??so what about "b-valued existence and equality predicates ..." (and so forth ...) and .... :

1 ??something about "kripke model" or something....

2 ??something about propositional theory ...

3 ??something about interpretation of quantifier ...

4 ??something about ... ??well, maybe nice simple example of #3, or something.... ??theory of "for all x there exists y st r(x,y)" ... or something ...

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

???something about ... ??"orthogonal partitions" and malcheff variety or something, and so forth .... ????...

something about mere oder up / downset , vs ... ??more ... ???...

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

??so suppose that we've got an open subset of the 1-point compactification of N ... ??then let's try thinking of it as "the set of sections of it (thought of as a subterminal sheaf) over _basic_ opens, equipped with the b-valued unary "existence" predicate given by "the basic open over which the section lives", and the b-valued binary "equality" predicate given by "the basic open over which the two sections agree" (??which i guess is just the intersection of the two basic opens ... ???or something ???....) ..." ??? ...... ???or something????....

??so... ??thinking of the basic opens as elements of a boolean ring, intersection corresponds to multiplication ???

???so the collection of basic opens contained within a given open u is ... ??a subset of the boolean ring ... ???the inclusion map being essentially the unary existence predicate??.... and ... ...

??well, there should be a number of different ways of thinking about it, and maybe good to try to keep them somewhat independent so as to be somewhat able to chekc then against each other ... ??or something ... ??

??so.. ??one approach is ...???something like ... ???"coherent geometric theory of a truth value" ... ??or something???...

??first, "coherent geometric theory of a set" ... ???or something ???....

set s equipped with unary and binary b-valued functions ("predicates") ... ??...

"reflexive"

?? "(x=x) <=> (x exists)" ???? ??so existence predicate seems redundant ???


"symmetric"

?? "(x=y) <=> (y=x)" ... ???


"transitive"

?? "(x=y) & (y=z) => (x=z)" ...???...


???then something about ???? "for x distinct from y, (x=y) is strictly falser than (x=x)" ??? or something??? ...??something about "efficiency" or something??? ... ??"irredundancy" ??....

???then something about "subterminal" ... ???... ??"(x=y) <=> ((x=x) & (y=y))" ???or something ????....

??hmmm, so for example, suppose that we have 3-element set {a,b,c} ... ??then suppose we have 9 truth-values aa,ba,ca,ab,bb,cb,ac,bc,cc ... ??? hmmm...

let's start even simpler ... 0-element set ...

???hmm, maybe we also need a sort of "inefficiency" condition ???? .... ???or something ????.... ??somethng about canonical presentation as not very efficient .... ????or something ????......

??what _about_ something about "basis-dependence" (or something ...???...) here ???....

???maybe try ignoring all in-/efficiency constraints for now ... ???

??so 0 generators always gives the initial sheaf ... ???or something ???....

so consider 1 generator a .... 1 truth-value "aa" ... could be anything...

??free boolean algebra on 1 generator ...

??now 2 generators a,b ... 4 truth-values aa,ba,ab,bb ... ??but really just 3 because ab<=>ba ... ??and then transitiveness says... ??what ??... aa*ab=>ab, and ab*ba=>aa, and ab*bb=>ab, and ba*aa=>ba, and ba*ab=>bb, and bb*ba=>ba ... ??? but by symmetricness this reduces to ... ???what ?? ... aa*ab=>ab, and ab*ab=>aa, and ab*bb=>ab, and ab*ab=>bb ...??or something ???...

??xy*yz => xz ...???

??but "p => q" means ...??what ?? ... p*q = p ??? or something ???

so something about "xy*yz*zx = xy*yz" ???

aa*ab*ab=aa*ab ... ???automatic??

ab*ab*aa=ab*ab ??ab*aa=ab

ab*bb=>ab ......................... ????...


??wait a minute, i think that i forgot the "subterminal" condition ... ???.... ??hmmm, which should hopefully bring it down to just freely choosing the existence predicate values, right???

aa bb ... ab <=> aa*bb ...

???so what _about_ simply an ideal ??? or something ???.... ??or complement thereof, or something ... ????....

???some confusion here ...

??something about .... ????collection of truth values that ... ???hmm, well maybe it is just like an ideal ... or something .... ??? ??"put in one global section for each element x in the ideal (??or its complement or something ???), but have its formal existence predicate value be equal to x" ... ???or something ???....

????something about ..... ????homeomorphism type of the open subspace .... ?????or something ??? .... ???hmm, or maybe of its closure or something ?????.....


????something about "system of boolean equations (or something) for which a solution (or something...) sort of amounts to having given stone space as closure (??or something??) of open subspace" ... ???or something ???....

???something about non-/degeneracy and in-/efficiency here ??? or something ...

???hmm, so what about something about case of "atomic truth-value" ?? ... or something ... ???.... ??is there a "the theory (??in what doctrine???) of an atomic truth-value" ???? ... ???or something ???.....

??is a "lawvere ultrametric" automatically symmetric?? ??if so then what's the proof??

??so consider the boolean ring presented by N's worth of generators with the pairwise products all 0 ... ??...

a "2-valued" solution to the system of equations is ... either exactly one variable takes the value 1, or exactly none does ... ???...

??now is this "complete" ???... ??no?? ... ??"gap" between "finite" and "cofinite" ?? ...

???so what about "non-principal ideal" here ???....

??what about "ideal class group" (??problematic because of lack of dedekind domain property?? ??or something ??...) and/or "algebraic k-theory" here ???....

???something about ideal as corresponding to zariski-closed subspace ... ??something about the limit singleton as closed but non-open ... ???...

??principal ideal gives open subvariety here ??? ???by closed complement <1 - the generator> ?? ... ???or something ???....

hmmm... ???what about converse??? (??or something??) .... clopen set ... ????.... hmmm... something about "orthogonal congruences" and "idempotent element" ... ?? ...

??so _are_ there lots of "complete stone algebras" (or something...), and are they a special case of "stone locale" ?? ... or something ... ???.... ???if so then _what_ special case ??....

??so _is_ it in general true that double negation topos of stone space is coherent boolean locale corresponding to the isolated points ???? .... ???or something ???....

??hmm, is there some funny back and forth (...) here where ... ??taking double negation topos of stone-czech compactification of the natural numbers removes the limit points, but then the "elementarization" (or something ... lawvere's bit baout "wallman compactification" or something ... ??...) bit puts them back ??? ???or something???

still haven't really investigated how whole "model of set thoery ..." issue enters into "forcing" idea ... ???.... or something ... ???? ...

??so... 1-point compactification of N ... double negation topos of the localic topos coming from the stone space ... ???as ... ??sheaves over boolean locale correspodning to boolean frame of regular open sets, which is essentially the complete boolean algebra of arbitrary subsets of N (with infinity cleaving unto the closed complement of the regular open set ...) ... ???suggesting that the infinity model gets removed ???? .... ??this example actually happens to be coherent boolean locale corresponding to discrete N ??? .... ??or something ????

???also suggesting that ... ???closed point of sierpinski space should be thought of as _domain_ of model morphism ?????? ..... ???or something ????.... ???because domain of model morphism gets removed under double-negation topology?? or something ??? can we check this quasi-independently somehow, maybe ?? ... ??something about ... sheaves over sierpinski space .... closed point as non-open ... ??stalk over it as sections over its minimal open neigborhood which is the whole space ?? ???or something ??? ...restriction map from such global sections to sections over the open point .... ...???so in the "exponent category", which i think of as the "category of finitary models" (or something ...), the open point is the codomain of the non-trivial model morphism ... thus retained while the domain (=closed point) is removed ... seems to fit ... ???

??but ... ??what about ???... ???possible non-relationship (or something ...) here between ... [???"point in top space as in closure of some other points" .... ????or something ...] and ["model of geometric theory as filtered colimit of finitary models" .... ???or something ...] ?????? or something ????? ??weird puns on "filter" and "limit" and so forth here ???????? ..... ????? or something ?????.... ???so what _is_ going on here???????...... ????something about ... ????locale (or something ... ???) created by ... starting with point "0" ... then putting in point "1" with 0 in its closure ... then "2" with 1 in its closure ... and so forth .... ????something about point "infinity" materializing as filtered colimit of 0->1->2->... ????? ???or something??? ... ???does this make any sense ????...... ????direct colimit of locales here, vs of top spaces ???? .... ???or something ????? ..... ??????..... ??also vs direct limit of posets ???? ..... ???also vs direct limit of k-coherent locales for various k, or idealized limiting cases of such ... ???...

???is there something going on here about .... ????a model being an actual filtered colimit of models, vs being .... ????some sort of ultraproduct (????or something??????) of other models ??????????? ..... ????is an ultrafilter an ultraproduct or something of vanilla models????? ...... ????or something ?????? ..... ????.....

????something about elementary equivalences and / or some sort of "elementary equivalences of many variables" (or something .....????....) relating to ultraproduct situations ....... ???or something ????? ........

??something about ... "generalized birkhoff theorems for various doctrines" ??? .... ???or something ??? ... and so forth .... ???.....

??hmmm... ??i think that there is _something_ like this (??...) going on here ... ???something about ... single-environment (??"classical" ???....) model theory vs multi-environment here ... something about .... ??some sort of very straightforward operation producing for example "model parameterized by 2" from pair of "models parameterized by 1", but then also something about ... ??ultrafilter on x (or something ... ???and so forth ... ???might have some different doctrines here mixed up, but .... ???....) as giving operation from "model parameterized by x" to "model parameterized by 1" .... ??? or something ??? ... and so forth ... ???....

??something about ... ???categorified lawvere (or something ...???...) theory here?? ... ???something about "doctrine" and so forth ??? ....


??also something about ... ??our example of double negation topos of stone space given by 1-point compactification of N .... ???something about ... ??being cautious about relating this to "poset of forcing conditions", including possibility of shoe-horning in "non-standard analysis" as special case of this with discrete poset ... because in those cases the topos of which you take double-negation topos might be pretty different .... our example(s?...) of double negations toposes of stone spaces was just for fun and educational purposes, or something ... ??including attempt to possibly dispel some confusion about "stone space vs boolean locale" and so forth ??? ....

