some sort of "orbit stack" ..... ??? .... ... ????or something ????.....

notes for next discussion with derek

questions ... ??...

maximal compact of so(2,3) ... and so forth... "branching rules" or something, and so forth ... ??...

??something about real forms of so(n) in general... in part for purposes of so(n-2,2) and penrose diagrams.... and so forth ...

??so what about maximal compact subgroup of so(2,3) ?? ... as so(2) X so(3) ... or something ... ???which numerologically seems to fit with the "anti-symmetrization wrt antipode involution of root diagram" (or something...)... ??but can we see it in more detail??...

???something about "extent to which cartan involution extends to reps" .... ???or something, and so forth ???? something about so(2,3) case, and so forth ... ???also something about g2 case for huerta??...

??something about ... ??adjoint rep of so(n) as exterior square of tautological rep ... using this to connect root diagram with "matrix picture" ... ??something about connecting this with some stuff that we maybe worked out about the matrix picture around the time that we were looking at the numerology of classifying invariant distributions on flag manifolds, and so forth ...

111
121
111

*1*
111
*1*

??something about "antisymmetrized" and "symmetrized" part of 5d rep as giving respectively 2d and 3d reps of maximal compact obtained as "antisymmetrized" part of adjoint rep ??? .... ???or something??? ??and so forth ?? ... ??but how does this relate to, for example, the "bottom row" of the adjoint rep as "flat 2+1 spacetime" ?? ... and so forth ... ???...

??so let x be a tag theory and t a commutatively monoided topos ... ??then consider... ???

??"commutatively monoided topos equipped with x-model in its action tag environment" ... ???...

???given a commutatively monoided topos t, consider its action tag environment ...

???given a topos t' over t, where t is a particular commutatively monoided topos ... ???....

t' -> t -> _comm monoid_ ...

??we can thus still obtain a tag environment ... ??? t' becomes commutatively monoided ... and so forth .... ???then what about some sort of adjoint to this process??? .... and so forth .... ????.....


???something about ...???fixed "external" commutative monoid m ... process of getting tag environment from arbitrary topos x by taking m-actions in it ... ???or something?? ... and so forth ... ??.... ??something about adjoint to this process ... ???.....

??something about ... ???"classifying topos for t-model over m" for t a tag theory ... ??? and so forth ....

??so what about something about "strong vs lax" here (... ) ??? ....

??so consider toric dimensional theories with "no morphisms"... identity morphism only, that is... ???...

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

??todd mentioned something about... ??separated presheaf as believing that dense inclusion is epi, sheaf as believing it's iso .... ????but then in the accidental topos of a projective toric variety ... ??something about not the map (e,s) : 1+1 -> L (here i'm doibg the case of the projective line...) being an isomorphism, but rather some sort of image inclusion for it... ??without however having automatic access to a systematic concept of "image" .... ???or something??? ....

??so.... trying to picture [the involutions giving different real forms of simple complex lie algebras] in terms of root diagram picture.... ??what about possibility that such involutions more or less correspond to ... ???nice ways of marking each a1 as being treated as either "compact" or "split" ?? ...??or something??? ??but then why for example does it seem like g2 has allegedly essentially just 2 real forms while b2 seems like it should have 3 ?? ... ???or something ... ??...

??what about something about second, "cartan" involution (corresponding to _maximal compact_ subgroup ...) on fixed lie subalgebra of first involution ???....

??hmm, so what about real forms of so(4) ??? .... something about so(3,1) vs so(3) X so(2,1) ?? and so forth ... ???.... ??also something about relationship to real forms of so(5), via "long root" relationship ??? .... ???.... and so forth ... ??...

??so what about so(3,1) as underlying real lie algebra of complex lie algebra sl(2,c) ?? ...

??hmm, so in general... ???the underlying real lie algebra of a simple complex lie algebra x is a simple real lie algebra?? and is a real form of the direct sum of two copies of x ??? so... ??to find the real forms of a semi-simple complex lie algebra it doesn't suffice to handle the simple factors separately?? ???but maybe "this is as bad as it gets" ??? ... that aside from the phenomenon we're sort of seeing here, it suffices to handle each factor separately ?? ...

??so am i claiming here that when you re-complexify a complex algebra (or something...) you get it's "double" or something ?? ...??is there some obvious argument for that?? ...

???hmmm... ??is "involution" really an appropriate name for what i meant to be talking about here ?? .... ????.....

??so... ??what _about_ the idea of locating the "toric quasicoherent sheaves over a toric variety x" as certain special objects in the mother topos (??and/or some of its daughters... ??...) of the toric variety ???? .... and so forth ... ???following a conceptual plan that we've been vaguely envisioning in the not-necessarily-toric case... ??but also perhaps relating to questions / ideas that we've been having about the "accidental topos" of a toric variety ... ???...

notes for discussion with todd this morning

???left- and/or right-universal property of cocomplete k-linear category of k-module objects in (??for example??? ...) a topos t ??...

"mother topos" of a toric variety ?? ... and so forth ... ??? ??"boilerplate" idea here ...???

(??maybe also something about "canonical grothendieck topology" and "best topos approximation" and so forth ... ??as peculiar here because of tag doctrine as maybe (?...) not very 2-topos-like (??or something???) ??...)

??something about variations on concepts of "grothendieck topology" and/or "lawvere-tierney topology" ... for various doctrines, and so forth ... ???...

??check semi-recent notes for other topics...

?? so let x be a category and y a full subcategory of _presheaf(x)_ with a left adjoint to its full embedding ...

??given a representable presheaf "[-,x1]" on x and a sub-presheaf s of it ... ??apply the left adjoint to the inclusion morphism ... ??and then consider the image inclusion of that ??? ??or something ???....

??maybe this doesn't generally give an endomorphism of the subobject classifier ???... ???...

??maybe it doesn't even come close .... ???....

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" ... ??? ....]