Tuesday, May 31, 2011

?? ... ??"quasi-crible" ... ????....

??down-set in posetization of slice category ... ???? .....

??some confusion here ... ???? level slips ... "covering" ... "sieve" .....
?? functor from _graph_ to _N-set_ given by "forcing each vertex to have precisely one arrow emanating from it" .... ????how close this comes to giving lex left adjoint ??? ... ??? whether it corresponds to some sort of "subregion" (???? ....) of subobject classifier, in way generalizing grothendieck topology ... ??? .... ??? ....
?? lex geometric realization of (??say for example "level 2" ??? .... "2-stage" ...) globoplexes ... ???

?? trying to really understand joyal(?? ...)'s ideas about ... whatever's going on here ... ???...

???model of (??nice???) geometric theory in something not quite a topos ... as with standard interval object in more or less usual not-quitr-topos category of "spaces" ... ????..... ???? .... presumably happening here as well ... ???....

??for simplexes, the key object to realize as "the 1-simplex" ... "generates" everything else ... ??so analogously seems like we should be particularly interested in the top-dimensional globe ...

???meanwhile general model as filtered colimit of objects of site^op ... ???

??which, in part, means that ... ??? to ["interpret as model in form of concretely structured set ..."] such an object m of site^op, we should ... ???? look at the globoplex morphisms from the walking top-dimensional globe to the walking

???so _is_ it really true that ... ???to interpret a simplex x as a model of the theory, you should ... look at the simplex morphisms from the 1-simplex to x?? ... ??or is it the other way around?? ... ?? ??slightly annoying to try to straighten out... at least if you try to do it by just guessing ... ???....

(???"sesquisimplicial set" ... ??? .... ?? "barycentric subdivision" (?? ...) .... ???? ... variants and relatives thereof ...)

??well, so... ???the tame (??... ?? "mundane") models are supposed to be ... site^op objects ... ???....

??and we're under the vague impression that "interval object" is supposed to be a reasonable name for this ....

???and i think that i'm expecting that a 1-point set would probably qualify as an "interval object", this being the case where "arrow between vertexes gets interpreted as equation" ... ????... ??but who knows?? ...

(??hmm, lex geometric realization really does require that "vertex" gets realized as single point ... ???...)

??well so anyway, let's try as a guess the category of "finite tosets with top and bottom" ... ???? ..... well, at first i was thinking "with" here as property, which made it seem like just plain "non-empty finite tosets" which annoyingly seems to be site rather than site^op ... but then i tried "with" as structure, and nicer things started happening ... like for example the decision about whether to include the so-called "[-1]-simplex" now seems to be the question of whether to allow case bottom = top, which moreover seems to correspond very nicely to that case where "arrows get implemented as equations", which threatens to fit in very nicely with mystical philosophy of "bar resolution" .... ?? the things being resolved as the case where arrows get implemented as equations ... ????....

(?? ... ??? globoplexes and "resolution" ... ??? .... ??? ... ??? "degenerate globoplex" ... ????...)

???so ... still using a lot of guessing, though in principle we ought to be able to work it out pretty much more systematically ... seems like ...

0-simplex as site^op object gives 2-point interval, just bottom and top ... so that seems like it's got to be [0-simplex,1-simplex]_site ... in particular rather than [1-simplex,0-simplex]_site = [0-simplex,1-simplex]_[site^op] which would be only one point ... so in general we turn a site^op object into a "model viewed concretely in the 1-simplex picture" by homming into the 1-simplex as a site object, or out of it as a site^op object ...

??? so in particular, modulo parity of number of sign mistakes here ... ????... ?? seems like i'm getting that ... ?? to see globoplex x as "model viewed concretely in top-dim globe picture", we should ... ??? hom the globoplex x into the top-dimensional globe ... ????....


???mental picture of ... ?? "generic (!!??....) random dynamically growing (via monos of site^op objects .... no non-monos ... ???...) converging to filtered colimit ..." ... ?? ... in fact not just converging to but reaching the (co...)limit ... model of double-negation subtopos ... ???... ??countable and/or continuum-sized ... ???? ..... ????....

??globoplexes corresponding to notable relations .... ??? ???hmmm , particularly ... ??? "n-stage trees with one excess ..." ... ??well, not sure exactly, but ...??? interesting possibilities ??? .... ??? ....

?? "horizontal vs vertical ... " ... ??...

??globe globoplexes of all dimensions ?? ... ??? ....
?? possibility of situation where "localization" acts as "universal flattening" process ??? ....
??vague feeling about ... ??this alleged comonad on the (2,1)-cat of cocomplete categories... and ... ??creating category of quasicoherent sheaves and/or maybe just plain old sheaves as (2,1)-colimit / "(2,1)-left-kan-extension" / "globalization" / "glueing" .... ???and ... some of urs's stuff .... ???? ..... ???? ???and other stuff???


?? "disjoint coproducts" as sort of "distributivity" ??? .... ???? .... ?? ... morphism into coproduct of s-indexed discrete (?? ...) family as ess ... ????? ..... s-tuple of slice category objects ... ??? .....

??? .... ???....

??glueing / globalization over non-distributive lattice ??? .... ??? ....
?? are we running into some sort of apparent paradox here, in connection with ... ???recently convincing ourselves that ... ??? "colimit-anti-reflecting cocontinuous functor between cocomplete categories" (??...) is equivalent to "isomorphism anti-reflecting cocontinuous functor between cocomplete categories", vs ... ??stuff learned from martin about how thick subcategories work ???? .... ???

?? maybe not, depending on which vague memories about what we learned from martin are more correct ... ?? ... ?? for example suppose that the main lesson was something like, yes, you can think in terms of invertibilization even if the projection is only half-bicontinuous, but you can't think in terms of killing off of objects unless it's bicontinuous ... ????

(??anything interesting here about ... ???how it works out in derived context ??????? .... ????? ... hmmm .... ???? .... ???maybe ... 2-place chain complex where boundary map is invertible ... ???? .... ??? ??? ... might be interesting ... ???? realtionship to ... ??? moral interpretation of serre's theorem and mistake of mine that martin pointed out .... ???and to other possible such moral interpretations in terms of "flatness" and/or "homotopy flatness" .... ???? .... )
?? topos : quasitopos :: ?? : cocomplete category ... ????...

??we seem to be suggesting here idea of ... ?? "quasi-crible" (??...) again ... ??? .... ??as (non-monic...) co-yoneda morphism ... ???? ??to what extent does this fit in with stuff johnstone says about quasitoposes ?? .... ??? ...

Monday, May 30, 2011

?? roughly speaking, "(one...) purpose" of "restricted yoneda embedding" as ... ??"to come close to being an equivalence, in which ideal case you actually recognize the original category as being equivalent to presheaves on the restricted domain" ... ... ???which is sort of saying (when you work it out...) that the prototypical functor along which to "restrict" (...) the yoneda embedding is in fact the yoneda embedding... in that orientation that in fact tautologically works out to the identity functor ... ??? .... ???so ... ??well, maybe this puts in an interesting light the idea that "isbell conjugastion" (...??...) is essentially just the "_other_ orientation" case ... ?? ...
?? universal [cocontinuous (by which i mean, in this context, between cocomplete categories ... ???)and universalizing some given cocones] as ... ???reducible to universal [cocontinuous and inverting some given morphisms] ??? ....

??by ... ???given co-yoneda morphism on cocomplete category, representabilizing its source presheaf ... ????? ....

(?? situation where might be reducible further to universal inverting some given morphisms ?? .... ... ?? ...)

??start with cocone ... get co-yoneda morphism from it ... get co-yoneda monomorphism from that as image inclusion ... convert back to cocone ... ??? .... for example some tendency to convert discrete cobase cocone to one with cobase "revealing overlaps" ??? .... ???? .... ???try example ??? say, co-span in _top sp_ ... (1,3) and (2,4) included into (0,5), say ... ???.... ??hmm, so maybe we should look for a nice presentation of the subrepresentable presheaf here ... though maybe think just a bit about what the canonical presentation looks like ... i guess, include a generator for any map into one of the "pieces", and include lots of relators ... anyway, seems like a relatively nice presentation would be ... about what you'd probably expect ... "pushout input" ....

(??something here vaguely reminding me of .... ???filter (??? ...) as "plateau" ??? ... plateau/s achieved by a net ... ???? .... ?????... ?? and / or ... ??? grothendieck topology where any open set of diameter > d gets covered ... limit as d approaches 0 ... ??? .... ??? .... ??? .... )



(?? "dense" (??) morphism of diagram schemes ... ??? ...)


(??general sketches in co-yoneda + co-co-yoneda form ... ???vague feeling about "isbell conjugation" ideas maybe showing up here ??? .... ??or am i making some level slip in trying to remember what isbell conjugation is like ??? ... ??or maybe not ??? ...)

???sheaves on 1-point space .... ????... site of open subspaces ... ?? "empty cover of empty subspace" ...

???more generally ....

?? is it true that ... ???if you take, say, a (nice?? ??or maybe completely arbitrary ??? ....) locale, and take the "co-yoneda sketch" where the site category is the syntactic poset of the locale and the specified co-yoneda morphisms are all the universal ones, then the syntactic category of the resulting colimits theory is just the syntactic poset again, while if the specified co-yoneda morphisms are just the monic universal ones (???and/or ... ???image inclusions of the universal ones .... ???? ....), then ... ???instead the syntactic category is the sheaves over the locale ???? .... ???

(???co-monad here ???? .....???? on _cocomplete category_ ??? ... ???? ....)

??does this actually make sense ??? .... if so then what's going on here??? ??relationship to mystical stuff about choosing grothendieck topologies where coverings are "covering-like" in some vague geometric sense ??? ... ??relationship to quasitopos ideas ??? ...

??? generalizations .... ???? .... ??? getting higher moore-postnikov factorizations involved ... ??? ....

??issue of ... ?? limits as well as colimits here ??? ...

??hmm, so maybe 1-point space really is good place to test the conjecture, to start with ... ??? ....

??total subspace as own co-square ... ???.... ??hmm, back to example theory "object with invertible codiagonal" i mentioned in other post ... ??...

???hmmm ... ??? so what happens when you apply this idea of "taking all the universal co-yoneda morphisms vs just the monic universal ones (??and/or image inclusion of the universal ones ... ???...)" to just a plain old semi-lattice instead of a locale ?? .... ??does this give something interesting, and/or secretly familiar in disguise ??? ....

??perhaps get quantales involved here ??? ...

i forgot to ask ... is image inclusion of universal co-yoneda morphism also universal ??? ... ?? and if not always then can we find some nice simple counter-example ?? ... ... ??and similar questions?? ... ??? ....

???maybe ... it's at least true for posets ???? .... ?????....