???what about something about boolean algebra given by something about ... ???regular open sets of unit interval ... ???.... "geometric realization of simplicial sets with orientation-switching ..." .... ???or something .... and so forth .... ????..... ????something about understanding boolean locale here ???????? ....... and so forth ..... ???????.......

??so what about something about .... ????... ??"derivation at a pair of points" ... ??or something ... ???as something about .... ?????isomorphism class of short exact sequences .... ????or something ?????? ...... ?????....... ????....

????...

???something about .... ?"hall algebra" ... ???? .... ???

???something about "symmetric" vs "anti-symmetric" pattern here ???? .... ???or something ??? .... and so forth .... ????.....

??something about ... ext(m,m) ... ????..... ????something about "derivation" .... ?????? or something???? ...... .... ??????.....

hmmmm.....

"deformation" ....

"normal bundle" .... ????.....

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

??so consider ... ???commutative k-algebra... ideal power filtration at a maximal (or something....) ideal .... corresponding to point ... ?? .... ????something about "maximal in maximal" ideal corresponding to tangent vector .... 2-stage fitration on quotient algebra ... ... associated graded... ???something about "fake tangent space" ... ??or something?? ... well, not quite, i guess, but ... ???.....

??hmm, something about the extension module as symmetric monoidal in the "tangent vector" case, but not the other case ... ?????or something ??? .... .... ??...

??so what about this concept of "derivation twisted by an automorphism" that baez mentioned?? ... hmmm... ??...

??so suppose that the automorphism is, for example, (x,y) |-> (-x,-y) ... ???

or also (x,y) |-> (-x,y) ???...

...and so forth ...

d(fg)=d(f)g(p)+f(q)d(g) ....

d(fg)=d(f)g(p)+f(-p)d(g) ... ??

d(f1)=d(f)+f(-p)d(1) ... ??

f = 1 ... ?? d(1) = d(1) + d(1) ... ??so d(1) = 0 ???

d("x^n" "x") = d("x^n") + "(-x)^n" d("x")

d("x^[n+1]") = d(x^[n]) + (-x)^n a ???

0
a
a - ax
a - ax + ax^2
a - ax + ax^2 - ax^3

???or something ??...

??hmmm... some idea that i had somewhere up above (and/or elsewhere...??...) seems confused now... backwards or something...

homomorphism from smooth functions on manifold m to upper-triangular 2x2 matrixes ... something about ... ??ext between skyscraper sheaves, or something ... ???...

??hmm, or_does_ it make some sense... ?? module of the algebra of smooth functions, with 2d underlying vector space ... ???and so forth, or something???...

??so what about something about ... ???stuff lawvere says somewhere about "wallman compactification" and so forth ???? ....

??something about ... ???boolean localic topos as spacial only in case of discrete space ... ???or something ??? ??is that what they said??? .... ??anyway, sort of seems to fit with some stuff ... ???something about "nonstandard analysis as special case of forcing" , so to speak .... ???? ... seems like "stone-czech compactification" (or something ... ??...) in this case, but ... ???....

??hmm, wpa says:

For normal spaces, the Wallman compactification is essentially the same as the Stone–Čech compactification.

???.....

??hmmm, so what about idea that ... ???maybe [bit about how to get plain old "2-valued" model from boolean-valued model ...??? ...] shows that we're right about sa... ???completeness of [the boolean-algebra over which a boolean-valued model lives] as being a red herring ... ???or something ....

??class of regular opens as closed under infimum of ordinary opens... which in finite case is just intersection ... ??or something ... ???....

??so what about double negation topos of localic topos corresponding to stone space?? ... ??for example cantor space ... ?? and so forth ... ???...

??relevance to conjecture about "t-model with every t-model morphism out of it an elementary equivalence" .... ???...

confusion ... ??...

regularly open sets in a top space form.... ???what??? ... ???a complete boolean algebra??? ... ??or what ???...


consider for example 1-point compactification of natural numbers...

??regular open =?= ... ??? finite not containing infinity, or complement thereof?? ...

(hmm, i guess that below we refute this... or something ... ???so some of the confusion here had to do with... ??not noticing that ... ???there's a whole lot of infinite but not cofinite clopen sets here??? ???or something ??? no, wait, that still seems wrong... still lots of confusion here ....??? ... ??what about whether clopen is equivalent to regular open here ?????..... ??something about ... if a subset is neither finite nor cofinite, then it's open xor closed... depending precisely on whether it contains infinity ... ???or something ???.... ??and thus not clopen??? )


supremum of the finite regular opens = ... ???non-existent ????

(note added later: not sure that makes any sense even on its own mistaken terms ... ???or something ???... )

hmmm.... ???....

?????.....

??closed = finite or contains infinity??

??open = cofinite or lacks infinity??

??every closed set x here is the closure of an open set ??? ???namely of "x minus infinity" ??.... ??no, wait... a finite set containing infinity is not the closure of x minus infinity ... in fact is not the closure of any open set... neither of a cofinite set (pretty obviously...) nor of a set lacking infinity (almost as obviously...)....

??so "closure of open" here = "finite xor containing infinity" ??? ??...

??so then is it true that the double negation topos here is the localic topos correponding to ... ???the (discrete) subspace obtained by removing infinity ??? or something ????.... hmmmm.... ????what about something about "perfect" (or something) here ???? .....


??????what about boolean algebra of "finite or cofinite subsets of the natural numbers" ???.....

???so does 1-point compactification of natural numbers have lots of regular opens that aren't clopen???

for example, consider the evens not including infinity ... the closure is the evens including infinity... the interior of that is ... ??the evens not including infinity ... so it seems regular open ... but certainly not clopen, right ???... because not closed ... ???

???at the moment seems really stupid to have imagined that for stone spaces clopen might be equivalent to regular open ... ???did we really imagine it then?? ???perhaps yes?? .... ??and could this explain some bad recurring confusion that we had ??? or something ????.....


??so maybe ... ???the clopens in a stone space form "the" boolean algebra .... ???not necessarily complete .... ??and the regular opens form ... ???a complete boolean algebra .... ???which is the global points of the subobject classifier in the double negation topos of the localic topos corresponding to the stone space ??? ????or something ??? is that right ???....

??regular opens as maybe obviously co-/complete by gabriel-ulmer stuff (or something... and so forth ... ???) ??something about ... any lawvere-tierney topology .... ???....

??in e-mail to toby from around january 2010 i seem to be suggesting stuff like "in a stone locale (??...), regular open implies clopen" or something ... ??and also talking about something called "stone algebra" and / or "stone lattice" ....

hmmm, some interesting e-mail to todd from around january 2010 ... on stuff like :

as for the more general question of whether the cokernel of a
dualizable object by a dualizable subobject should be dualizable, i'm
beginning to wonder whether it might even be possible to find examples
"in nature" (in some loose sense) where this fails.

[end of quote]


and so forth ....

??hmm, but also (???less impressiely ... ??or something...) .... :

i suspect now that it _is_ easy within the original doctrine to show
that the cokernel of an n-dimensional object by an m-dimensional
subobject is [n-m]-dimensional, by using "categorified arithmetic of
chern classes" or something like that. (and this probably should have
been "obvious" in some sense.)

[end of quote]

??so what about double negation topos of localic topos corresponding to stone space?? ... ??for example cantor space ... ?? and so forth ... ???...

??express "unique path-lifting" (or something...) property of "covering projection" by poullback diagram ... involving path spaces (using "interval object" or something ...) and evaluation at first endpoint ... ??or something ... and so forth ... ??...

??david ullrich seems to suggest thinking about lifting of path-homotopy along these lines... ??or something ...

??what about something about gabriel-ulmer approach to grothendieck topology (which approach we've already used a bit, in for example thinking about double-negation topology for "decidable toset" ...and so forth, or something ...) in understanding something about ... ??creation of pointless boolean locale from pointful one ... ??or something???? ... and so forth ... ??...

??still things that confuse me here ... ???....

??something about locally presentable categories, and mere right adjoints between them vs those preserving filtered colimits, or something ... ??and why i don't see something like that showing up in stone space context, or something ... ????.... ????...

?? "derivation at a pair (p,q) of points" ... generalizing "derivation at a point p" ...

d(fg) = d(f)g(p)+f(p)d(g)

d(fg) = d(f)g(p)+f(q)d(g) ??? ... ??or something ?? ...

??for example d(f) := f(q)-f(p) ??...

d(fg) = f(q)g(q)-f(p)g(p) =?= (f(q)-f(p))g(p)+f(q)(g(q)-g(p)) = d(f)g(p)+f(q)d(g)

??seems correct ??? ...

??what about something about ... ??"every t-model morphism out of it is an isomorphism" ?? ... ???or something ... ???... ??something about ... of a flat presheaf ... ???or something ???....

hmm... ??with this "existence and equality predicates" business, seems like... ???the "actual universe" is a "subquotient" ofthe "nominal universe" ?? or something ??? ??does that make any sense ???....

??well with a general map, neither domain nor co-domain accounts for all the stuff, right?? ... both contribute some ... ??? or something ???...

hmmm...

??what about something... ??functor from boolean algebra to subquotients of nominal universe ... ??or something?? ...??...

??what about some stuff here reminding us of ... ??some semi-recent stuff about zariski topology for boolean ring ?? ... or something ... ??where was that?? ... ??notebook358, or something ??... (maybe right around p 207, which is where i just happened to open to ... ??..) ... mixed in with some stuff that didn't make that much sense, i think, but .... ??... (??something about ... ???apparently incorrect idea that i had about ... ??some sort of "stalk-wise" version of booleanness of a heyting algebra... ??? or something .... ????....) ... hmm, but stuff around p 207 does seem surprisingly close to stuff we're trying now ... or something ... then weird maybe interesting stuff before that ... ???...

??something about ....???promoting an idempotent to invertible as promoting it to 1 ... ??or something ???....


??open sets of a sober space form _what_ kind of frame ??? ...

??something about ultrametric spaces ... ??? or something??? ...

??so what _about_ relationship between two (??...) aspects of ... ??whether boolean algebra of "boolean-valued model" (or something ...) is co-/complete ... ??...

1 ??something about quantifiers ... extending [assignment of truth values to generators] to quantified statements as well ... ??or something ....

2 ??something about ... ???pretopos vs topos, and stone space vs boolean locale, and so forth ... ???....

??so... let x be a boolean algebra ... and y a stone open for it ... ??construed as the canonical "set with x-valued existence and equality predicates" obtained from it ... ??...

??something about ... ???"model of [the geometric theory of a truth-value] over x, with existence realized as x-genuine existence and equality as x-genuine equality" ... ??or something ... ???something about lawveresque "quantifiers as adjoints to substitution" approach .... ??something about case x = {true,false} ... ???.... ??as maybe only case where "genuineness" constraint is non-vacuous ??? .... ??or something ???... ??or maybe what i mean by that is that ...??x-genuineness means just that ... ???? "if exist(s1)=exist(s2) then equal(s1,s2)=exist(s1) precisely in case s1=s2" (??or something ... ??...) ... and ... ??? ?? ???or something ???....

??does that (??...) make any sense ????.....

??so what _about_ looking at non-canonical realizations here ??? ...

??something about ... "existence" : s -> x .... ???.... as being monic in canonical realization of stone open y ... ??thus identifying y with certain sort of subset of x ??? ... ???or something ???....

??so let s be a subset of x st .... ???what ???...

???something about ... ??the clopens contained in an open ... ???....

?? ... stone open ... "b-valued existence predicate, tw b-valued equality predicate as coarse as reasonable relative to the existence predicate" ... ???something about subobject classifier and injective resolution ... ??? ... and so forth ... ???....

??lots of reasonable stuff ...

de-enrichment.... factorization ...

t-enriched topos ...

t-big limits ...

t-elementary equivalence ... verging on automorphism in what limit exactly???

k1,k2-pretopos ... ??when subobject classifier materializes???

canonical model of t over it's boolean alg of nullary predicates, for t "classical first-order" os... asf os ...

???...asf .... ????....


??something about ... ??? "[x=y] <= [x exists]" ... ????or something ... ???

??what about ... property of being canonical "set with b-valued existence and equality predicates" corresponding to given stone sheaf" ... ???including case of subobject of 1 ... ??elegant treatment?? ... something about subobject classifier and injective resolution and so forth ... ??...

??so what about whether "nonstandard analysis" (or something...) can be thought of as "forcing with discrete poset of forcing conditions" ?? ... or something ... ?? ??also whether this is essentially what lawvere says ... ???....

so consider "natural structure of tangent space of identity element of coalgebraic monoid" ... or something ... but then something about version where -1 is not assumed ... ???....

??so... ??what about ... N-modules and/or "P"-modules as sort of "cones", and relationship to something about "tangent cones" and so forth ... ??....

???also something about "affine modules" ... ???and / or "projective modules" ???? .... ??and so forth ... ???....

??so let x be a boolean algebra, and consider the category where an object is "a set with x-valued existence and equality predicates" ... ???or something ???... ??maybe some sort of localization here ??? .... ????....

??so given such a set, consider ... ???assigning to each element x1 of x the set obtained by "promoting x1 to true" ... ???or something ... ???....

??relationship to sheaf over corresponding stone space ?? ... ??... and so forth ... ???something about "boolean-valued model", or _something_ ??? ....

??so what about going the other way?? ... starting with sheaf over stone space (or something...) and obtaining "set with x-valued existence and equality predicates" ... ???in maybe obvious way??? ...???something about ... vague feeling about ... ???something about ... ??"injective" or something??? ... something about "injective resolution" ??? ... ?????or something???? .... ???what about _heyting_ algebra case ??? ... or something ... ??and so forth ... ???...

??so what about "isomorphism" as special case and/or limiting case of "k-elementary equivalence" ??? ....

??"boolean-valued model" ... ???....

something about ... ??"t-model in not-necessarily-coherent boolean topos" ... ??vs ... ???something about boolean algebra ...

??"boolean space" (or something... "stone space" ...) vs "boolean locale" ... ???....

??something about which of these is relevant to "forcing" ... ??and so forth ...

???something about nonstandard analysis ... "ultrapower" as involving adjoining of generic element, sort of...

???bit about ... "ultrapower as still elementarily equivalent to original" ... ??though something about slice category... extra constant ... ??? ... anyway, vs something about "boolean-valued model" and "forcing" ... or something ...

??something about ... with "boolean-valued model", is "the boolean algebra" in question just an external (or something...) such, or is it an object in a topos (??such as the underlying topos of a sufficiently good epsilon-universe ?? ... ??or something???...) ... ???

??remembering some confusion (and so forth...) about something like this ...

??something about ... ??the truth-value object in a boolean topos ... such as a boolean localic topos ... ??

??what about something about ... ??instead of passing to double-negation topos, considering internal double-negation boolean algebra of truth-value heyting algebra ??? ... or something ... ??... and so forth ... ??...

??hmmm... wpa on "boolean-valued model" specifically says something about _complete_ boolean algebra .... ??? .... hmmm....

??so what _about_ non-standard analysis (or something... and so forth ...) as involving a different sort of "boolean-valued model", based on just plain boolean algebra ??? ... ???or something ??? .... ??or is that really a good way of thinking about it ... ??? ... not sure ... ???...

hmm ... ??...something about ... maybe actually ... ??completely distributive boolean algebra... simply power set boolean algebra ... ??but what should we think of it as "being treated as" ?? ... ??boolean algebra... ??complete (or something...) boolean algebra ... ??completely distributive boolean algebra ... ?? ....

??for one thing, is the "boolean-valued model" supposed to preserve infinite disjuctions??? ... or something ... ??wouldn't particularly expect to see such a requirement (??or maybe it's obvious that we won't see such a requirement?? ... ??would be too restrictive ???), but then why bother mentioning completeness??? ... or something ... ??? yes, could have to do with "examples showing up in practice" or something ... but... still confusion ...

????hmmmm, they say this:

"The completeness of the Boolean algebra is required to define truth values for quantified formulas."

...really have to think about that ... not at all sure that i get it yet .... ????????.....

well, i sort of get where they're coming from (was trying not to read their detailed discussion yet but accidentally saw word "supremum" ...), but i still have to think about it a lot ... ??.....

??something about ... ???attitude of something about "interpretation preserves certain relationships", vs "interpretation is given by values at generators, with generating process needed in codomain to flesh it out ..." ...??or something ... need to think about this ... is there really some nice lawveresque way of exploiting adjoint functor interpretation of quantifiers here ... and so forth ... ????....

??something about idea of "coherent boolean topos" approach to classical first-order theories ... ???as making it seem like completeness isn't very relevant here (??...) ... ???or something ???...

??something about ... ??"quantifier as infinitary operation vs a unary" and something about "actual variable vs formal variable" .... ???or something??? ....

??double negation topos of presheaf topos on exponent N vs N^op ...

??in both cases you get the terminal topos, but in N case the model is an "ideal model" corresponding to +infinity, vs in N^op case it's "genuine model" corresponding to 0 ... ??or something ???...

??that is... ??"taking colimit" in N case, vs "evaluation at 0" in N^op case ...

??so consider ... ??"forcing x to embed into 2^[natural numbers]" ... ???or something ...

x >-> 2^N ...

x X N -> 2 ...

??...

??so what _about_ relationship / interaction between various (??partial ...???) "booleanization" processes ... namely, "double-negation topology" and ... ???adjoining of complements ... ??also "keeping only the complemented things", i suppose ... ??what _about_ weirdness of "double-negation" process... ??as not an adjoint functor in any very obvious way ... ??or something ...

something about examples that we've been considering where first we throw in some complements, then take double-negation topology ... ??... ??roughly, "because if you didn't throw in those complements then the double-negation process would be too destructive... too many "collapse morphisms" ..." ....???...

major topics to talk about with todd next time...

1 "toric"-related doctrines ...

2 catalog of toposes arising in algebraic geometry ...

??something about ... full-and-faithfulness questions ... technical meaning of "quasicoherent" and so forth ... ???"structure/semantics" ... "stacks" ... ... and so forth ... ?????.....


3 points along "kz" spectrum ... ??something about "epistemological collapse" ... ??something about when morphisms are automatically homomorphisms ("being a homomorphism as vacuous property" ...)... and so forth ... ??something about case of semilattice monad on _poset_ ...

??minor topics to mention...?? ...

??"topos-building operations" ... "minimal syntax" ... ??"t-geometric doctrine" ... ??"t-ag geometric doctrine" (??not to be confised with "tag doctrine"; see "toric" above ...) .... ??....

??"forcing semantics" ... ??point behind it ... ??it = trying to make connection to "forcing" ...

double-negation toposes and so forth ... forcing ... probabilistic stuff ... rado graph, dense order ....

??something about morphism of ag theories from finite-group-like to scheme-like ... ???and so forth .... ???....

??so what about "boolean locale" corresponding to ... ???"cocomplete boolean algebra"?? ... ??or something ???... ??what about such coming from a top space, for example an alexandroff space ?? ....

??something about "coherence" here... ??something about lots of examples that we have where double negation topos of a coherent topos seems to be coherent ... ??but then there should be lots of examples where it's not, right?? ...??what are some nice prototypical examples of this ??? ...

??regular opens in a top space (??or something??) as forming what ... ??... regularization as adjoint ... ??? or something ??? ....

??"boolean frame" ... ???...

?regularly closed subsets of one-point compactification of natural numbers ...

??something about... ??what's going on with forcing ... ???... ??boolean frame?? ... ??poset of forcing conditions ...???coherence ?? ...

??so what _about_ ... ??point of talking about general idea of "forcing semantics" ??? ...

??so what _about_ something about ... "freely distributively adjoining colimits up to a certain nice cardinal, then completely freely adjoining them up to another certani nice cardinal" ... ??and so forth ...

so let's consider boolean algebra coming from a poset ... ?"regularization of a filter" or something??...

??what about relationship between .... [???lawvere's ideas about topos where object is space, vs topos which is space ... ?? ...??and so forth ???....] and [something about ... confusion (and so forth ...) between "stacks" and "algebraic stacks" ... ??? ??and so forth ???...] ??? ... ??