??over a cocomplete poset, a co-yoneda morphism as universal precisely in case it's surjective ??? .... ???? .... ???if correct, then generalizations ?? ....

did that make any sense at all ??? ... ??then a monic co-yoneda morphism is ... ????.... ???? .......

??maybe what i meant to say was that .... ???in poset case, co-yoneda morphism as universal precisely in case its image inclusion is universal ... ????.....

???the basic non-distributive lattice ??? ... ???? .... ???"triple of sets, equipped with isomorphisms between the product of any two .... ??? ...." .... ????? ..... ???? ...

??vaguely reminds me of ... ?? "latin square" ??? ... ???? ....

??so is category of latin squares "modular" / "malchev" ??? ... ???...

(??any interesting sense in which "distributive" category is "modular" ?? ... ???confusion ?? ...???...)

??and what about the primeval non-modular lattice ??? ...

?? actually, if it goes the way that i'm imagining at the moment ... ?? "sheaves over a modular lattice" ??? .... ??doesn't it seems likely that this (...??...) stuff is already well-known ???...

??? "sheafification" of "pre-latin square" subobject classifier ????....


??so... let's consider the subobject classifier for pre-sheaves on the primeval non-modular lattice ... ???

?? objects a,b,c,d,e .... morphisms a->b->c->d and a->e->d ... all diagrams commute ...

walking a-elt ... 10000

walking b-elt ... 11000

walking c-elt ... 11100

walking d-elt ... 11111

walking e-elt ... 10001

??...

?????"representabilization" ... ????

???only subobject of d that gets promoted to true is true ???? ???or wait ... ??not true???... ... ???? .....

??empty subobject of a gets promoted to true ... ??so a is basically out of the game ...

?? now it does seem to come down to which subobjects of d are dense, in fact ... ???...

??the important one seems to be that 11001 is dense ... ??? is it true that 11101 doesn't matter that much ?? ... ???? ....

???maybe .... ??? "either restriction from c to b is invertible, or e is empty" ... ????.... ??something about this as maybe seeming vaguely familiar ??? ....

??? vague memory about "constant" ??? .... ???? ....

???"each element of e gives a bijection between b and c, but e might be empty so there might be no bijection" ... ???? but i'm confused ... ??is it always the same bijection that's given ???..... ??maybe yes??? .... ???maybe they're all 2-sided inverses for the restriction from c to b ... ????....

?? don't forget could be of limits doctrine rather than just products ... ?? ....


?? how close this comes to giving a grothendieck topology ???...

???canonical grothendieck topology as geometrically smallest for which representables are sheaves ????? ...... ??????..... ???vs ... "representabilization" ??? ....???? ...

??trying to understand canonical and / or subcanonical ?? ...) grothendieck topology in terms of _models_ ... of .... theory ... in some doctrine ... ???....
?? so... ?? harvey friedman says statement something like "every sequence of rational numbers has a subsequence converging as fast as s" (for some reasoanble particular s, apparently) implies consistency of peano arithmetic, if i understood correctly ... ?? apparently inherently a "second-order" statement, thus escaping becoming celebrated as intuitively clear and difficult-to-doubt first-order statement presumably unprovable in peano arithmetic ... ??? .... ?? ...

??idea that ... ??? maybe there's a sort of demonstrating of usefulness of hilbert's idea of "formalism" here ?? ... "usefulness of reasoning about ideal objects that you might be ambivalent about actually believing in" ... ??? ....

?? negation of first-order, though ????? .... ???? well, that's obviously not the right way to say it... what was i trying to say then??... ..hmm, i may have lost the thought here, or it may not have made any sense ... ???vaguely thinking about ... turing machine ... witness ... statement which if true can be proved, but not necessarily for its negation ... ?? ...

i still have a fair amount of confusion about some stuff here ... which would be interesting to try to clear up some day, probably ... can't articulate (not surprisingly perhaps) my confusions too well at the moment ... ?? interaction between "ground floor equivalence of statements" and "talking about itself" interpretation and "referential opacity" .... ??? ... ???? ....

??well, but what about very clever (in bishop's sense...) sequence ... very cagey about making you think that certain conspicuous subsequences of it might be converging ... ???? .... ???....

Sunday, May 29, 2011

??well so for example, let's continue playing around with set-pair families construed as geometric realization schemes for bipartite graphs ... ??to get some of the flavor of various possibilities ... ???for example ... ??well, for a real simple example, realize the sources and targets as points, and realize the arrows as equations ... ???this as vaguely reminding me of something ?? ... ?? maybe various things ... ???....

anyway, this seems to give the set of "components" of the bipartite graph ...

i was going to give a more involved, random silly example ... something like realize a source as 5 points a,b,c,d,e and a target as 4 points f,g,h,j, and an arrow as identifying , say, the d of the source with the f of the target ... ??? ...

???let's see... ?? is this example (of a site category ...) the right example for ... ??? putting in axiom saying that a model co-span is a co-product co-span ??? ... ??perhaps yes ?? ... ??so then what does the (??) axiom look like ??? .... ??hmm, might be able to think of this particular example as simply a grothendieck topology ... ???...

???hmm, this example reallty does seem to relate to .... ??? some of those "naive intuition" constraints that we mentioned... about what a geometric realization scheme might be like ... ???

???co-product injections as in fact monic ???... ?? monicness being something expressible in _limits_ doctrine, rather than colimits ... ???? .... ??? ...

???so ... ???a geometric realization scheme here as ess just a co-product co-span
precisely in case .... ???? .... the co-yoneda morphism from walking source-target-pair to walking arrow gets geometrically realized as an isomorphism ... ???....

seems to hang together reasonably well so far ...

let me switch for a moment to a different very simple example ... site category terminal ... co-yoneda morphism from walking point-pair to walking point ... ??this co-yoneda morphism being non-monic ... ???? and ... ??so when is this co-yoneda morphism realized as an isomorphism?? ???precisely when the "model point" is initial??? .... ???so that this is "the theory of nothing" ??? .... ????

??how about ... co-yoneda morphism from walking nothing to walking point ... ???.... realized as isomorphism when model point is ... ???? initial ??? ... ??? ....

?? do formulas still form a topos ??? ..... .... ????...

didn't have time yet to check last few examples more carefully ...

??and in fact there's probably something a bit screwed up there ....

???but ... ???if quasitopologies really did involve non-monic co-yoneda morphisms in a key way, then wouldn't they call them "quasi-cribles" or "quasi-sieves" or something??? .... ??... so... ??do they??? ....

??? theory of object x with invertible codiagonal ... ????.... ???"non-disjoint coproduct" ... ???....
"canonical saturated colimits sketch structure" analog of "canonical grothendieck topology" ??? ... ???also "subcanonical" ???? ....

??hmmm.... ??so perhaps i've been somewhat unhelpful / wrong in sometimes telling people that "lawvere-tierney topology" parses very differently than "grothendieck topology" ... ??well, or maybe not that wrong ... just that there's this sort-of funny way of interpreting "lawvere-tierney topology on topos t" as "subcanonical grothendieck topology on topos t" ... ???and ... question whether for example allan adler might actually have been trying to get me to understand this ??? ....

??question whether concepts of "canonical" and "subcanonical" grothendieck topology were introduced only / mainly for application in case of grothendieck topology on topos ?? ... ... hmmm, but then ... ???idea of using canonical topology on category that's somewhat but not completely topos-like, to "toposify it" .... ??or even toposify in this way a category that's _not_ particular topos-like ... ??? ...

?? syntactic vs semantic here ... ???maybe shift in viewpoint as to which is which here ?? ...
??? model = co-simplicial object .... (?? in cocomplete category ???) ...

?? formula = simplicial set .... ??though "basic formula" = simplex ??? ....

?? co-yoneda morphism into basic formula = ... simplicial set over walking j-simplex ...

??? "simplicial set with 0-simplexes bi-colored, with all edges going from earlier color to later color" ... ????

?? "geometric realization scheme that makes this simplicial set look like the edge, via the hopefully obvious projection" ... ???...

?? bipartite graph vs set-pair family ???

???try taking bipartite graphjs as the "things to be geometrically realized" and set-pair families as the "geometric realization schemes" ....

?? notate a set-pair family something like this: {(2,0),(1,3),(1,1)}

??notate a bi-partite graph something like ... ???a rectangular matrix i guess???

2 5
0 1
2 3

???so what's the geometric realization / tensor product in this case?

??sew in 2 211's abd 3 031's ???....

??? not much "glue" to hold things together here ??????? ..... ??? .... ???? ....

???some conceptual mistake here ??? ..... ????

????hmmmmm....ok, i guess that i'm supposed to do more glueing than i was realizing at first ... ???when you "glue in an edge from a 211 to a 031" (??? ...), .... ????you're actually identifying the 2+0 to a single point, and also the 1+3 and the 1+1 .... ???? ??? fits naive intuition better if ... the "inclusion of the model source into the model arrow" is in fact a genuine "inclusion" ... injective and non-surjective .... ????? ???maybe should start with such an example first ???....

(?? again, "model" pun here ... ???? "staunton chess pieces" .... ??? ??as "abstract concrete (???) models of idealized abstract chess pieces" ??? ... ??? ... ???perhaps chess position as sort of thing that can get "realized using staunton realization scheme" ... ?? ...)

??? "classical-valued" model vs non-such .... ????..... ????...

???so ... ???now we seem to be looking for ... ???a set-pair family where ... ???? well, amounts to "2 disjoint subsets with non-total union" ... ??? ... say, {(1,0),(0,0),(0,1),(0,1)} .... ?????

?? then geometric realization of that 2-by-3 matrix above is ... ???1 point for each of the two sources... 2 points for each of the 3 targets ... ?????and 1 point for each of the 13 arrows... ??again, still (??...) not much "glue" to, for example, allow discernment of which source and which target an arrow is glued to .... ?????....

??anyway, seems to be a total of... 21 points in the geometric realization here ??? ....

???idea that what slightly _less_ naive intuition calls for is .... ????? yes, having the points of the model source not bump into each other outright when included into the model arrow, and not bump (outright ...) into the points of the model target either... but ... ??that "bumping into each other" (in both cases just alluded to...) should in fact happen in a more "subtle", "weak" way... (??discernible only in "variable" environment ???? ...) .... amounting to a sort of delicate but strong "glue" ... ????? ..... ??? ??? vague sense of interesting "homotopy-theoretic" ideas here .... ???? .....


(???? elevating finite limits theory to geometric theory, vs elevating arbitrary colimits theory to geometric theory ?????????? .....)

(??? possibility of getting those "naive intuition" constraints on geometric realization scheme to be enforced as axioms expressed under this or some extended doctrine .... ???? ... ????)
??having endomorphism of infinite order (?? ...) as tending to give lots of cocones / co-yoneda morphisms .... ???? ....