??so... the sort of topos that lawvere seems to want to think of as having its objects be spaces ...??how do _i_ think of its objects ??? ... ??as formulas of geometric theory ... ??? ... hmmm... ??but something about ... special case ... rather boring theory ... ??or something???... ??hmm, maybe only "relatively boring" ... ???.... ???theory of local comm ring ... ???...

"the ring" ...

"the ring squared" ...

"the invertible elements" ...

hmmm...

??something about "the functorial viewpoint" ... ???....

??something about... "turning the yoneda crank" ... ??...

???hmm, what about decategorified analog "improving t-algebra to frame" ... ??? or something??? ...

??so let x be a commutative ring...

now consider cosimplicial ring where .... ???...

??so what _about_ property (or something .... ???...) of grothendieck topos of inverse image functor from _set_ being logical?? ...

??includes "2-valued" or something ??? ... (??or _is_ that correct ??? ...) ???anything else ??? .... ???....

(1+....+1)^(1+...+1) ... ???... ???something about 1^1 ... ???....

?? _set_^(1+1) ... ??...


??so what about ... ???doctrine (or something ...) of "geometric theories over" an elementary topos ?? ... or something ... ??and so forth ?? ... also ag theories over elementary topos, or something ... ???...

??something about subcanonicalness (or something ...) of double negation topology on presheaf topos and monicness of all morphisms in exponent ?? .... or something ...

??what about other correspondences (or something ...) here ?? ??something about ... arbitrary double negation sheaf as sum of representables ... ??or something ... ???...

??"boolean-valued model ..." ... ???...

something about "every morphism out of a double-negation model is an "elementary equivalence"" ??? ... or something ... ??? ??and characterizing double-negation models that way, or something ... ??....




"elementary topos as set theory... ???equipped with generic model of ... " ... ???or something ??? (??does that mean that ... ??a logical morphism (or something...) between elementary toposes is something like a morphism of set theory models, picking out in the codomain model a model of a first-order theory correposnding tothe domain ... ???or something??? ...) ...and so forth??? ... ??so what _about_ _elementary_ topos theory as all about "forcing" ??? .... ... or something .... ???....

??so what _about_ factoring geometric morphisms into logical and "anti-logical" part ??? .... or something ... ???....

??so what _about_ something about ... stuff lawvere says about something about "big vs small topos" (or something .... ???) and something about ... ???certain big topos as slice topos of ... other big topos ... ???or something ???? .... and so forth ... ???.... slice topos of "big topos" over object corresponding to some "space" as big topos of that space ... ???or something ??? .... and so forth .... ????....

??so what about quasitopos vs topos in connection with doctrine morphism from tag to g ?? ...

??and what about this doctrine morphism, vs one to "g slice commutative monoid" ?? ...
and so forth ... ??something about (2,1)-full-and-faithfulness here??? ... and so forth ... ???...

so let x = k[a,b]/ab=1 ... then consider comm x-alg x[c]/c^2=a ... ???or something ...

???consider underlying module ... ???or something ???....

??something about basis 1,c ... ???so free module ??? .... or something ?? ...

hmmm....

??hmm, so what about morphisms between "self-inverse objects" in an ag theory?? ... vs between just ordinary invertible objects ... ???or something ... and so forth .... ????.....

??so what about whether there's a whole dimensional category (or something like that...) of self-inverse objects, in general ??? .... and so forth ...

??what about relationship of self-inverse objects to something about "real" and/or "bosonic", and so forth ??? ....

??what about something about ... ??possibility of ... ??interpreting "universal coefficients theorem" (or something... and so forth ...) in terms of ... ???something like ... ?property vs structure vs stuff (and so forth ...) analysis of things like "self-inverse object" ?? ... ???...

??so what about z/2-graded comm alg here ?? ... ...and so forth ... ???..

??hmm... ??what about strictness of commutativity here ?? ... ??? ... and so forth ... ???....

??so what _about_ laurent polynomials graded by Z/p according to residue of exponent ?? ...??what _is_ going on here ???? ....

??so what about removing multiplie points from affine line ?? ... ??...

??what about something about divisors and so forth here ?? ....

??so what about double negation topology on commutative ring classifier ?? ... ????.... ??hmm, perhaps trivial (??os??) because of exponent having terminal object ?? ??or something ?? ... ??hmmm, so what about sa old-fashioned custom of defining commutative ring to be non-empty ??? ... ??or something ...

??so what about something about ???random total order on natural numbers being dense and bi-unbounded ?? ... or something ... ??analogous to erdos-renyi or something???...

??what about "continuous geometry" here ?? ... or something ... long-shot ...

??so what about double-negation topology of presheaf topos on _filtered_ (or something ... ??...) site ?? ... and so forth ... ??... ??as "atomic", or something??

??what about ... ??something todd mentioned ... classical models of presheaf topos as forming free filtered colimit completion of exponent, or something?? ... ??relationship to ... ??diaconescu's theorem ... and gabriel-ulmer duality, and so forth ... ???....

what about stuff that lawvere says about ... ??non-standard analysis vs forcing ...
??something about set vs poset .... ???.... ??something about bit about "...as cohen showed... destroyed by passage ... even though "elementary" in technical sense ..." ... ???... ??also something about using forcing to show independence (or something...) of axiom of constructibility as involving poset of forcing conditions given by "basic open sets in cantor space", while for continuum hypothesis it's "basic open sets of a big generalized cantor space" ... ??or something ... (??something about site category and basis (??and/or sub-basis ??...) ...something abotu scott and solovay ... ??... measurable functions modulo sets of measure zero (or something...) ... ???....

???hmm, so what _about_ something about ?? ... poset of forcing conditions ...??maybe filtered, or something??? .... or what .... ???... and so forth ... ??...

??so what _about_ (2,1) (or something...) (??partial ?? or something??...) full-and-faithfulness of certain way of getting stacks over some site ... ??and how this might relate to full-and-faithfulness of way of getting t-modeled topos for certain geometric theory t ?? ... ??and so forth ...

??something about ...??possible weirdness of doctrine interpretation being (2,1)-full-and-faithful on large class of theories ... ??or something ??... ??except something about ...??when codomain (??or something...) doctrine is deliberately tailored to this ... ???.... hmmm... ??what about decategorified (??or something??) analog here, maybe ?? ...

??what about how "localness of hom between local rings" issue plays out in "stack" context ?? ??or something??? ... and so forth ... ???hmmm, or is this just a "small (zariski? ...) topos" issue in the first place?? ... or something ... ??...

??so are we saying that there's hope (??modulo localness of hom issue ...??) for arbitrary module in etale (or something...) ringed topos to correspond to object in original ag theory ?? ...?? or something ??? ....

??so this example of theory of dense unbounded linear orders arising as double negation theory seems to involve pun on all possible meanings of "dense" ... ??so did lawvere-tierney get terminology "dense" from classical double-negation case???

??what about double negation aspect of rado graph, and something about ... ??probabilistic aspect ... ????.....

[from 2010/12/5]

was talking to john huerta about g2 and related stuff this evening...

got to wondering about... hmm, this idea is changing as i'm trying to describe it... i guess that what i'm really saying is... consider a g2 "line" as a projective line...

i guess that what i'm really saying has something to do with blowing up the basepoint singularity of the 2d schubert variety at a point of the g2 "line" grassmanian... ??or something like that??... ??and it's getting blown up into its projective line of g2 points?? or something like that??...

so... since the "base of the crown" consists of four dots, that should mean that we're interested in the line bundle over the projective line bundle whose space of sections is four-dimensional... ??or something like that?? ??and we're interested in viewing that four-dimensional space of sections as a representation of the g2 line stabilizer... ??or something like that??...

??so what _is_ this general idea that we're exploring here?? ... in the context of something like blowing up the basepoint singularity of an arbitrary schubert variety ... ???or something like that???....

at first i didn't even realize that this curve that i was getting had to be the g2 line itself... ???.... i was imagining that it might be some other weird curve... ???.... for example an elliptic curve... and then i was contemplating perhaps watching the elliptic curve vary as the ratio of the radius of the rolling ball to that of the stationary ball varies... or something like that... ??seems like we should still try to get some sort of variation like that going here?? ...hmm, but maybe that's problematic... in ways that huerta was hinting at... that for arbitrary values of the ratio you don't even get the rolling trajectory to nicely close up... and he was also hinting at some sort of "quantization condition" being involved in getting thw rolling trajectory to in fact close up... and this seems very suggestive now... that there's some sort of continuous variation going on here, but that what's really crucial is some sort of discrete variation related to that in a "quantization" way... ??or something like that?? ... have to try to work this out...

is there something going on here about ... ??interpreting parabolic subgroups as automorphism groups (involving both base and fiber... morally like a wreath product or something???....) of certain natural vector bundles, or something like that??? ???something about... ???the mysterious or at least confusing "extra stuff" on which a parabolic subgroup acts?? ??or something like that??....
??something about ... "quasi-tautological bundle over partial flag variety" ... ???or something??? hmmm.... ???and so forth.... ???.....

so how does this (??...) fit into the whole "invariant distribution" game?? ... or something... ???...

one of the themes of the discussion was that john wanted to talk a lot about the 2d schubert variety (and its infinitesimalization at the basepoint) on the g2 point grassmanian while i wanted to talk a lot about the 2d schubert variety (and its infinitesimalization at the basepoint) on the g2 line grassmanian... and it took us a while to remember that these are dual to each other and to contemplate exploiting that duality in certain ways... so maybe i should consciously try applying that duality here now... so... ??what's going on here?? are we getting some alleged 3d rep of the point stabilizer?? ... and so forth... ??seems like we're getting some pretty vanilla line bundle over the "dual" projective line of a g2 point... the g2 projective lines through it... ??....

??so what about "structural" / "axiomatic" (or something ...) description of cartanian geometry modeled on particular g/h ??? ...

??seems like g-torsoroid that's h-flat and k-flat _is_ [h sup k]-flat ... ???so does that screw up ideas about "natural geodesics" ??? ... ... and so forth ...

??what about ... ??getting graded nilpotent lie algebra by ... somehow ... ???starting with "bi-cartanian geometry" ... meaning reciprocal relationship between kleinian geometries sharing same stabilizer rep on tangent space ... ???or something .... ???relationship to "associated graded ..." ??? ??or something ??? ....

??what about "composition of rolling relationships" ??? ... or something ... ??relationship to some sort of span composition ??? ... ???? ... and so forth ... ??...

??so what _about_ "rolling one geometry on another" vs creating a new geometry this way ... ???or something ??? ....

??what about something about ... ??g-torsoroid that's h-flat, where downward normalization of f is non-trivial?? ... or something ... ??as peculiar because of ... ??usual bit about "making the g/h picture viable" as weird ... ??for example consider case g = h ... ?? ...???...

??so what about ... ??for example, non-geodesic light-like curves in ordinary "flat conformal space-time" (or something...) vs ... ??non-[conformally flat] spacetime ... ?? ... ??or something ... ???...

??so what about trying to use 1+1 penrose diagram to understand 2+1 flat conformal spacetime by "spacial radial symmetry" idea ?? ...

toric... ??sa doctrine morphisms ... asf os... for todd ...

rado... ??explicit description of polyadic boolean algebra ??? ???vector space objects ... ???and so forth??? ... ??wa sa where lies on "classification" spectrum ??? ...???os???.....

contact.... ???sa whether arbitrary contact manifold can be thought of as parabolic cartanian geometry in the hopefully obvious ways ??? .... os... asf os... ????.... ????sa "natural geodesic" asf ... os... ??which leads to sa ... :

"bi-flat" .... ??sa simply... ???g-torsoroid that's both h-flat and k-flat ... ???os??? ... asf os... ???.... hmmm.... ????? ... ???sa whether this implies [h sup k]-flat, os, asf os... ???also sa "generalized rolling" and "hamster ball" and so forth ... ???...


kenji ... ??sa ... ??"combinatorics"... and ... ??"human relations" ... ??applications of symmetry ... ???os... asf os... ???.... ??sa "symmetry vs structure" ... ???os... asf os... ???...

todd...

kz vs idempotent ... ??morphisms between algebras...

??something about whether "collapse" of epistemology amounts to kz becoming idempotent ... ???or something ...

??something about what goes wrong with excessively naive attempts to get good "epistemology" ...

toric stuff ... ??on wish list ??...

??something about getting beyond "foundational" aspects of ag .. or at least to somewhat different sense of "foundational" ... or something ...

??something about validity of technique i've been using in ... ??"imposing colimit axioms" ... sometimes in context of doctrine with other ( (2,1) ...) structure with compatibility with colimits ... and so forth ...) ... ??something about "coherent topos" case ... ??or something ?? ... and so forth ... ??something about "compact object" vs "finitely presented" and so forth .... ?? ... ??sa bit martin seemed surprised about ... ??full and faithful right adjoint ??? ... or something ... and so forth ...


derek...

???something about ... ???integrable invariant distributions on flag variety... also on apartment variety ... ???and so forth ...

...not sure i said that right ... something about whole bit about "relative invariance" ...

but anyway ... something about nilpotent / graded lie algebras and non-integrable distributions and polynomial exponential map and affine structure of bruhat cells and so forth ... in particular how flat space-time fits into its conformal completion ...

??somewhat perverseness of ... penrose diagrams as applied to globally non-trivial space-times ... vs current focus on merely "kleinian" as opposed to "cartanian" aspect of conformal space-time ... ???so then what _about_ more cartanian approach here?? ... ??something about "generalized legendrian submanifold" and so forth ??? ... ???something about "geodesic light-like curve" and so forth?? ???what _about_ "geodesicness (or something... and so forth ... higher dimension ... ???) of curve (or higher dimension ...) respecting a distribution" ... ??purely intrinsic to the distribution ... ??or something ?? .... and so forth .... ??? ....
??something about "cartanian duality" ...

??hmmm, so what about something about "geodesicness" (and so forth...) here and something about ... ???when torsoroid admits various different "figure-space pictures" ... or something...

??something about "hidden heisenberg" and so forth ... ??

??so what _about_ something about "conformal non-/flatness" here?? ... and so forth ... ???

??maybe also mention cartanian stuff here to huerta ? ...

??also something about ... further detail about how (??d2 and/or b2?? ...) apartment fits into classic penrose diagram ... ??well, i guess that that's part of stuff that we already mentioned above ... about using polynomial exponential map to understand how flat space-time fits into its conformal completion and so forth ... but maybe worth emphasizing as being a particularly simple thing to try to straighten out... just by drawing picture superimposed on vanilla penrose diagram ...

all right, i'd better try to read and understand this "caution" in hartshorne ...

Caution 5.13.1. If S is a graded ring which is not a polynomial ring, then it
is not true in general that F^(?x) = S (Ex. 5.14).

??might be ok ??? ... ??the "sheafification" (or something...) process might affect the unit module itself ?? ??or something ?? ??try to come up with an example ??? ....

what about the vague idea of "lie groupoid for which its lie algebroid is a poor invariant because the lie algebroid is the linear hull of a more constrained tangent cone ..." ?? ... or something ...

??light-like curve in 2+1 conformal spacetime ... ???... ???something about ... ???nails and gimbal moving simultaneously... instantaneous swiveling, and instantaneous

??arrow of time on gimbal .... ??helping to visualize cone shape ... ??or something ??

???something about ... for a particular event, "time" as the tangent space of the gimbal at the nail(s??) (??which maybe we can also sort of canonically (???) identify with the "most distant" diameter of the gimbal ?? ... ???hmm, what about whether certain identification here is conformal ... ???or something ??....) and "space" as the tangent space of the globe at the nail ... ???or something??? ....

??ideal class group of the laurent polynomials ...

??ideal class group of Z[1/3] ... ???...

??something about "branch point" here ??? .... ????....

???ideal class group of Q ... ????....

??absolute galois group of Q ... ????....

ideal class group of ... ???...




??divisor class group of dense open affine subvariety of elliptic curve ... ??...

??"zariski topology hasn't got enough open sets... use coverings instead ..." ... ???or something ... ???....

??so what's an "idempotent monoid" ?? ??monoid x where ... ???multiplication is isomorphism from x tensor x to x ... ???or something ???... ???... and so forth ...

??walking idempotent monoid ... ??? ... and so forth ... ???....

??idempotent monoids in monoidal category of abelian groups under tensor product, for example ... ??being a module of such a ring as a mere property ... ???or something?? ...

??so what _about_ relationship between [??indications that we're maybe seeing of relationship between "double-negation topology" and "generic" ... or something ... for example with "decidable toset" ... ??...] and [... ?? something about concept of "generic" in "forcing" context ...?? or somethihng ... ???....] ???

????bunch of side issues here ... ??? ... ??what about something about "baire category ..." vs "open dense ..." and/or "regular open" ... ???... or something .... ??? ... ??and so forth ... ? ...

(for martin)

hi... let me mention one thing that confused me a bit...

??hmmm, let me think outloud about this a bit ...

inverting a morphism in an ag theory which is not a coequalizer ...

??hmmm, not clear how to include case of removing higher codimension variety here ... ??or something?? ... ?????hmmm, what about idea of ... ???removing by [blowing up, followed by removal by localization] ?? ... ??or something ?? ... ????hmmm, any relationship to "adding extra stuff" ??? ... hmmm... not clear ... ???? .... because we only seem to be adding a line object under this scenario ... ??or something ???...

??what about something about top exterior power here, or something ?? ...

??something about funny tradeoff, or something ?? ??extra stuff, for being able to remove higher co-dimension subvariety ... ??or something ?? ...

anyway, back to ... ???something about ... ??"trying to force at least one of a pair of functions to take on an invertible value" ... ??? ??or something ???....

??ideal power filtration ... ???....

?? "extra line object with structure amounting to a canceling scale except on a closed (and perhaps higher co-dimension ...) subvariety " ... ??...

???hmm, might there be an "infinite regress" problem here ??? ??or something ??? ... ??each new line bundle that you introduce as requiring fixing ... ??is this just a silly way of pushing the problem over the horizon, or what ??...

well, so let's try it anyway... ??maybe to try to find out whether there's an infinite regress or something ...

polynomials in 2 grade 0 variables x,y ... ??...

??then consider ideal power filtration for
... ??...

??something about simply ... ???adjoining inverse for underlying module of ideal ??? ... ??or something ???? .... ...?? we're supposed to know all about thsi already, right ?? ... ???...

??"a pair f,g of functions equipped with an inverse for the cokernel of (f,g) : 1 -> 1+1" ... ??? or something ... and so forth ....

??i'm getting a bit confused about what this is supposed to do ... blow up a sub-variety, or remove it, or what ... ???

??also are we taking advantage of special aspect of 2d case here ??? ... and so forth ... ??...

???something about inverse for _ideal_ .... ???hmm, so how do you get your handson that??? ... and ... ???this as probably pretty much how martin solved the problem ??? ... ???or something .... ??? ....

??so... if adjoining inverse ideal is supposed to remove the sub-variety (or something...), then what _about_ the other thing that's just supposed to blow it up ?? ... well, so it's got something to do with "taking the ideal power filtration and interpreting the filtered object _as_ a graded object with extra structure ... " ... ???but there must be a way more conceptual way of thinking about it ... that at least on good days we must already have sort of understood ... ??...

??something about ... ???"inverting the inclusion-in-next-stage operator" (?? =?= "scaling parameter" ??? ... =?= "scale" ??? or something?? ...)... ???as ... giving the "scaling-deformed" object ??? ... ???and so forth ... ???... ??what about inverting vs setting to 1 here ??? or something??? hmm, maybe sort of doesn't parse, or something... or vacuous.... something about inverting it as what allows you to "pretend that it's 1" ... ???or something ... ???...

??what about something about "epsilon-treed topos" ... ??or something ... and so forth ...

??forcing ...

??snow-globe ... ??...

"model of set theory" vs "theory ..." ... ???...

??consider models of the category classifier... and morphisms between those models ... ??and then some sort of morphisms between those ???....

??something about natural transformation classifier and so forth??...

??what _about_ gray's "formal category theory" ?? ... and so forth ... ??...