??vaguely reminding me of ... semi-recent ideas about theories relating to "peano arithmetic" ?? ... ??? ....
??? generalizing from flat co-presheaf (??by "co-presheaf" here i really mean ... ???that it could be valued in any co-complete environment ... ???? ?? well, perhaps before the generalization i assume the environment is actually a topos, but not afterwards ... ???? ... ..... ??? vs ... ???actually using galois connection involving just actual "classical-valued" co-presheaves .... ???? ....) to non-flat, vs generalizing from monic co-yoneda morphism to non-monic ... ???might part of the point be that to get non-trivially different galois connection you need to make both of these generalizations ??? ... ???? ....

???try examples ... ????....


(??? any sort of "ternary galois connection" (???) going on here??? ... formula, model, co-yoneda morphism .... ???? ....

??? ?? "gabriel-ulmer triality" ???? .... ??? ..... ?? though ... ?? extent to which (??some?? ...) stuff that we're trying to work out here is covered under "gabriel-ulmer duality" heading as ... ???maybe affected by ... "size issues" ... ???? ..... ??? .... ????? ??? 2-topos ??? .... "doctrine" .... ??? ... (??? idea of trying to _define_ doctrine as special kind of 2-topos ??? ..... ???? and doctrine morphism as .... ???? special kind of geometric morphism ??? ... ???? .... ??extent to which this is already part of what lurie (or maybe others) worked out ... ?? maybe for somewhat similar reasons to why i'm trying to work it out .... ??? one reason for wondering about this being ... ??possibility of cheap way of getting help with working out some "size" issues ??? ... ??? .....)

??? truth-value-valued pairing between model and co-yoneda morphism ... ??
??? set-valued pairing between model and formula ... ??
??? "codomain" projection from co-yoneda morphisms to formulas ... ????

??? actually, probably still fair amount of sloppiness / confusion here from "grothendieck vs lawvere-tierney" issues .... ???

?? "candidate for closed" ... ????formula ???

?? "candidate for dense" ... ??? or "candidate for dense in given candidate for closed" ... ??? "co-yoneda morphism into formula" ???? ....


)

(??generalizing from co-yoneda morphism to arbitrary morphism between presheaves ... ??? .... ???grothendieck vs lawvere-tierney ?? ... ??? "auxiliary types" ???? .... ?? ... hmmm ... sensible morphism from one formal colimit to another ... ??? if we argue that this can be reduced to co-yoneda morphism (?? using "auxiliary types" trick or "lawvere-tierney" trick ... ???? ...), then why not argue further that it could be reduced down to just ordinary morphism ??? ...... ???? .... ??? hmm, or is there something in the obviously "asymmetric" nature of colimits that actually makes it sensible that ... ?? you should only attempt the one of the two "symmetric" sorts of reduction here ?? ... ???? .....)
??hmm.... danger here (...) of categorifying certain mistake i sometimes make with cocomplete posets ... ?? or _is_ it a mistake ??? .... "subsemilattice" ... as automatically corresponding to "closure operation" ... vs ... ???

i knw that we had some confusions here, but i don't completely remember yet what they were... which perhaps is a bit like still being confused ... ??...

"kuratowski vs anti-kuratowski" ... ???...

?? various 3-elt (??) quantales here ?? ...

modules of quantales ... ????....

?? "semilattice equipped with closure operation" vs ... ??module of certain 3-elt quantale ??? .... ??? what _is_ the difference here, if any ??? ...

???whether closure operation is itself cocontinuous ??? .... ???and/or continuous ... ????? ......

?? different "gauge-fixing" conventions here ... ??...

(?? "gauge-fixing" and "wlog"/"mawa" ?? ...????)

?? might as well assume ... (sup) semi-lattice with endomorphism ... ???... which could then also be monadic or comonadic ... ??? .... ??vs some other convention where ... ???? might as well assume monad on (sup) semi-lattice ...


???anyway, more or less seems like ... ??it was a mistake to think that i might be categorifying that mistake here ??? .... ?? full (?? ... ??consider decategorified case again ... ?? ...) subcategory being closed under limits as more or less enough to get reflector left adjoint to full inclusion ... ???? ...




?? speaking of quantales .... ?? decategorified version of "combined doctrine" phenomenon here ?? .... ???relationship to "boolean hecke algebra" and "hecke quantale" ???? .... ... ????any direct connection between hecke quantale and toric varieties ???? ..... ????? .....

??maybe sort of in line with something todd was suggesting .... ??? ....


??vague memories of ... ???at one point feeling like we'd pretty nicely categorified a bunch of stuff about sub-semilattices and so forth ... ??riding to box mountain (???) park ... ???....


???categorified version of "kuratowski / anti-kuratowski situation as module of 3-elt quantale" ????? ..... hmmmm .... lawvere-tierney topology ... ??? ....


?? hmm ... having remembered a bit more, and/or thought things through a bit more, it seems like if we avoided categorifying those mistakes then it was probably only through some sort of compensation oferrors ... ???...

??? ... confusion between reflector and monad carrier ... ?? monad carrier as composite of right adjoint full inclusion and left adjoint anti-[cocone-reflecting] reflector (?? ...) has nice properties of neither ... ??... so not even cocontinuous, let alone also continuous ... ??? .....

(??so one thing to straighten out here a bit further is that stuff about 3-elt quantales ... the nature of the confusion about them, and possibly categorifying all that (including the confusion ...) ... ????... ??and it should be fairly easy to straighten out, assuming that we actually get the time and motivation to do so ...)

?? but still seems to be true that, morally, (?? full and) closed under limits = has left adjoint reflector .... ??? ....

??so for example, consider limits-closed full subcat of _graph_ generated by some edgeless graph ... ??evidently the reflector takes any non-edgeless graph to the terminal graph ... ??? ....
?? reflective (???full ???? ...) subcategories of _graph_ ??? .....

??? "codiscrete graphs" .... codiscretization left adjoint to inclusion functor ... ????....

??? closed under limits ???....

?? fullness does seem intuitively to help a lot in narrowing down the list of closed-under-limits subcategories ... ???....

??? some sort of "birkhoff theorem" ideas here ??? ....

?? some idea about "secondary law" ... ???vs "secondary operation" ?? .... ?? "implication" ...??? ... "horn clause" ??? .... ???? .... ??? .... ??? is composite of monadic functors between posets again monadic ??? .... ??? ....???? ....


??again, grothendieck quasitopology as possibly interesting special case here ... including case where quasisheaves form a complete heyting algebra (??...) ... ???....

???full complete (??...) continuously included subcategory generated by single graph ... ??? might want to consider even simpler examples... ??n-stage trees ??? .....
??recent insight into terminology "reflects" (with sort-of relationship to "reflective subcategory" ... ?? ...) as still leaving mystery of terminology "creates" .... ????.... ??don't see any relevant concept of "creative subcategory" (??? ...) offhand ... ???...

"preserve colimits" =?= preserve colimit cocones =?= preserve universal co-yoneda morphisms ... ???

???? universality of co-yoneda morphisms as appropriate concept (??for certain purposes ?? ...) even when only some co-yoneda niches are universally filled ?? ...

"reflect colimits" =?= preserve non-universal co-yoneda morphisms ???....

"create colimits" =?= ... ???... ????preserve non-universally-occupied-ness of co-yoneda niches ???? ..... ???? ....

??preserve failure of specific something to be universal niche-occupant, vs preserve failure of there being any universal niche-occupant ... ???...

"preserve failure" and "create" ... ??? "superman" phenomenon here ??? ???... ??downstairs universal niche-recipient ... ????... as superman candidate for image of upstairs universal niche-recipient .... ????? ..... ??at least narrows down filed of upstairs candidates to some extent ???.... ??? .... in "superman" way .... ??? meaning that maybe there isn't actually any satisfactorily qualified candidate... ??? .... ... not sure i've got this quite straight here yet ...

????"reflection" onto _non_-full subcategory ???? .... ???should at least sort of exist, though maybe somehow not really important for certain present purposes ??? ??"correlation between moore-postnikov-like factorizations of adjoints" ... ?? seems like might become more complicated (?? ...) in "higher-dimensional" case ... as usual i guess ... ???....
?? co-yoneda morphism vs "cocone where base diagram is a discrete (op- ?????? ... ?? ...)fibration" ??? ... ?? ...

?? relationship to ... monicness of co-yoneda morphism .... ??? .... ??? ....

???? ..... and / or to ... ??... way commutativity conditions "don't affect" (co-)limit of a diagram .... ???? .... ????......

.... discreteness ... ??? ... (op-??...)fibration-ness .... ???? ..... ... ???beyond discrete to truth-value ???? .... ????? ......

??confusion / relationship here between ... ?? truth-valueness of fibers of ("fibrational" ???) diagram (???...), and truth-valueness of fibers of .... ???? ???further fibration on top of that ... ???

??? ??? "tower of fibrations" ??? .... ????with fibers getting lower-dimensional as you go up the tower ??? ... ????? ..... ???vaguely reminds me of .... ??? certain extended moore-postnikov-like (??? ...?? ... iterated conception of ... "purpose in life of space of homotopy dimension n ..." .... ??? .... ) dreams .... ???? .....

???hmmm, but middle level of tower as "representable" ??? .... ????? .....

??partial "2-out-of-3" rule for fibrations ??? ... ??? ....

Saturday, May 28, 2011

?? 2' : 1' :: grothendieck topology : lawvere-tierney topology ... ???

?? "lawvere-tierney topology" =?= "sub-canonical (????) grothendieck topology on grothendieck topos" ????.... ??? meaning of "sub-canonical" here as perhaps not so hard to remember now ... "geometric" meaning ... ?? ...

???compatibility between co-yoneda morphism and formula, vs between co-yoneda morphism and model, vs between formula and model ... ???? ..... ????? .....

formula as presheaf, model as co-presheaf ... tensoring gives "realization of formula in model" ... ???so a stretch to consider as "compatibility" ... unless you're into "categorify truth-value into set" ...??? .... ???...

??well, might actually be some funny but semi-plausible way to actually get truth-value here .... ???? ......

??? "generic model" ... as identity functor of formula category ... ???

???"tensoring" co-yoneda morphism with model-candidate co-presheaf ...compatible = "results in isomorphism" ... ??? .... ??? relationship to "implication" ... ??in topos case, but then more generally ??? ..... ???? ....

??? "auxiliary types" .... ???? ..... ???? ....

?? in "lawvere-tierney" situation, can use ordinary morphism realization of co-yoneda morphism, instead of co-yoneda morphism .... ????_is_ that ess what actual lawvere-tierney idea is using ??? ....

??various examples of theories ... old stand-bys ..."walking epi" ... "walking co-category" ... ???...

???presheaf on _finset_ that's finite-valued but not finitely presented ??? ... ?
??try some example of a nice small site category, the sort where it's easy to survey all the grothendieck topologies (...??...) ... and instead survey all the reflective subcategories ... ??? .....

??how quasitopos / quasitopology fits in here ... ??? maybe hinting how some other stuff fits too ... ???

?? "cocomplete category obtainable by sketch with underlying category c" vs "reflective subcategory of _presheaf(c)_" .... ????....

??so let's consider .... for example, "colimits theory of an object x with endo-morphism f, with axiom saying ..." ????..... ??we can specify a co-yoneda morphism into the representable functor .... ???which indeed seems to amount to a presheaf over the element category of the object ... ??? .... ???and then insist that ... ??the representabilization of that formal colimit be ??????????????....


???how does all this relate to stuff that we think we know about "gabriel-ulmer duality" ?? ... ?? .....
??i didn't quite catch whether todd's "epistemological paradise" was supposed to make simultaneous totality and co-totality _possible_, or was it rather that it was supposed to make it impossible for only one of them to hold ... ???.... ?? ...
??relationship between moore-postnikov-like factorization of functor and that of its adjoint ... ??particularly in connection with ... degenerateness of some of the factors ... ???moore-postnikov-like factorization _of_ adjunctions ...

??seeing something like this in context of... ???reflective subcategories ... inclusion vs reflector ... ??? what sort of "surjectiveness" (??...) property reflector has ... ??? .... ?? "does nothing but universalize some cocones / co-yoneda morphisms ..." .... ?????relationship to "localization" / "invertiblization" ... ?? ??hmmm... ?? "conservative" orthogonal complement .... ??? "cocontinuous functor that reflects universal co-yoneda morphisms" ... ????....

?? _is_ motivation for "co-yoneda lemma" anything similar to why i've suddenly been thinking about "co-yoneda morphism" ?? ... ??namely as more "invariant" ("eliminate the middleman") version of "cocone" ?? ....

?? "universally universalize all of the co-yoneda morphisms that are universalized by cocontinuous functor f" ... ????..... ???as giving "reflector" ... ????left adjoint to inclusion of ... ???

???so _is_ this concept essentially just what people mean by "reflects colimits" ??? .... ??and how many puns on "reflect" are there here?? ...

(??several perhaps ??? ?? "reflect isomorphisms" ... ???... well perhaps that one's not really a pun, but ...

??["reflects colimits" as implying "reflects isomorphisms"] as aspect of what was striking me as weird last night ... ??? ....)

(??? monicness of co-yoneda morphisms .... ????what _is_ going on there ??? .... topos ...

???hmmm... i just tried looking at representabilization of presheaves on a finite toset, aka "n-stage trees" ... ??seems like the universal co-yoneda morphisms are all epi .... ?????is that maybe just an accident in this case ??? ... ???all reflective subcategories being distributive lattices ... ???.... ?? .... well wait, seems like they're not quite _all_ surjective ... ??? representabilization of empty presheaf ??? ....)

???colimit diagram schemes that are finitely complete ?? ... ???....


full-and-faithful right adjoint ... colimit-reflecting left adjoint ... ???? no wait, i was trying to match things up but i think maybe i got the wrong ones matched up ... but anyway, i think that setting up a correspondence like this, but correctly, is part of what we'd really like to do ...
?? cocones (or co-yoneda morphisms ...) of terminal type in colimits sketch ??? ... ???seems weird??? ... but ok ?? .... "invertiblization" ... ???....

??topos case ??? .... ????

?? ... example... ?? "product span" ... ???

c :=

"source" -> "arrow" <- "target" ??walking arrow vs walking source-target-pair ... ??? ??co-yoneda morphism .... ??? walking source-target-pair -> walking arrow

functor f : c^op -> _set_ .... "span" ...

co-yoneda morphism in c goes to co-yoneda morphism in _set_^op under f ...

actual sources X actual targets <- actual arrows ... ???.... ??? some confusion here ... ??formal vs actual in _set_^op ??? .... ?? "c respects given co-yoneda morphism" ??? .... ????..... ??co-yoneda morphism coming from a cocone ... ????.... ???example ??? ???on _set_^op here ?? ... "[actual arrows,-]" <- "[actual sources,-]" + "[actual targets,-]" ???confusion here .... try again ... co-span y -> x <- z in category c ... ?? push particular special co-yoneda morphism forward from walking co-span to c ... special one = walking y + walking z -> walking x ???.....

"[-,y]" + "[-,z]" -> "[-,x]" ...??yes, seems right ?? ....

pushing "[-,y]" (for example) forward ...

??say for example along "yz,x,-" ... ???

???well, pushforward of representable is representable, right ??? ...

???so maybe now we're ready to pretend that x,y,z live in _set_^op ... ??? ??thus forming a span / bipartite graph ... ????..

"functions on sources" + "functions on targets" -> "functions on arrows" ....

?? "you can get a function on arrows either from a function on sources, or from a function on targets" ... "function on arrows that only depends on their sources, vs one that only depends on their targets" .... ??plenty of _other_ functions on arrows though ... ??? even if it's a complete bipartite graph ???? ..... ???confusion again ???.....

???_the_ b-cobased cocone in walking b-cobased cocone as a colimit cocone .... (??why???? ...) ????but the corresponding co-yoneda morphism doesn't seem to be invertible ?????? ..... ????confusion ??? .... but so sharp that we ought to be able to resolve it ... ????...

(????general colimits in walking b-cobased cocone ????)

??let's consider as analog of "cocone with given cobase", "co-yoneda morphism with given (presheaf) domain" ... ??but then .... ???maybe the key thing that we've been screwing up is in fact just that ... ??? "universality" of a co-yoneda morphism is not equivalent to invertibility ??? .... ??but then what _is_ it like ???...

??? "a morphism from the co-domain to another actual object t is essentially the same thing as a co-yoneda morphism from the (virtual) domain to t" .... ???....

??? "representablization" and "total category" ??? .... ...??that's what it's like ??? ..... ????....

(there _are_ invertible co-yoneda morphisms, but those aren't merely universal .... ???they come from co-bases which have a terminal object .... ????..... diagram scheme where all you have to do is evaluate at the terminal node to get the colimit ... ???.... ??? "vacuous colimit" in some sense ??? ....)

???so what are we saying??? .... something like .... ???corresponding to a reflective subcategory (??to begin with, let's say of a presheaf category ... ??? .. well, hmm, actually this is causing me some confusion ... blurring distinction between "grothendieck" and "lawvere-tierney" viewpoints ... which is why all those ?'s here... but i'll try to clear it up ...) is, for each object in the site................??????????????? ??? "promoted to true" ... categorified true ... ???kind of thing that people probably do categorify a lot ???

???there _is_ some sort of "image/kernel duality" (??...???...) here, like i think i was vaguely feeling .... ???? "dense vs closed" .... ???? .... ???? .....

Friday, May 27, 2011

?? grothendieck topology as special case of limits sketch ??

?? how to get cocone from "covering" ... ??? ....

??saturated sketch ??? ??hmm, well, first, is there maybe some better concept than "cocone" to use here ?? ... ???something like ... ??? "weighted diagram : co-wedge from weighted diagram to object :: weight : ??? from weight to object" ??? .... maybe .... ????element of presheaf ???? ...... ????? new improved "sketch" as maybe something like ... ??collection of presheaf elements to become universal ??? ..... ??saturated such as collection of presheaf elements closed under .... ???? ..... ???or is it maybe somehow ... subobjects rather than elements ???? ..... ??no, that sounded a bit annoying / ugly and now fortunately it also sounds a bit wrong ... ????.....

??? "structure on weight w and object x amounting to cocone from canonical representative diagram for w to x" ???? ??? ???yoneda vs co-yoneda .... ????? .... ??? ..... ??from representable to non-representable functor as easy direction ?? ....

???weight as "virtual object" .... ???? ....

????morphism from presheaf to representable presheaf vs monic such ???? ....

???might it be that .... ????a morphism from a presheaf to a representable presheaf is "colimit-like" precisely in case ..... ??? well, there's two seemingly pretty incompatible things that i was thinking of saying here ... :

1 it's invertible ... ???...

2 its image inclusion is also "colimit-like" .... ????

???... for each object, a full subcategory of it's slice category .... ???? .... ??? ....

???functor taking given formal colimit to given .... ??? .....

???functor promoting given "co-yoneda morphism" to isomorphism ... ????...

???functor promoting given injective co-yoneda morphism to epimorphism .... ??????????

example ... ??? "discrete sum" ... co-yoneda morphism from some given discrete sum of representables to a representable .... ???? as amounting to a cocone for that discrete sum ???? ??moreover sure _looks_ like a cocone ... in category of formal colimits .... but from diagram living in category of actual objects ... ??? ...

???slice topos of presheaf category over representable presheaf ... ???as ... ???... presheaf category over slice category ??? .... ????? .... ???

????? ..... really try to straighten this all out ... "nice, saturated version of colimits sketch described in terms of weights instead of in terms of weighted diagrams or canonically weighted diagrams" ... ??? .... ???? .... ????relationship to ... ????reflective subcategory of presheaf category ??? ..... ????? ..... ????....

????a presheaf "respects" a co-yoneda morphism iff homming into that presheaf takes that co-yoneda morphism to an isomorphism .... ????and this parses ??? ...

??? a presheaf "respects" a pre-sheaf on a slice category iff ... ??? ... ???? ...

???by the way should it be slice or co-slice here ??? .....

??weird how the grothendieck topology analog here seems sort-of clearly more "complicated" than the corresponding (...) sheaf subtopos analog .... ???? ....

???morphism _from_ colimit as nice ... ??but ("yoneda" ... ??as opposed to "co-yoneda", which would be _from_ ...???...) morphism _to_ _formal_ colimit as nice ???? .... ????... ?? relationship to ... "morphism out of colimit is nice, but cocone as made out of morphisms going the other way" ... ???? .....

(???some stuff here vaguely reminding me ... not even that vaguely now ... of ... ??? syntactic vs semantic category of accidental topos ... ??? ... ??? .... manifestation of cone / cocone (!!pun?? ... fan /co-fan ...) as object therein ... ????)

(?? "enriched grothendieck topology" ... ?? over _ab gp_ does this amount to re-invention of concept of "cocomplete abelian category" ??? ... ??? ....... ??? also enriched version of this alleged generalization that we're trying to work out here ...)



??clement berger ... ??? ... ?? maybe sort of mixture of grothendieck topology special case with kleisli category special case ... ???....