??so what about relationship between "localization" and "local" ??? ... ???something about "extreme ..." ... ???...

notes for next discussion with todd

??lots of classifying topos examples...

???double negation topology for "decidable toset" or something?? ??also "decidable set" ... ?? ... ??something about "boolean (??or something...) object classifier" ??? .... ???so what _about_ double-negation topology for object classifier ??? .... and so forth ... ???so what _about_ conceptual meaning of "double-negation topology" ??? .... relationship to some kind of "generic", or _some_thing ?? ...

??inverse pair of real vector spaces ... ??... and so forth ...

??zariski ...large vs small ... ??"not a property of the carrier ring alone, but of its relationship to the model" ... ?? "good element" and so forth ... ??minor goal concerning "cuboquadratic algebra" ... ???or something ...

??_maybe_ also stuff about other doctrines ... ?? ... ??"abelian diaconescu's theorem" ... ??gabriel-ulmer duality ... ??as applied to g and / or ag doctrine ??? ... ??something about "compact object" vs finite colimit of such ... ??or something ?? ... and so forth ... ??...

??something about situation where double-negation topology is maybe particularly simple... "everything not outright false becomes true" ... ???or something ???.... ??something about when double-negation topos is "complete" ... ??or something ... ??? ... "trivial pi_0" or something ... ??...



??also, kz background to epistemology theory ... ???.... !!! ....??start with this ...

?? ... small zariski topos ... ?? condition on model that's _not_ just condition on carrier ring ... ??so what about, for example, "good element of commutative ring" ?? .... and so forth ... ?? ??"good invertible submodule of internalization of v over commutative ring" ... ???and so forth ... ??....

??so what about the geometric theory of "dense" intervals ?? or something...

??hmmm, not just putting lawvere-tierney topology on topos of simplicial sets ... ?? ...

??consider classifying topos for inverse pair of real vector spaces ... ???and so forth ... ???

??vs classifying ringed topos for inverse pair of modules ... ???and so forth ... ???

??what about "classifying topos for invers pair of real vector spaces, where real vector spaces are interpreted in terms of usual topology ..." ... ????

??... classifying topos for truth value ... ???....

??so what about interval objects in _space_ (or soemthing...), and projective plane objects in _space_, and so forth ... ????....

???so.... consider _set_^[ne fin toset, with injective order-preserving maps] ... ??? or something... ???.....

??so what about ... "compact object" vs "finite colimit of compact objects" ???... or something... ???and so forth ... ???....

???something about models of boolean (or something ...) toposes, vs models in boolean toposes ... ??? and so forth ... ???...

??something about "cantor orders" ... ??or something ...

morphism from "strictly ascending j-tuples" to "strictly ascending k-tuples" ... ???as given by order-preserving injection k -> j ... ????....

???so seems like we want _set_^_"decidable toset"_ ... ???but then just the projective objects, or something ??? ........

1 1 1 1 1 1 1 1 1 1 1 ...

0 1 2 3 4 5 6 7 8 9 10 ...

0 0 1 3 6 10 15 21 28 36 45 ...

0 0 0 1 4 10 20 ...

.
.
.




0 1 4 9 16 25 36 ... =

0 1 2 3 4 5 6 ... +

0 0 1 3 6 10 15 ... +

0 0 1 3 6 10 15 ...


???....

???hmm, so we visualize the building blocks here as simplexes, but with only the degeneracy maps available ??? ... ???or something ??? ....

??1-simplex squared as 2 2-simplexes plus the diagonal 1-simplex ??...

1 * 1 -> 1 + 2 + 2

1 <- 1+2+2 -> 1


j^k = ????




???presheaf topos as always coherent??? ???os??? with the compact (??or something...) objects being ... ??? or something??? ... and so forth ... ???...

??so what about ... ???a chain complex as a nice way of getting a vector space object from an interval object ... ???or something ???....

??so what about diaconescu's theorem in the simplicial set case ??? ??what's the flat thing corresponding to an interval object ?? or something?? ... ??is this something that we've thought about before ?? ... ??well for one thing, it's the "good cosimplicial object" ... with geometric realization preserving products ... or something ... and so forth ... ???...

??so what about the underlying "degeneracy set" of a simplicial set?? ... ??...

??degeneracy site maps to simplex site, or something?? ???so expect to be able to extract model of simplicial topos from model of degeneracy topos ... so expect geometric morphism from degeneracy to simplicial ... corresponding to algebraic morphism from simplicial to degeneracy ... ??? ...

??what about something about euler characteristic here???? ??or something ... ??? .... ???schanuel ... ????....

??so what about something about "boolean" boolean algebras ?? ... ??or something .... ??something about surjections between finite sets??? .... ???injective homomorphisms between finite boolean algebras ... ????...

??so what about for example underlying degeneracy set of nerve of Z/2 ?? ...

?? hmmm ... ???...

1 * (1 1 1 1 1 1 ...) +
(g-1) * (0 1 2 3 4 5 ...) +
(g-1)^2 * (0 0 1 3 6 10 ...) +
.
.
.

??so what about subobject classifier for degeneracy sets ??? ....

??hmm, maybe subobject classifier here is very big ???? ???? or something ???...

???hmmm, so what about something about ... ???degeneracy set consisting of ... ???two 0-simplexes whose entire degeneracy parts (or something ...) agree ??? .... hmmm .... ???...

???so what about something about ... simplicial set vs augmented simplicial set in some of above discussion ... ???...

???and so then what _about_ some sort of grothendieck topology maybe amounting to denseness or something ??? ....

???and what _about_ "booleanness" (or something ...) here ?? ....

???what about idea that ... ???grothendieck topology here might have suspiciously similar to ... ???something about ... ???what happens automatically as result of coming from augmented simplicial set ... ??? ... ???or something ??? ....

??hmm, so what about something about ... ?? "promote anything not absolutely false to true" ????? ?? or something ?? ...

??hmm, so what about idea that ... ??maybe it makes some sense that adding denseness axiom to theory of "decidable toset" causes booleanness (or something ...) as side effect ?? .... ????...

???what about relationship between "completeness" and "booleanness" of theories ... ??? or something??? .... ????....

??so what _about_ vector space objects in this topos ??? .... or something .... ????....

??so what about double negation topology here?? ... ???and what about something about "coherence" ??... ??extent to which "coherence" lives at truth-value level ??? ... or something ...

??so what _about_ lawvere-tierney topology associated with model ??? .... or something ... and so forth ... ??...

??relationship to "structure /semantics adjunction" ??? ....

??hmmm, so what about double-negation topology for "decidable set" ?? ... and so forth ... ??? ... what about "tannakian" take here ??? ....

??? ....

so i'm getting a bit confused here ... about locally finitely presentable categories, vs their opposites ... and stuff like that ... ???...

??consider a small finite limits theory t ... and consider its models in the not-so-small finite limits environment _set_ ... ?? ...

?? but also consider the free arbitrary limits theory on t ... ????....

let's try an example; say t is the theory of categories ... ??...

??so what is "the free arbitrary limits theory on t" like?? ... assuming that it actually makes sense ... ??does it make sense??? ...

let's see, it seems plausible that taking the contravariant set-valued functors on x which take a given class of cocones to limit cones amounts to taking the universal cocompletion of x in which those cocones are colimit cocones ... ??or something ??? ???something about case of x a kleisli category, or _something_ ??...

??consider "graphs with a unique edge" ... ?? =?= "bi-pointed sets" ??...

??hmm... "simplicial sets with a unique d-simplex" ... ????....

??well so, let's consider small categories ... ????as ... "freely adjoining colimits to the finitely presented (??os????) categories, except respecting the existing finite colimits" ... ???or something???

??ok, so i guess that i'm hopefully less confused now ... ???something about ... the locally finitely presentable category of models of a finite limits sketch can also be thought of as... ???the opposite of the free small limits theory on the
sketched finite limits theory ... ???or something ???.... ??rather than some sort of extra "duality twist" (or something...), as i was imagining for a bit ... ???

??so what about ... ?? "cocomplete category which is canonically the cocontinuous set-valued contravariant functors on it" ... ??or something ... ???something about "cocomplete category where representable pre-sheaf = pre-sheaf taking colimit cocones to limit cones" ... ???or something... so let's define a "continuous pre-sheaf" (on a small-ly cocomplete category???) to be one taking colimit cocones to limit cones ... ???then are the continuous pre-sheafs on a small-ly cocomplete category always precisely the representable ones??? ... ???maybe some "set-theoretic problems" even in just precisely formulating the question ??? ...

(for martin)

so let x be a symmetric monoidal cocomplete k-linear category and m : y -> z a morphism in x. then to universally require m = 0 (without disturbing the symmetric monoidal cocomplete k-linear structure) is equivalent to universally requiring 1 : z -> z to be its cokernel, which in turn is equivalent to universally requiring the quotient map z -> cok(m) to be invertible. and the universal symmetric monoidal cocontinuous k-linear functor in question can be explicitly constructed as the left adjoint reflector onto the full subcategory of x consisting of just those objects that "believe" that all of the tensor translates of the diagram y -> z -> z -> 0 are exact; where the arrows in the diagram are respectively m, 1, and 0; and where an object w "believes" that a diagram d is exact iff homming into w turns d into an exact diagram.

i think that i can prove the above claim, though as usual there are details that i need to check carefully. the general principle is that given a cocomplete category x, and given some class of cocones in x, the universal cocontinuous functor that takes the cocones in that class to colimit cocones is the left adjoint reflector onto the full subcategory consisting of those objects that "believe" that all of the cocones in that class are colimit cocones.

(i _think_ that that principle is true in general, without extra "set-theoretic" assumptions; but i'm not certain of it.)

so for example take x to be the graded modules of polynomials in n+1 degree 1 variables, and take m to be the projection from the unit graded module to its minimal quotient... this may give a useful way of thinking about the quasicoherent sheaves over projective space ...

i hope that i'm not making huge mistakes here... i'm pretty awake now, but still capable of making pretty big mistakes ...

??so... given an ag theory t and a morphism m : x -> y in t ... ??suppose that we "want the identity morphism of y to be the cokernel of m" ... ???or something ??...