???is one or other of these special cases special case of the other?? ...
old discussion with urs (where at the time it was pretty difficult to understadn what he was saying, though by now it seems like it should be somewhat les difficult...) ...


me:

[my interest in trying to understand these ideas has been motivated in large part by my interest in trying to understand algebraic geometry. i know that a nice algebraic “stack” or “scheme” can be construed as the moduli stack of models of a “theory” expressed in the “doctrine” of symmetric monoidal finitely cocomplete algebroids; the objects of the theory are known as “coherent sheaves”.]

urs:

[I am wondering how this is related to the following general abstract conception of coherent sheaves of modules:

For C⊂Alg op the ∞-site of formal duals of ∞-algebras in question, the ∞-stack of quasicoherent ∞-stacks is the almost-tautological one that assigns SpecA↦AMod.]


there's more to try to decipher, but maybe i should start with trying to decipher the above... which now seems like it might be more or less clear, though i should probably check to see if it actually hangs together the way that i suspect that it might ...

?? hmm, i guess that urs's point here, or at least one perhaps minor one, is supposed to be a sort of "site-independence" idea (??allegedly same such idea promoted by johnstone in baby elephant preface ?? ...) that he alluded to somewhere nearby ... ?? that there's some sort of topos whose syntactic category can be thought of as sheaves over site s for (of course) various different s, for one of which the underlying category is _affine scheme_, while for another of which it's c = category of some sort of more general schemes, and there's a certain stack over this topos which can thus be construed as a pre-stack with extra property over either _affine scheme_ (in which case it assigns to each affine scheme the category (???....) of modules of its affine coordinate algebra) or c (in which case it assigns to each c-object the category of quasicoherent sheaves over it) ... which you could try to philosophize as saying something like "from a sufficiently site-independent viewpoint there's no essential distinction between the concept of module and the concept of quasicoherent sheaf" ...

(??i should worry here for at least a moment about whether it's really true that for a pre-stack to be a stack is really just a mere property (rather than structure ... ??), just as in case of pre-sheaf being a sheaf ... ??? ...)

?? but i'm not at all sure to what extent i would agree or disagree with such a philosophy ... even though part of what he's getting at is evidently related to things that i sometimes say... along the lines of "defining quasicoherent sheaf concept as globalization of concept of module" ... not clear to me how similar / different these philosophies are ... could be subtle or not-so-subtle differences, migth be worth thinking about ... ?? ...

?? "no essential distinction ..." vs "one concept arising by conscious act of globalization of other concept" ... ??? .....

?? urs's formulation as perhaps involving / depending on grothendieck topology ... ?? vs mine as not so much... ??maybe except ... ???more like trying to conjecturally extract grothendieck topology from it ... ???? also ... ???syntax vs semantics ... ??????? ...... ????? ......
i wrote:

[this “restoration of exactness”, the fact that cauchyness spreads from merely the finite discrete colimits to much more general (homotopy-)colimits when you pass from abelian groups to (“2-sided”) chain complexes, means that the (weak) cauchy completion of a dg algebroid is almost the same thing as either its free completion or its free cocompletion.]

but even though i think that i remember a lot of what i was trying to say here, i'm confused at the moment about why the free completion and the free cocompletion (which indeed should be equivalent to each other) should also be equivalent to the "(weak) cauchy completion" ... ??oh, wait, no, maybe it's obvious ... that's the whole point ... that the definition of "cauchy-completion" is that it should include just those limits and colimits which can be indifferently thought of as either one ... ??? ....
?? cauchy completion of commutative monoid ??? ...

didn't notice at first how non-trivial this might be ??? todd more or less pointed this out when i brought up the idea of trying to recover a toric variety from its accidental topos alone ....

(??non-toric analog here ??? ? ?? stuff simon willerton told me about ... ???"correspondences" between algebraic varieties, and situations where there aren't too many, so that you can actually recover the variety from its mere abelian category of quasicoherent sheaves ??? ... ???? ....)

??so what _is_ the cauchy completion of a semi-lattice like, for example ??? ..... ???how about the "half-decategorification" of such?? ???

??very vaguely reminds me, again, of ... ?? cocone category of cofan, and of things that that reminds me of ... ??? ..... ??closed category ... ?? ....

?? ??? _other_ parsing of "cauchy-completion of semi-lattice" ??? .... ????? ....
?? "relatively free cocompletion of cocomplete category x, relative to existing colimits, as x itself ..." .... ???but ... ???question of totality here ???? .... ????? .....

???once again confused between "small-valued" and "small-ly presented" ... ??? ???latter has better chance of implying former than vice versa ... ?????....

???? "how total is total" ??? ..... ?????.....

???maybe for total categories, small-valued really does imply small-ly presented ???? .....

???todd also said something about .... ??? between total categories, cocontinuous functors have right adjoints ???? ..... ???again making me wonder about relationship to locally presentable ... ??? ....

todd's "epistemology" paradise-dream as .... ??? (??in part?? ...) allowing simultaneous totality and co-totality ??? .... which according to silly analogy i was trying to make seems like ... ??? "reducing quantum world to classical world" ??? .....

??idea of ... some kind of table ... cardinality parameters for ... ??size of diagram scheme ??? .... ???size of target category ... ?? size of values ... ??? .... ??? showing relationships between / constraints on ... ??? ..... resulting categories of formal colimits .... ??? .... ??? .....
?? relationship between "locally presentable" and "total" ?? ...

?? "total" here as ... ??having left (?) adjoint for yoneda embedding ?? ... ???? .....

???relationship between "complementarity" phenomena for (maybe?) both ?? ... category and its opposite can't both be nice ... ???unless all diagrams are commutative ... ???? ..... .... ?? ...

?? "total" vs "partial" ... ??? ...

?? did todd say that locally presentable should imply total? ?? whatever it was, it sounded plausible at the time ...

notes for discussion with todd this morning

lot of topics ... not sure where to start ...

1 ??? "accidentalness" of topos as property ?? ... (?? clarify meaning of "property" here ... ?? ??vs structure, stuff, ... ??? ... ?? (2,1)-cat vs (2,0)-cat ... ??) ?? start by taking double negation subtopos ?? ... ??? "naive geometric interpretation" of objects in resulting boolean topos ... ??? ....

(?? maybe other places where we've run into booleanness lately?? ... ??or maybe that was secretly more or less just this ... ???)

2 ??? maybe discuss some of martin's questions ??? ....

3 ???homotopy type of accidental topos ... ??? make it clear that i'm not sure yet where we might go with this ... ???..... ??? TAG structure ... ??? ... ??? toric ideal class group ??? .... ??maybe see this ?? ...

4 maybe this stuff or this ... or this ...

5 ?? any follow-up from last time ??? .... ??which i guess was sort of about mysteries of "flatness" and "combined doctrine" ... ?? ... ?? sort of approached in certain way ??? ... ?? last (??) "this" above as maybe somewhat different approach ?? .... ?? other approaches ... ??again, probably good to be clear how i'm flailing aorund here ... ??? taking seriously trying to understand universal properties of theories in combined doctrine ..... ???? which i guess would more or less include stuff like this ...

Thursday, May 26, 2011

?? being accidental topos of toric variety as mere property ?? ....

?? double negation subtopos ... ???? ... torus ...

???trying to seriously visualize (??? ...) toric quasicoherent sheaf over torus .... ??? relate to lie theory and kaleidoscope-as-fan .... ???? .... ???? .... ??? ....
?? intuitively weird for "classical" AG morphism to fail to preserve kernels ?? ... so let's try looking at nice simple example ... ???

vector space with operator ... ??taking kernel of operator ??? ...

??already confusion here ... ??? this as tensoring with something, so left adjoint ?? ... ... ??? but "kernel" sounds right adjoint-ish ... ????? ..... ???? ... ????? ...

ok, blecchh, this is some weird mental glitch of mine ... happened the other day too, though i forget exactly who i was talking to at the time ... (?baez? ... something about "eigenspace" ...) anyway, i should have said cokernel in the first place .... ???? ...

???so we're thinking about "formal limits of quasicoherent sheaves" (?? ...) and seeing what they evaluate out to in domain vs in codomain ... ??? in part to try to figure out whether there's something "special" about how they evaluate out in the domain ..... ????? ......

so ... specific example ... of kernel (or maybe mono??) not preserved by taking cokernels ... ???

x >-> y

???we want x to have an element x1 which is not in the image of the operator ... ??and we want y to have an extra element y1 whose sole purpose is to make x1 in the image after all ... ??so it sounds like we want our mono to be a 1-by-1 matrix whose entry is the generator representing the operator .... ?????....

so then .... can we flesh this out to a clear statement about how formal limits evaluate out in the domain, and then try to come up with some sort of "geometric interpretation" of it ???? .... ?????.....

??well so for one thing, plain old generic number has now acquired property of ... ???being monic ???? ..... ????.....

???moreover, adding any (??...) constant (??? ...) to it still gives a monic .... ????......

hmmmm.... ??? "spectrum of operator" ... ????? ..... ??? ....
?? schubert interpretation of conformal structure (cone field ...) on g2 quincunx grassmanian coming from ... ??being projective light cone ... ??? ...

???hmmm, so this is a pretty general question actually?? ... lots of cases of (generalized) grassmanians nicely interpretable as "projective light cones", which systematically get those conformal structures, so can ask about schubert interpretation of those cone-fields ...
???homotopy type of _spectrum_ (...??) coming from commutative monoid ... ?? ...

topos vs TAG theory ... ??? .....

Wednesday, May 25, 2011

?? cocones of cofan ... vs cones of corresponding fan ...

?? various funny sorts of "double duality" here ....

???might cones be related to toric quasicoherent sheaves ?? ... ??? ...

??? ... "idempotent" commutative monoid ... ??? .... ???vs co-monoid ??? ....

??vague feeling of resemblance of hom-sets between co-cones to ... ??internal hom in enriched poset ?? ... ??? ... ??? ....
?? monoid presented by generators g_abc with a,b,c in {0,1} and relators given by ... ???

?? 4 relators??

g_000 * g_100 = g_001 * g_110

"g_0ab * g_1bc is independent of b" ??? ...

??most obvious (for me at the moment) example of monoid with geometric realization having non-trivial higher homotopy groups ... ??? ....

?? morse-segal category (in this case poset ...) of generic ellipsoid ... ????... ??? "2-globe" / "bihedron" ...

??pre-sheaves on morse-segal categories ... ???.... hmmm ... ??? lawvere ... joyal ... ???... freyd ... ??...

???morse-segal categories (??of certain posets especially ??? ...) and ... a-series schubert-galois connection ... ????.... ??? ...

pi_1 as ... ???free on 5 generators ??? ... ???hmmm, so is it sort of like there's a redundancy among the 4 relators that manifests as an element of pi(2) ??? ...
?? "relatively free cocompletion of a category c, relative to specified cocones becoming colimit cocones" ...

??vs ... ??? "universally compelling specified cocones in a cocomplete (??up to a point?? ...) category to become colimit cocones" ...

???latter as special case of former, via ... ????.... ??seeing cocomplete category as its own "relatively free cocompletion, relative to the existing cocones becoming colimit cocones" ??? ..... ??? ....

???classification of reflective subcategories of cocomplete categories ... ??? ....

???moore-postnikov-style classification of cocontinuous functors .... ??....

?? "promoting cocones to colimit cocones" ... ??

??limit sketches ??? ....

???small-ly presented presheaves on a large category, vs small-ly valued such ... ??? .... latter doesn't imply former??? ??does former imply latter ??? .... ???maybe yes ???? ..... ??well, only for large categories with nevertheless some reasonable smallness condition ... ????

?? finitely presented presheaf on finite category is finite-valued ...

?? finitely presented presheaf on finitely presented category is not necessarily finite-valued ... ???but maybe nevertheless, situation is "better" for certain higher cardinalities ??? .... ??maybe fairly straightforward ...


?? relatively free co-completion of ... ????....

??for sufficiently nice k, the k-small-ly presented presheaves on a locally-k-small category c form the free k-small cocompletion of c ... ??? hmm, that seems to work even for k = omega ... ??? ... which i didn't intend, i think ... ??have i got something particularly screwed up here ??? ...

?? k-small-ly presented presheaves on locally k'-small category ... ???.... ???with k''-small collection of objects ??? ....

k-small diagram scheme ... ???k-small-ly presented diagram scheme ... ??? ... ???"relations" in diagram scheme as superfluous, _sort of_ ... ???though not to "what it means to have limits of that kind" ... ??? ....




for sufficiently nice k:

1. the free k-small cocompletion of a category c is the category of k-small-ly presented presheaves on c.

2. the relatively free k-small cocompletion of c, subject to the constraint that a class x of k-small cocones in c become colimit cocones, is the k-small-ly presented presheaves on c which take cocones in x to limit cones.

3. as a special case of #2, if c is k-small-ly cocomplete and x includes all of the x-small colimit cocones in c, then this allows us to universally compel a class of k-small cocones in a k-small-ly cocomplete category to become colimit cocones.


??? analogy here between tensor product of ab gps (?? or ... ?? comm semilattices ?? ... ???? ....) via preentations and tensor product of cocomplete categories via sketches .... ???? vague worries about whether there's some sort of duality screw-up in this analogy ??? .... ???? maybe not ??? .....
reading martin's note ...

???so let's consider ... ??functor f : _set_^op X _set_^op -> _set_ which is continuous in each variable separately ... ????... ??using _set_^op as free complete on 1 object, seems pretty clear that category of such functors f is ess category of sets ??? ... ???which is pretty much what we were expecting ?? ....

?? so suppose that x and y are symmetric monoidal cocomplete v-enriched categories for some nice v ...

then consider tensor product of x and y as cocomplete v-enriched categories ... "universal bi-v-cocontinuous ..." ....

x#y # x#y -> x#y

....

let's try another approach for the moment ...

x symmetric monoidal category ... (secretly x = (2,1)-category of v-cocomplete v-enriched categories ... ???)

x' category of commutative monoids in x ...

x' inherits tensor product operation from x ... but in x' it's actually now the coproduct .... ????why ??? ....

m1,m2 commutative monoids in x ...

m1 # m2 gets commutative monoid structure how ????......

(m1 # m2) # (m1 # m2) -> (m1 # m2) .... ??coming from:

(m1 # m2) # (m1 # m2) -> (m1 # m1) # (m2 # m2) -> m1 # m2


??so ... martin's "F" and "G" live in m1 # m2 ... ??? ....

?? "m1 # m2" here is the v-enriched bifunctors from m1^op X m2^op to v, v-continuous in each variable separately ??? ....

?? so then first we should cook up the v-enriched quadrifunctor from m1^op X m1^op X m2^op X m2^op to v, v-continuous in each variable separately, taking (a,b,c,d) to F(a,c) # G(b,d) .... ????
??situation where ideal carving out subspace is "automatically" principal ?? ... ??for some good reason ..... ???vs contrasting situations .... ??? ....

???"characteristic function" .... ???? ... (??"boolean" case ??? ...) ... ??"normalization" ??? ...

???pun on "character..." here ???? .... ????? .....

?? any interesting case of combining this (...) situation with situation where subspace carved out is subgroup ??? ..... ????? .... ??? "fourier analysis" ???? .... ????? ....

???"root of unity" .... ????

"x^n" .... ???? ?? vs "x^n-1" ... ??? ... ???? toric ????? .... ????....

??? "log(x^n)" ... ????? ..... = x*n ... ????....

Tuesday, May 24, 2011

"ordinary concept of zariski open cover as inadequate ..." ... ??????? .....

???to what extent do we understand what this is really supposed to mean ?? ....

??in some ways as seeming somewhat adequate ... ??? .....

hmmmm ..... ????? ......?????

???grothendieck topology on poset site category as ... ???no more than "grothendieck [-1]-topology" on it ??? ..... ??? ....
i want to think outloud here about the "tensor product" functor on the co-cones of a co-fan ...

one reason that this functor should exist is as the manifestation at the model level of the alleged essential geometric morphism from t^2 to t for which the tensor product of toric quasicoherent sheaves (preceded by the universal bicocontinuous functor) is the extra left adjoint, where t is the accidental topos of the corresponding toric variety. that's a bit circular, though, since my reason for guessing that such an essential geometric morphism exists is mainly because of its manifestation at the model level.

anyway, i have a couple of further guesses about how to think about the alleged tensor product of co-cones:

1 as a sort of categorification of the intersection operation on the poset of cones of the fan ...

(???confusion/problems here ??? .... ??? unit element for intersection as "total subset" ... ???? whereas that doesn't seem to qualify as a cone, even in affine case ???? .... ???? .... ????hmmmmmm .... ???maybe slightly tricky ???? .... when there's a largest cone (that is, the affine case ... ??? ...), it can play (at ?? ...) the role of "total subset" ... ???sa... ?? this as maybe sort of nice, in suggesting something fishy about unit element for intersection of cones even in those case where it does exist ... ???sa ... ???as not preserved by inclusion into semi-lattice of all (??? ... ???even those including non-trivial antipodal pairs ??? ... ???) cones ... ????)

(hmm, so what about co-cone viewpoint on above parenthetical discussion?? ... span of co-cones ... ??? binary such as "safe", nullary such as ... ???? ..... ???? .... ...hmmmm... co-slice ... ?? ... ??vaguely reminds me of bit about "every slice cat is a topos / distributive lattice" ... asf os...)

2 as "tensor product" of categories enriched over the whole lattice ... ??? ...
?? topos t which is "best approximation to" such whose syntactic category is given category c, vs whose semantic category is c ... ??? ...
?? "descend from a covering" ... ??


??? "lift to ... " ... ????

?? "extend to ... " .... ????

Monday, May 23, 2011

?? "spectrum" process (?? "localization" ?? ...) associated to "globalization" of "module AG theory" as (2,1)-functor on category where only "localizations" are admitted as morphisms ... ??? ... ???? .....

??? .... ????factorization into localization followed by conservative morphism ... ??? localization factor as ... ?? "substitute" for original point of spectrum ... ???? .... ????? .....

???big zariski topos as maybe involving combined doctrine ... ??? .....
??? "structure sheaf" ... "unit quasicoherent module" ... ?? "genuinely semi-monoidal pro-monoidal" picture here ??? ....
x pro-monoidal category ...

x1 # x2 as presheaf on x ... formal colimit of x-objects ... so ... element of presheaf x1 # x2 ... over x3, say ... as x3 -> x1 # x2 ...

???so ... ???binary and nullary operations in concrete operad of yoneda image objects in opposite monoidal category .... ??? ....

?? "associativity" ... ???..... ???isomorphism of pre-sheaves (?? of 4 variables ... ?? 1 of them of different variance ?? ...) between certain "tensor product" presheaves ??? ... ???more concretely ... ???.....

a#b -> c .... c#d -> e .... ????

??? getting quadratic operad from arbitrary operad at expense of introducing compound types ??? .... ??? maybe either via getting pro-monoidal category from operad, or just sort of directly ... ???....

??possibility of ... ??? 3(?...)-codiscrete simplicial "nerve" of pro-monoidal category and/or of quadratic operad? ... ??with "something extra" amounting to mere property in both cases ?? ... ???reflective in quadratic operad case ??? ...

??possibility of just "facial" nerve ??? ... ??maybe in "quadratic operad" case ?? .... ???? ....

??in "untyped" case, just set of 2-cells tw 4-ary relation ("form a 3-cell") ?? ...

?? homotopy type of such simplicial set ... ??? ... ???in "accidental topos" case ??? ....
?? ... pro-monoidal category ....

?? ... quadratic operad ....

"deformation cohomology" ... reverse-engineering example from ordinary algebraic variety .... ???? .... ???singularity ??? .... ...

?? ... "family of commutative monoid operations" ... ??? ....

action ... ??? ....

?? plenty of quadratic operads that don't come from pro-monoidal categories ... ??? .... ??? other direction ??? ..... ... ?? "something extra" here ??? ... ... ??if it works at all ... ??? .....
?? presheaves on a finitely generated (?? or maybe presented???) category ... ?? topos of "finite diagrams" of a sort ... ???.....

Sunday, May 22, 2011

?? what to talk about at this conference that i'm apparently being invited to ... ???

??? singularities of schubert varieties ... ??? .... ??? ......

??? "sesquicoherent sheaves" ??? ......

??? kaleidoscope as fan ??? ... ???"maximal torus embedding" ... ???.... ??? toric stacks ???

?? ag theory of torsors of algebraic group ... ???



??? reply to alex soon about arrangements ... ??? airfare and so forth ?? ....
?? homotopy type of accidental topos of toric variety .... ??? ... ???affine and non-affine cases .... ????? ..... ??how TAG structure relates to it ??? .....

co-/homology groups of this homotopy type ... ?? as relating to abelian category of abelian group objects in the topos ... ??? .... which is of course (we think ... more or less ...) the quasicoherent sheaves over the toric variety ....

?? this direction of thought ... and/or perhaps other vague thoughts vaguely related ... or at least impinging here ... ?? ... ??? as tending to make it more obvious how likely that all this stuff is already well known in some form somewhere ...

????vague ideas ... old ideas that never really managed to get to work ... ideal class group ... homotopy type / groups of geometric realization of monoid .... non-unique factorization ... octahedron /generalized diamond ... ?? .... ??? ....

?? ... thomason ... ??? ...

Saturday, May 21, 2011

?? (toric ...) theory with object pair "inverse only up to given idempotent object" (in some sense ...) explicitly put in ... ??? .... ???relationship to "explosion wrt sequential direct limits" ??? .... ... ???? ....