???so... ???we want x -> y -> y to be a cokernel diagram ... but moreover also xz -> yz -> yz for any object z ... ??or something?? ...

??so consider those objects w st ... ?? ??the kernel of [xz,w] <- [yz,w] is the entire domain ... ??so that is, for any morphism yz -> w, xz -> yz -> w vanishes ... ?? ??or something ?? ...

??let's try an example ... modules of k[x] ... morphism -> ... ???object w st ... ????

??might be confused but not working out the way i think i want it to yet ... ???...

all right, so let's try pairs of vector spaces, and (1,1) -> (1,0) ...
object w = (a,b) that "thinks" that that map is zero, in the sense that ... ?? it looks like zero under homming into w ... ??so (a <- a, b <- 0) looks like the zero map ... ??so obviously a is zero ... ??or something ... ???so is this making sense ?? ... ??but so is the conceptual interpretation making sense here ?? ... ?? martin suggests that "modding out by an ideal" and "localizing wrt a function" should fit in here ... which seems sensible, except that i seem to keep getting confused when i think about it ... something seems backwards ... let's see... ideal (1,0) >-> (1,1) ... ???so i guess that i did have it backwards??? ??also composite function (1,1) -> (1,0) -> (1,1) ...

??so it seems good now ?? ... ???...

??so what about map from unit sheaf to skyscraper sheaf ??? ....

or ... map from unit graded module of [polynomials in n+1 degree 1 variables] to minimal quotient ... ?? or something ... ??hmm, or did i sort of screw up here ?? ... the coequalizer is zero ... inverting the map from the minimal quotient to zero ... making the identity map of the minimal quotient the coequalizer ... setting the original map to zero ... ???...

1+1 -> L -> cok(1+1 -> L) -> 0 ...

??something about setting cokernel map to zero vs setting cokernel object to zero ... ???maybe equivalent ??? ....

so did i mishear martin, or something ?? ... ???...

(for todd)

i think that you noticed that one of the first items on my wish list was "galois shapeshifters", so i might as well take a stab here at trying to say what i mean by that phrase... probably i don't have to bother warning that the ideas here are pretty uncertain in my mind ...

part of my experience of learning about topos theory was to hear the ritual recitation of stories about the role in its historical development of grothendieck's attempts to prove the weil conjectures...

(the ritual nature of this recitation is tied in with the "estrangement" that i've sometimes mentioned: "...an estrangement that i felt during my earlier and less successful experiences in trying to learn algebraic geometry. i’d heard rumors of something called “toposes” which according to an often-repeated story played a significant role in advances in algebraic geometry associated with alexander grothendieck and the proof of the “weil conjectures”. then i learned topos theory, especially through the inspiring (sometimes a little _too_ inspiring) lectures of bill lawvere, and although i found it beautiful and highly applicable it didn’t actually seem to help me much with learning algebraic geometry...")

so anyway, the question of "why do people make such a big deal about the weil conjectures?" has continued to bug me for a long time; and semi-recently, i've formulated a tentative answer: roughly speaking, the big deal is secretly because of what i call "galois shapeshifters" ...

roughly speaking, a "galois shapeshifter" for me is an algebraico-geometric object which lives over an algebraic number field k and is therefore subject to the action of the absolute galois group over k, and which "changes its shape" under that action. in particular, the plain ordinary homotopy type of the space of complex points of a projective algebraic variety (with the subspace topology relative to the usual topology on the ambient complex projective space) may change.

apparently serre in particular made somewhat of a project of making examples of this phenomenon as nice and explicit as possible ... explicit, dramatic, simple, low-dimensional, palpable, picturesque, etc ... i haven't gotten to the point of actually really understanding any of serre's examples yet, but it's somewhat of a project of my own, not very developed so far, to understand serre's examples, or more likely to find my own such examples and understand those...

the point of such examples, insofar as i understand it, is to develop a sense of knowing what you're up against when you try to develop functorial processes for extracting invariant objects from algebraic varieties defined over algebraic number fields which specialize in certain cases to usual homotopy-invariants of subspaces of complex projective space... which, i have the vague impression, is part of what the weil conjectures (or at least certain ways of trying to prove them) are all about ...

(again, as usual, i could easily have some crucial details mangled here, but i think that the actual situation is vaguely similar to what i'm trying to describe here ...)

thus there's a tension between the change that the homotopy type may undergo and the nevertheless apparently somewhat powerful invariants (like "etale cohomology groups" ... or something like that ...) that resist that change ... which makes me uncertain and curious as to how dramatic the examples of shapeshifters can be, and to what extent you can sense the intrinsic aspect that doesn't shift ...

it seems to me a reasonable (if unfortunate...) rule of thumb that when the word "cohomology" appears in a mathematical exposition that's the place where it changes from exposition to obscurantism conveying only the author's having given up on finding a conceptual understanding of what they're talking about... a partial exception in the case of actual cohomology groups of reasonable spaces, as opposed to the more obscure sorts of "cohomology groups" that i associate with "homological algebra" ... i guess that i hope that a better understanding of the "galois shapeshifter" phenomenon might help me in trying to develop a conceptual understanding of some things for which a conceptual understanding usually seems to be lacking ...

at some point i should try to read the n-lab article on "cohomology", which i'm imagining was written by urs schreiber ... i'm undecided so far as to whether there might be an interesting conceptual understanding buried in that article... i think that i sometimes learn interesting things from urs, though maybe always second-hand rather than directly, so far... i've noticed that he generally doesn't seem to get what i'm saying, judging from the way he usually tries to interpret it in terms of things to which it seems to be essentially unrelated ...

so let x be a topos, r a commutative ring in x ... ???... and suppose that y is a "topos over x" ... ?? meaning that we have a geometric morphism from y into x ...??that is "algebraic morphism from x to y" ... call it f ... and suppose that we have a ring hom from f(r) to the underlying ring of a local ring r' in y ... ???then (??according to tierney??) we're supposed to get a unique ... ??... ??geometric morphism from y into the "small zariski spectrum topos of r", st / tw ... ??? or something???....

so suppose r is a commutative ring... and y is a topos, and r' is a "local commutative ring" in y, and h is a ring homomorphism from the "internalization of r in y" to r' ... ???then this is supposed to give us (with some essential uniqueness??...) a geometric morphism f from y to the "small zariski spectrum topos of r" ... and ... ????a "morphism of local commutative rings from the pullback under f of the main local commutative ring in the zariski thing to r' ... ??

??well, so let's think about the corresponding algebraic morphism from the small zariski topos to y ... ?? or something ... ??

??an algebraic morphism from the small zariski topos of r to some topos y picks out a local commutative r-algebra with a certain property ??? ...???namely (??though somewhat tautologically?? ??or something???) the property that ....

??? "simulated annealing" ... ?? as "brainstorming - shakeout cycle for robots" ... or something ... ???einstein ... imagination vs knowledge ... "type 1 vs type 2 error" .... ??"pruning" ... "let a hundred flowers bloom, then kill 99 of them" ... ???... ???something about breadth vs depth ?? ....

??brainstorming / barnstorming ... ???....

"stabilizer subalgebra of point of coadjoint orbit depends only on what wall of weyl chamber point lives on, whereas stabilizer subalgebra of corresponding point of conjugacy class depends on what wall of weyl alcove the original point lives on" .... ???or something like that ...

???here talking about in the compact form, for the moment?? ???perhaps the compact, connected, simply connected form??? ... or something ...

??so what about something about ... ???the homogeneous spaces in question ???

???the coadjoint orbit ones as just the partial flag varieties of the complex form ??? or something ??? .... ???but what about the conjugacy class ones??? are those still just certain partial flag varieties of the complex form ?????? .... and so forth ... ???...

??something about "structure / semantics adjointness" in somewhat general doctrine setting ... ???

d doctrine ....

e "family" of d-environments ... ?? (2,1)-fr into semantic (2,1) cat of d ...

s pre-stack on e ... ???or something ????....

.... ???....


???something about ... "universal example of theory of which given "potential model" is actual model" ... ??or something ...

??decategorification "universal example of system of equations of which given "potential solution" is actual solution" ... ???something about "cyclic subalgebra generated by x" or something, and so forth ... ???

??something about "clone" of x ... ???...

???something about ... ?? given doctrine d extending the doctrine of categories, and given functor into underlying category of d-environment ... ???...

??what about something about "diagram theory" here ?? ... or something ... ???...

notes for summer course ...

gunnarsen invited me to talk to his students sometime this summer for about a week or ten days... so i should try to plan out stuff to do / talk about ...

he suggested connecting with material about modular curves... child's drawing viewpoint ... which seems somewhat reasonable, i think ... though i was originally idly imagining "galois theory" (in very general sense ...) as topic... might fit together pretty well, perhaps ... ??....


??

covering spaces ... advent ... bread crumbs ... "structure vs symmetry" ... ... and so forth ... ???...


(??what about something about ... ??my experience with ... ??at first thinking that 2-sphere might have some weird covering space ... ??or something ?? ... ???also my experience with ... ??at first thinking that "any symmetric object must be bilaterally symmetric" or something ... yes, that one seems to have been particularly stupid, but ... ???... ??something about my maybe not even really having precise understanding of the question at that point ... ??or something ??...)


???branch points ???? ...... ????? .... ??"moduli ... ? " ... ???...


???sa kuga ... "galois's dream" ... island picture ... ??... !!woozle ... asf, os... something about my woozle experience; see something about "advent" somewhere here...

??"...as an added bonus, each of these methods of problem-solving can also be used as a way of screwing up situations really badly when misapplied ... " ... ??or something ... ??...

??just happened to think of a possibly interesting minor example of "symmetry reasoning" ... sa "wear/carry" vs "ser/estar" ... ???or something ... ???and so forth ... ??? something about vague idea of "broken symmetry" here, maybe?? ... something about ... abstract pattern with perfect symmetry ... concrete manifestations of it with less ... ??or something??? ... hmmm ...

??try thinking of other examples like this?? ...

??something about bit about ... ???microscope ... "double blind" ... ??something about "respect group" of a partition ... ??or something ... and so forth ... ???... ??hmm, maybe relating also to something about "separate but equal" ???? or something ??? ... ... ????... ??hmm, in weird ways ... ???or something ...