??baez suggests proving triviality (= principality ? ...) of invertible objects in certain context by ... ??? "set of atoms" taking tensor product to cartesian product ???? .... terminality of object invertible wrt cartesian product ... ???....
?? tensor square of range of "opposite color bishop" on "checkered" infinite quadrant ... ???

a,b,c .... ac = bb ....

x,y .... xb = ya, xc = yb ... ???....

d "xx"
e "yx"
f "xy"
g "yy"

????? .....

??hmmm, so is this the example where ... ???the tensor square is not quite principal, the lizard's tail being a bit chewed off ??? .... ???.....
discussions with kenji ...

?? mention idea of ... invariantization of decoration by superposition of translated copies ... ???? .... ??as "proof" of "every subgroupcomes from some decoration depicting some feature" ... ??? feature concept as abstracted from decoration concept, sort of ???....

??maybe ... ??wallpaper patterns ... ??? ... ???...

??/equivalence relation up to which they're classified ... ???

?? "pathological wallpaper pattern" ... ???

??? "wallpaper pattern" as "feature" of wall ??? .... ??or perhaps as "decoration" "depicting" feature ???...

?? "symmetry-breaking" ... ???....

??so ... ??not actually too sure about what traditionally qualifies as "wallpaper pattern" ... ???famous 29, or is it 37 or 23 or something??? .... ??cocompact?? ...????discrete ??? .... ??? ... ??certainly for purposes of talking to kenji want to consider features like "point", "line", and so forth .... which, ok, would get into non-discrete stabilizers ... probably a good idea ... ???? ..... ???some confusion here ??? .... ???? ..."all possible decorations of wall" ...???? .... co-compactness, vs .... ???? ..... ??? .... ?? "frame" ... ????.....

??in pattern p picture (some typical p ...), wall "looks like" ... ???certain compact manifold ... same dimension as group ... ???... ???not exactly my usual mental picture of a wall ... ????? .... ???....

??? ??"blank canvas" ... spectrum of unwritten writings that can be written on it ... logician's existence ... ??? borges's universal library ... ???....

???"pixie dust in kaleidoscope" ... ?? section on coxeter groups in those on-line notes on group theory by milne ....

?? "picture yourself as one of the specks of dust ..." ...

??? "sort-of classifying all concrete groups by sort-of classifying all abstract groups and ..." .... ??? ... subgroups and their cosets .... ??? .....

??kenji's question about "cosets" of non-subgroups ... ??? hmm, so what subgroup _is_ showing up here ??? ??subgroup generated by "(??right??) differences" between elements of subset ?? no... just ... subgroup of all elements under whose corresponding translation action the subset is closed ... ?? ?? "periodicities" ... ??? ... ... ???proof that for a subgroup this gives itself ?? ...

??hmm, subsets of Z as good playground here ??? ....

?? "characteristic functions of subgroups as representative of all symmetry (?? ...) classes of functions" ... ???.... ???? ... "characteristic function of set of periodicities" as idempotent map on function space ... ess taking each function to its symmetry class ... ?? .... ???

(?????situation where ... ideal carving out closed subgroup is principal for some "automatic" reason ??? ... ????? .... ???)
co-cone category of co-fan giving projective plane .... ????......

Friday, May 20, 2011

??pro-monoidal category ... discrete with 2 objects 1, x ... "x tensor x = 0" ... ?? ....

1 # 1 -> 1

1 # x -> x

x # 1 -> x

????? commutative monoid ... ??? ew action ... ???? ...

??????????.....?????


???quadratic operad ... ????....
?? so let's try cooking up an example of day convolution where the (binary...) tensor product doesn't have a "second right adjoint" ... ???which should be pretty easy, if i'm thinking about this the right way... ???just need to pick a "generic" (...) example ?? ... pro-monoidal rather than genuinely monoidal ....

???also... got into discussion with todd about ideal class groups and dimensional categories and so forth ... ??follow up on this ... ??blow-up and so forth ... ??but also ... ???toric ideal class group ??? .... ??? .... ???? .... kummer .... ??? .... ??? gunnarsen ???? ...... rng ... ??? ....

???? "checkerboard corner" ... "downward decorrelation" ... "localization" .... "ramification" .... ????.... idempotent vs invertible ... ???? ..... checkerboard edge ... ???..... ????? ...... ?????? ........... ???? filtered colimits vs ... ?????? .... ????? .....
?? 7-month-old unanswered math overflow question about "homotopy-flat dg module" ... ???....
??confusion between "algebraic" and "geometric" direction in connection with ... ??? geometric interpretation of "flat" ... ??? ....

??? "smoothly (??? ...) varying fiber ..." ... ???trying to interpret such condition more "in terms of domain ..." ??? ... ??? .... ????? ......
?? so the left-universal property of the accidental topos of the affine toric variety corresponding to a commutative monoid m wrt the (?? ...) combined doctrine is ... ??? "action of m on the unit object, making the unit object into a flat m-object" ... ??? ....

??again, attempts to give "flat" "geometric" interpretation here ... ???....

Thursday, May 19, 2011

baez is here for a few days so i got a chance to ask him for ideas about "toric dimensional theories" ... i mentioned my idea about "gereralized currency exchange system" (?? ...)... he mentioned the idea of considering for example say "burning of methane" as a toric dimensional theory where the grading group is free abelian on {water, carbon dioxide, methane, molecular oxygen} (i think that that's what he said ...) and there's one generating element in ... ??well, whatever grade would be appropriate for burning of methane ... ???...

... have to think about this ... ?? ....

if we get more of a chance to talk i should ask him about the bit about product of dimensional theories ...

???toric serre's theorem for P^2 ... ???? ..... ???? .... glueing scheme ... ??? ???triangle ??? ... ???? .... ???? ....

also... mentioned an idea about "dyson series" ... quantum case vs stochastic case.... he thought that it was peculiar the way in the stochastic case the division into diagonal vs perturbation doesn't leave the diagonal part as itself being infinitesimal stichastic ... whereas i think that it's peculiar that in the quantum case it _does_ leave the diagonal part as itself being infinitesimal unitary .... ??? ....

??actual "torus" aspect of toric varieties ... ??? "re-scaling" ??? ....

??maybe try calculating invertible actions of "bishop range infinite quadrant" commutative monoid ??? ....
?? faint attempt to start writing the paper ...

toric quasicoherent sheaves

in this paper we develop a concept of "toric quasicoherent sheaf over a toric variety" analogous to that of "quasicoherent sheaf over an ordinary variety".

[of course, this is probably already a well-known concept under some name somewhere, so even if there's still enough additional material to make the paper worthwhile i'd have to re-write the above to take the well-known-ness into account...]

just as the quasicoherent sheaves over a (decent?) variety form a cocomplete abelian category equipped with a nice tensor product, the toric quasicoherent sheaves over a toric variety form a grothendieck topos equipped with a nice tensor product.

the conceptual motivation behind this development is mostly not described here, though some of it may be relatively obvious. (variety : toric variety :: vector space : set.)

1. the cocone category of a fan

a toric variety can be considered as a so-called "fan", made out of "cones" which are conical subsets of a lattice (in the sense of fg free abelian group) which have "faces" which are other cones of lower dimension. more convenient for present purposes is the dual "co-fan" made out of the "co-cones" dual to the cones in the fan, bearing a "co-face" relationship where the dual cones bear a face relationship. (the co-fan may at first however seem less intuitive than the fan because the concept of "co-face" may be less familiar than that of "face"; a co-face is bigger than the original where a face is smaller than the original, and the way that co-faces "lose" a dimension is by becoming translation-invariant in that dimension.)

the cocones of the co-fan of a toric variety v form a category v# where the morphisms are translation maps. the main purpose of v# here is that the category v% of "toric quasicoherent sheaves over v" is defined as a certain subcategory of the category of set-valued functors on v#. the name "toric quasicoherent sheaf" is justified in part by the fact that abelian groups in v% are essentially the same thing as ordinary quasicoherent sheaves over v.

the cocone category v# has some notable structure which is of use in constructing the toric quasicoherent sheaf category v% and in establishing corresponding structure on it: v# has filtered colimits (a co-face is the union of an increasing sequence of subobjects isomorphic to the co-cone of which it's a co-face), and a semi-monoidal structure compatible with the filtered colimits. there's generally no unit object for the semi-monoidal structure except in the case where v is affine, though.

(in fact, in the case where v is affine, v% is the topos of actions of the commutative monoid given as the "coordinate monoid" of v, while v# is the category of models of that topos. this may serve as a guide to the way in which v% and v# are, like v itself, "glued together from affine pieces" in the general non-affine case.)
??? .... "diaconescu's theorem for pre-sheaf topos equipped with day convolution symmetric tensor product" .... ???and generalizations and adaptations of this .... ??? ...

??relationship to ... ???vague feelings about ... "combined doctrine" ... in non-toric case ... ??"flat model" .... ??? with "flat" here trying to have some sort of naive intuitive interpretation .... ???? .....

??? ... monoidal (or perhaps pro-monoidal ...) site category ... for example direct limits of principal N-sets ... "geometric morphism whose lex left adjoint part is symmetric monoidal" .... ... ??? ...

??i think that i vaguely remember that we've already thought about "trying to generalize diaconescu's theorem from the pre-sheaf case to the sheaf case" ... ???....

Wednesday, May 18, 2011

?? possibility of topos of "toric motives" ??? ...

??abelian groups therein ?? ...
??so consider TAG theory of toric quasicoherent sheaves over projective line ... ???... and consider ... ???models into TAG theory of k-modules vs into that of [mult monoid(k)]-sets ... ??? ....

??seems like they probably _are_ different in general ??? ... ?? "ideal class group vs toric ideal class group" ?? .... ???? ... ??? ...

??so maybe it really is true that ... in affine case they're the same, but pretty much not beyond that ... ????.... ??? ....

??analogy set : ab gp :: a-module : b-module here, for comm ring hom h : a -> b ?? ...

?? models of [a-AG theory of quasicoherent sheaves over projective line relative to a] into [a-AG theory of "k-modules over b" ???? .....] vs into [a-AG theory of "k-modules over a" ... ???? ...... ???? ...] .... ????

??having trouble seeing difference here??? .... ????work out more explicitly ??? ...

commutative ring k : commutative b-algebra k :: .... ????? .....


????square .... ???? ....

commutative monoidal ab gp .... comm monoidal set ....
ab gp .... set ....

??????? ...... ???? .....
???"globalization" as "structure/syntax" and "localization" (in certain sense that i think that i used at some point ... ?? ...) as "semantics" adjoint to it ?? .... ??? ..... ?????? .......
???so is it clear what we mean by, for example, the pre-stack over _comm ring_ of torsors of the borel of gl(n) ?? ... ??? ..... ??? ....

??and then what about getting AG theory here by globalization ... ??? .... ??? ...