??something about integration of functions with symmetry properties... or something... and so forth ... ??something about "overkill" here ???... ??or maybe i'm just over-reacting to what that one guy said that time ... ??... ??hmm, something about hints about _continuous_ groups here ... ???and so forth ... ???...

!!!special relativity ... ???...and so forth ... ???....



??something about vague general phenomenon of ... ??"z/2 as test case / launching platform / or something "... ?? ... ?? here i'm conflating "mirror symmetry as prototypical symmetry" with

??something about "hall of mirrors" and "kaleidoscope" ... ??and so forth, or something .... ????maybe something about non-abelianness here ??? .... or something ... ??something about bug/feature aspect of this ... complicated/fun ... ??or something ... (??what about slightly meta- idea here ... "one person's bug is another person's feature" ... role reversal there ... ??...) ??something about "coming into focus" ...?? ... as angle between mirrors varies ... ??or something ... and so forth ...

??hmm, something about weird memories that i have about playing with mirrors when i was a kid ... also something about that album cover ... ??maybe i should bring some mirrors ?? ... or something ... and so forth ... ???what about where to get kaleidoscopes ??? ... ??a2 vs b2, and so forth ??? ...



(??something about my experience with hall of mirrors ??...)

????hmmm, so what _about_ something about ... ????actually linking together stuff like ... a2 kaleidoscope (and some others...) and "actual galois theory" (...) and covering spaces and modular curves and so forth ... ????what _about_ something about mystery of branch points here ??? .... and so forth ... (something about arnold ... ?? ... ??hmm, something about "singularity" here ... ??)

(??something about ... ??even if you to some extent think that it's a good idea to avoid too much focus on branch points, might get dragged into it to some extent because of something about ... ??understanding how "child's drawing" works ... ??? or something ... ?? ...)

??something about ... "they're as afraid of you as you are of them" ... (or something ... and so forth ...) ??perhaps good as example of symmetry principle that's genuinely useful and powerful where applicable, but then also dangerous where not, as for example in case of man-eating tiger or something ...

religious examples ... ??hmmm, besides "permutations on class of religions" (and so forth...) stuff, there's also ... ???something about ... group theory ideas _in_ some religions ... ??something about ... reincarnation ... golden rule ... relationship therebetween ... "whatsoever you do unto ... " ... ??or something ... and so forth ... ??? ????something about ... cake-cutting ... mule ... and so forth .... ???....


nation examples ...

??what _about_ historical vs contemporary examples here... ??especially something about difference between boundary between them for me vs for my audience .... ???...

??"political philosophy" examples, or something ?? ...

??"social situation" examples ... (which i already hinted at somewhere above i think ... see "man-eating tiger" or something ... ??...)

??see (??for context ? ...) bit i mentioned about big julie's "dice with no spots" somewhere ...

????something about "cutting problems down to size" ... and so forth ... ??or something ... ??something about "dual" ways this can happen?? ... (?? something about "n birds with one stone" vs "don't bother with the non-symmetric birds" ... ??? or something ... and so forth ...) ...or something ... and so forth ... ???something about relationship to "mathematical induction" ... ??something about orbits (??as fewer of them to worry about than of elements ... ???or something ... ) ... ...and so forth ...

??other places to check for where i may have written down relevant examples / ideas ?? ... and so forth ... ??...

??something about _me_ learning... ??something about modular curves os, asf os???...

??something about galois's usage of terminology "group" ... and so forth ... ???.... ??sa lack of time ... (??to perfect terminology, for example ??? ...) .... ???something about "ambiguity" bit .... ???.... ...my translation career ... ??? ...

??also something about "abstract vs concrete" ...

??soemthing about "logic" (and so forth ... "structure" ...) interpretation of cayley diagram (since gunnarsen mentioned something about them.... ?? ...)

??something about ... "keeping track of levels of structure" .... "varying reality" ... ???something about "imaginary numbers" here ??? well, maybe don't try to push that metaphor _too_ hard, since ... ???something about "imaginary" vs "non-real" ?? .... os... asf os... ????might still be able to push it a bit, though ?? ....






ok, i'd like to start with a game...

(most of my games will probably end up being too easy, or too hard, or both simultaneously... or something ... ???... ??something about me being out of practice ... ????...)

i guess this is really a whole bunch of games that are all pretty similar to each other... not sure that i have a name for this game / these games... maybe i should call it "mathematical 3-card monte" or something like that ...


??variations on game ...

???defining one thing in terms of another... "syntactically" ... ??then also something about "semantic" comparison ... ??and so forth .... ????....


??sa summer course ... sa religion ... meta ... os...asf os... sa reincarnation asf ... ???... golden rule ... ??os... asf os... ??sa cake-cutting ... ??...


???sa... "when i was here last time we discussed (or something...) this puzzle about how to express one rational function in terms of another that's at least as asymmetric... in a sense everything that i want to tell you about now is an attempt to explain the secret behind the solution to this puzzle ... which as it turns out is the secret behind the solution of a _lot_ of puzzles ..." ??? ... ???....

??something about "hedgehog" ... (???"hedgehog theorem" ??? ...???) ... ??? ...

??"theory of ambiguity" ... ?? ..."theory of undefinability" ... ??? .... ??nominally (??...), "ambiguity" as "about" word, "undefinability" as "about" thing, but ... ???sort of describing same idea ... ??... ??"irreducible ambiguiity" ... "irreparable" ... ?? ... "inherent" ... "inherent ambiguity of word as related to indefinability of thing under its umbrella" ... thing can't be characterized by word ... ??? .... ???? .... ??"umbrella term" ... "inherently an umbrella term" ... ???"who's buried in x's tomb?" ... ???_is_ that where i first started with the "galois theory = ambiguity theory" slogan ??? ... ... ???"tomb-mate" ... ???.....


??"galois'sprinciple" .... as ["x is definable in terms of y" = "y breaks symmetry more than x" ... with slight subtlety about meaning of "more" here... "qualitative rather than quantitative" ... ?? ... ??also my tendency to often get the principle backwards ... ???probably avoid psychoanalyzing me too much in this respect, for fear of infecting others... ??...]

(sa proverbial shop teacher with missing finger ... might in fact be able to learn from them how not to lose your finger ... or at least, one way not to lose it ... ??unfortunately .... "way to not lose it" vs "way not to lose it" ... ??? ... ??? .... ????? .......)

??so what _about_ which versions of game to start with / middle with / end with ... ????... ??? ... ???....

??? ..."breadcrumbs" ... "structure as breadcrumbs" .... "featureless expanse ..." .... advent ... piglet ... ??as "stupidest person in world" .... undercurrent of nastiness .... ??? .... ... ............ .... ....kuga .... island ... ???"conceptual un/wrapping" ... ????"imagination vs knowledge" .... ????.....

??advent as _punchline_ here??? ... my turn ....

volleyball .... reincarnation ... "my/your/their turn"

???cartoon ... tom biting own tail ??? .... ??? .... .... cake-cutting ....

????double coset interpretation of "moduli space ... " ... cartesian product of homogeneous spaces as non-homogeneous .... (????"separate but equal ...." ..... ?????? .... ???? "diagonal ..." .... "degenerate ... " .... ????....) .... homogeneous components thereof ... ??? ... "classify" ... ?????.... "joint ... " ... ??? ....



???? "at first it seems like they're completely different from you / in a completely different situation ... but then suddenly you realize that if just one little (??? ...) thing is changed .... ???"switched" .... ??? ..."shift of perspective" .... ??? ... then they(??or their situation ... ??? ...)'re really the same ... ???.... ?? ... then you can use this information in different ways .... example : [1["i understand what they're thinking / feeling / experiencing, so we should be able to live in harmony ..."] vs 2["i understand what they're thinking / feeling / experiencing, so i should be able to trick them ..."] ... ] ... another example : [1["you find a complete stranger lying hurt in the middle of the street (??road??? ...), and you think : if this were my friend, ..."] vs 2["you find your friend lying hurt in the middle of the road, and you think : well, if this were a complete stranger i might just leave them lying here, so probably it's ok if i just leave my friend lying here ..." ...]] ..... ???in such examples, either way is a characteristically human way of thinking (... ???...) ... ????and the secret of math (???? ....) is to take this characteristically human mode of thought and exploit the hell out of it in extreme (??? ....) ways .... ??? .... ???? .....

????choosing "bad" symmetry group ... ??? "ok, today i'll try permuting good and evil " .... (???"there are some episodes of the tv show "seinfeld" sort of like that ..." ... ????too dated, or maybe not ??? ...) ???"too big" ??? ... "inappropriate" .... ????not a symmetry of certain particularly relevant structure (??despite counterpoint about how non-symmetry permutations / groups thereof (??? ...) can be very useful in certain ways ... "broken symmetries" ... ???? ....) .... ???? .... ???"in fact good/evil (??also true/false ??? ...) switching is a pretty good (perhaps prototypical ??? .... ???....) example of a "symmetry" probably broken for most practical purposes, but nevertheless often interesting and even potentially useful to think about, though of course (???) dangerous ... ???prototypical (??? ...) example of situation where you should think before you act" ???? ..... .... ???? ..... ??"misere transform" of a "game" ... ??? ... ???? .... funny tendency as to how strategies are affected ... ???idle speculation of mine about "intrinsic / invariant (??under "change of goal") "powerfulness" ("position of control ..." ... ???) of move" ... ???? .... ??? ....


???symmetry as the more useful for you "the more exciting your life is" ... "the more you face the unknown" ... ???os?? ... ????sa "entropy" ??? ....

???... ?? people who say stuff like "it's a dead field ... everything's already been discovered ... all that's left is calculating the nth decimal point ..." ... as idiots .... ???? ...

?? experience of ... ??generation who think they had it hard trying to sympathize with next generation but not quite appreciating how much (or at least in what ways... ??) harder things have gotten...

??? kummer's chemistry analogy ??? (???connection to hilbert(??)'s rng (??) analogy that gunnarsen told me about ??? ...) ...

???kronecker's youth-dream ... ??? "turning ideal numbers into actual numbers" ... ??? ....