Tuesday, May 17, 2011

?? free AG theory on TAG theory t1 -> AG theory t2

=?=

t1 -> underlying TAG theory of t2

??? .... ???....


???underlying TAG theory of AG theory of modules of comm ring k ... vs ....

TAG theory of actions of mult monoid of k .... ???? ....

???some confusion here ?????? ........

???????????????? .....

??? affine variety case ???? ..... ???? ...

??right adjoint process of taking underlying mult monoid, followed by ... ????is taking actions a left adjoint process ?? ... ??? .... ???

???............


??? 2 very different ways to get from a toric variety v and a commutative ring r to a set of something sort-of like "points of v over r" ... ??? .... via taking mult monoid of r, vs via taking underlying TAG theory of AG theory of (syntactically) r-modules ... ??? .... ??with some further confusion about possible ways in which maybe this distinction gets somehow blurred in "affine" case ??? ... ???? .... ????....

???check certain e-mail i was writing to todd to see to what extent i might have gotten this stuff screwed up there ... ??e-mail about moduli stack of elliptic curves as toric stack ... ??? .....

???getting pre-stack over _comm ring_ directly from pre-stack over _comm monoid_ ... ????....... ???? .....

???maybe distinction really does vanish in "affine" case ??? ??then what about "close to affine" ??? ... ??? ....

???comparison functors between ... ??actions of mult monoid of comm ring k, and k-modules ??? .... ?? ... ??? and what doctrine such comparison functors might qualify as theory morphisms wrt ?? .... ???? ....

??underlying [mult monoid(k)]-set of k-module ... free k-module on [mult monoid(k)]-set ... ??? ....
m commutative monoid....



?? essential geometric morphism _[m^2]-set_ -> _m-set_ corresponding to "tensor product of m-torsors" ???? ..... ????....
??my naive "explosion" idea was that a point of [the spectrum of the TAG theory of toric non-quasicoherent sheaves over a toric variety x] over a toric affine scheme y shuld be "a point p of x over y tw a toric open u of x to which p belongs" (?? ...) ... ??whereas at the moment i seem to be getting some sort of annoying parody of that instead ... ?? "point p of x over y tw a toric open u of _y_ to which p belongs" ... ???wait a minute, that doesn't seem to parse too well; it doesn't make sense (as far as i can see...) to say that a point belongs to a subspace of its domain of variation ... ???..... ????? ......

hmmm....

well, let me try to put that aside for the moment... even if it's more or less impossible to ... and say what i was going to say, anyway ...

which is... ?? maybe this at least fits with already more or less knowing that the category of direct limits of principal actions of a commutative monoid doesn't simply fall to pieces without its filtered colimits structure .... ????....

??and .... ??? imagining toric variety x over toric variety y ... ??as maybe exploding into pieces corresponding to toric opens of y rather than of x? ... ?? but there seemed to be problems with that even before noticing the non-parsing problem above...

??and ... ??? ??maybe contemplating all ways in which alleged presheaf (??or maybe pre-stack?) (??and also somewhat similar examples?? ... ??including non-toric analogs?? ...) in question is "bad" ... ???.... ??for one thing, to see how "bad" spectrums of _any_ (...) kind can get, let alone this kind ... ??? ... ?? having idly conjectured at various points that spectrums of even the most general TAG theories (or perhaps not quite so general ...) are somewhat good in certain ways ... ?? ...

?? maybe ... ?? not having enough (?? ...) models over multiplicative monoids of fields as a sort of "bad"-ness here ???? .....

?? had some idea about "non-separated" ... ???? ....

??? "morphism x : 1 -> 1, tw [??object u ... with multiplication and/or comultiplication operation??? ... ??? ... maybe epi-ness ... ??? ??? .... ??well, maybe all that i'm trying to say here is "coherently idempotent" ... ????... ??no wait... ?? "idempotent" commutative monoid ??] on which x becomes invertible" ?? .... ???? ....

??this as theory which really "wants" to have category-valued spectrum ??? ... ???...

?????? Z/n as idempotent commutative monoid in TAG environment _Z-set_ ?? .... ????? ......

??todd reminded me ... ?? universal property of day convolution over actual symmetric monoidal category x .... ?? ... "symmetric monoidal functor from x" ... ??affine vs non-affine case here .... ???? .....

???cartesian product of TAG theories .... ???? .... ??? ....

?? "explosion" ... ??? ....

????? ..... ???? ....

Monday, May 16, 2011

?? huerta seems to be saying something about ... ???clifford jordan algebra showing up in jordan algebra associated to division algebra ... ??? ....

1-by-1 hermitian ... ???

2-by-2 hermitian ... ???

??? ... ???? ....
_N-set_ as TAG theory ... ?? "every object as colimit of unit object" ??? ... ?? = "no stuff" ?? ....

?? whereas with explosion, "torus object" as maybe (?? "a priori" ... ?? ...) giving some stuff??? .... ????? ...... ???? .....

?? category where an object is a commutative monoid and a morphism is a functor between the cocone categories of their affine fans ... ???

hmm, that as adequate way to say only in "torsion-free" case ?? ... ??? ... ?? "direct limit of principal actions" ??? ... ????...

(non-toric analog?? ...)

hmm, so let's try homming N into N here .... ???? or N into the multiplicative monoid of a nice field ... ??? ...

??? hmmm, but what about symmetric monoidalness here??

??? "endomorphism e of 1, tw idempotent (??...) object on which e is invertible" .. ???....
??? me : them (= those who emphasize "stacks over (2,1)-sites" ... ??? ....) :: doctrine meta-doctrine : (2,1)-topos meta-doctrine ??? .... ????.....

(some other meta-doctrines in the fray as well ... some just minor (?) variants/variations, perhaps...

?? "environment" as twice-categorified "point" ??? environment of 2-topos ... model of 1-topos ... point of locale ... ??? does "locale" here deserve to count precisely as "0-topos" ??? ..... ??"theory" as _not_ twice-categorified point ... ??rather twice-categorified "propositional formula" ? ... ??? .... "theory" here as ambiguous between "object in syntactic (2,1)-category of doctrine d" and "object in syntactic (2,1)-category of (2,1)-topos coming from d" ... this latter syntactic (2,1)-category being the pre-stacks on the former ...

?? "doctrine of toposes" .... (???clarify theories vs environments here... (???also for all our other doctrines???? ....)) also of t-modeled toposes ... ???but then glorify these doctrines into 2-toposes ... ??try to get explicit about what's going on here ... ???


??? size issues ... ??? ......

?? lex theory has "big" moduli stack ... ??? "unbounded" ?? ...

?? whereas topos or AG theory (??for example??) may have more "manageable" moduli stack ... ???

?? doctrine has big moduli 2-stack ...

??whereas 2-topos (or ... ??? ....) may have more "manageable" moduli 2-stack ...

??what about "backing up to get a running start" here?? ... ??? .....

?? doctrine as determined by it's (2,1)-category of environments, but not so for (2,1)-topos ??? ... ??? .... ??? ... ??need whole 2-stack of environments ??? ... ??? .... ??? ... ???...
?? tensor product (as non-quasicoherent...) of toric quasicoherent sheaves is again quasicoherent ... ????.... this as linked with "flatness of localization" ??? .... ???? ??or something different ??? .... ??? ....

a,b N-sets ... (a tensor b) tensor Z, vs ... ??? (a tensor Z) tensor_Z (b tensor Z)

???relationship between "flatness of localization" and "idempotence of localization" ???? .... ???? ......
?? "unlimited dimension moduli stack as bad, unlimited number of pieces ok" ... ???....

??? "you wouldn't really want to consider the modulli stack of models of an environment" .... ????..... ????....

???underlying lex theory of free geometric theory on lex theory ??? .... and categorified analog ... ??? ....

Sunday, May 15, 2011

?? formal vs intuitive approaches .... ??? (??possibly other variations / mixtures as well ...) ??student / teacher conflicts along these lines ... could be in either (...) direction .... how to deal with .... ??? either approach _can_ be legitimate in appropriate circumstances, but ... ????.....

??contrasting sorts of paranoia ... ???or is it more like... ??paranoid aspect vs schizophrenic aspect ... ????.....

?? programming .... ???....
??? algebraic limits (??especially _products_ ?? ...) for toric dimensional theories vs for dimensional theories ... ?????? .... ???? ..... (??also for TAG theories vs AG theories ... .... ???...) .... ??? as "geometric colimits" .... ??? "glueing" ... ??? .... ?? "explosion comonad" .... ??? .... ???? .... ????vs "direct sum of toruses" ??? ..... ?????? ..... ??? .....

??? whether "explosion" seems insufficiently exploded ??? ..... ??? ... ??? doesn't seems like "disjoint" (?? ...) pieces .... ???.... ???? .....

??? does explosion comonad here (...) maybe give something "non-separated" ?? ??? .... ???? .....

...try to figure this out...

?? so consider explosion of affine toric variety associated to N ... ??....

?? what _is_ it's TAG moduli stack ?? ... assuming that actually makes sense ... ??...
"tensor product of cocomplete categories" ... ?? "universal bi-cocontinuous" ...

"tensor product of k-modules" ... k comm ring ... "universal bilinear ... " ...

"measures on u(x) X u(y), mod bilinear relations" ... ???

x,y cocomplete categories ....

???presheaves on u(x) X u(y) .... ???

??? "object o that believe that cocone c is a colimit cocone " ... ????

?? "homming into o takes c to a limit cone" ???? ...


??? when does/n't universally imposing colimit property give "reflective subcategory" ??? ..... ????....

??? "presheaf p on u(x) X u(y) st for any object x1 in x and colimit cocone c in y, the cocone yoneda( "x1 X c" ) gets taken to a limit cone by homming into p" ... ????

u(x)^op -> _presheaf_(u(y) ... ???....
??? "explosion comonad" for filteredly cocontinuous categories and/or toric varieties ........ ???? .... ????.... ???....

???"cohomology" here ????.......

???hmmm, but does TAG structure really "survive explosion" ??? .... ???.... ??? "day convolution wrt tensor product of classical models" ??? .... ??? ....

?? non-toric analog ??? ..... ...???....

Saturday, May 14, 2011

??? abelian-group-graded commutative quantale ... ???? ..... ????
not sure how many posts from this blog may have been lost in recent site-crash... evidence of at least one from may 12th ... if i manage to remember anything about any posts that were lost i'll try to write something here about it...

Friday, May 13, 2011

??so is the topos of set-valued functors on the fan category of a toric variety v essentially just the topos of toric non-quasicoherent sheaves over v ??? ... have we ever thought this before ?? ... ??? ....

how good (??) of a TAG theory (??...) does this (?...) topos itself qualify as ?? .... ???? ....

??in particular do we (??obviously) get that (...) same compatibility (??) relation between the topos and TAG structures ?? ...