??so let x be a dimensional category, and then let x' be the dimensional category of x-objects equipped with actions by abelian group g ... ???or something?? ... ???what's this like??....
??is an x'-model maybe esentially an x-model together with a g-torsor?? ???or something?? ...??? ...
??maybe in some cases, but ... ???shouldn't it be a bit different from that, conceptually???....??? or aomething ?? ....
??well, so what about sort of same idea with ag theory or g theory, for example??
hmm... geometric theory case seems like it should be straightforward, no ?? ...???...
???something about ... ????a g-rep equipped with an action of h as a gXh-rep ... ???? ... ??or somethihng??? ...
??so consider ... ???the dimensional theory of "a t-model together with an a-torsor" where a is, for example, some finite abelian group ?? .... or something .... ???.....
???somethihg about ??"using gabriel-ulmer duality" (or something....) to simulate tneosr product via homming" ???? .... ???or something??? ... ???....
Friday, March 4, 2011
Thursday, March 3, 2011
johnstone p 204 :
"in the suggestive terminology of tierney, we say that the topology j _forces_ f to be an interpretation of l"
hmm, l here is just a "language", but they go on to consider the case of a theory t as well ...
f here is... ???something like... ??the generic model of a "stuff-level" theory, to which forcing conditions / axioms / coverings are being added ... ??or something??
??something about ... maybe the "l" bit involves shoe-horning "structure-level" in with "property-level" to some extent ... ??or something ...
"in the suggestive terminology of tierney, we say that the topology j _forces_ f to be an interpretation of l"
hmm, l here is just a "language", but they go on to consider the case of a theory t as well ...
f here is... ???something like... ??the generic model of a "stuff-level" theory, to which forcing conditions / axioms / coverings are being added ... ??or something??
??something about ... maybe the "l" bit involves shoe-horning "structure-level" in with "property-level" to some extent ... ??or something ...
(for martin)
hi...
i'd like to try to state some questions here that i'm interested in ...
we've already talked about many questions of the general form:
for some specific algebraic-geometric theory t, can we give a nice description of its universal property; that is, of what it's the "classifying space" for, or of what it's the "moduli space" of?
(where "algebraic-geometric theory" is my terminology for "symmetric monoidal cocomplete k-linear category"; sometimes i use "finitely cocomplete" instead of "cocomplete" but for now i'll stick with "cocomplete".)
thus for example we've talked a lot about the case of t = quasicoherent sheaves over P^n, and we've explored possible answers in that case such as "t is the theory of a line object L equipped with a good embedding into the direct sum of n+1 copies of the unit object".
but the new questions that i'm interested in (actually i've been thinking about them for a while, but i don't think that i've gotten a chance to explain them to you very well yet) are the same kind of questions, except dealing with so-called "geometric theories" instead of "algebraic-geometric theories". and just as "algebraic-geometric theory" is a synonym for "symmetric monoidal cocomplete k-linear category", "geometric theory" is a synonym for "grothendieck topos".
(roughly speaking, a grothendieck topos is a category which has all finite limits and all small colimits, and where the finite limits "distribute over" the colimits in the same way that they do in the category of sets.)
thus for example, at there's a brief discussion of many different ways of associating a topos to a scheme:
"More exotic examples, and the raison d'être of topos theory, come from algebraic geometry. To a scheme and even a stack one may associate an étale topos, an fppf topos, a Nisnevich topos..."
my main idea here is that when we create a topos from a scheme (or stack) in this way, the universal property of the resulting topos (or "geometric theory") should be very directly related to the universal property of the algebraic-geometric theory of quasicoherent sheaves over x.
thus for example consider ...
??something about "strictly local ring" and/or "henselian ..." or something ?? ...
i'd like to try to state some questions here that i'm interested in ...
we've already talked about many questions of the general form:
for some specific algebraic-geometric theory t, can we give a nice description of its universal property; that is, of what it's the "classifying space" for, or of what it's the "moduli space" of?
(where "algebraic-geometric theory" is my terminology for "symmetric monoidal cocomplete k-linear category"; sometimes i use "finitely cocomplete" instead of "cocomplete" but for now i'll stick with "cocomplete".)
thus for example we've talked a lot about the case of t = quasicoherent sheaves over P^n, and we've explored possible answers in that case such as "t is the theory of a line object L equipped with a good embedding into the direct sum of n+1 copies of the unit object".
but the new questions that i'm interested in (actually i've been thinking about them for a while, but i don't think that i've gotten a chance to explain them to you very well yet) are the same kind of questions, except dealing with so-called "geometric theories" instead of "algebraic-geometric theories". and just as "algebraic-geometric theory" is a synonym for "symmetric monoidal cocomplete k-linear category", "geometric theory" is a synonym for "grothendieck topos".
(roughly speaking, a grothendieck topos is a category which has all finite limits and all small colimits, and where the finite limits "distribute over" the colimits in the same way that they do in the category of sets.)
thus for example, at
"More exotic examples, and the raison d'être of topos theory, come from algebraic geometry. To a scheme and even a stack one may associate an étale topos, an fppf topos, a Nisnevich topos..."
my main idea here is that when we create a topos from a scheme (or stack) in this way, the universal property of the resulting topos (or "geometric theory") should be very directly related to the universal property of the algebraic-geometric theory of quasicoherent sheaves over x.
thus for example consider ...
??something about "strictly local ring" and/or "henselian ..." or something ?? ...
Tuesday, March 1, 2011
??so what _does_ it mean to have a model of a classical first-order theory over the stone space given by 1-point compactification of N? ... ???.... ... somewhat concretely and explicitly ... ??.....
??hmm, so what about something about ... ???stone space (or something...) of subquotients (or something) of a set x ???? .... ???... ???hmm, what _about_ something about "orthogonality of partitions" (or something...) here??? .... ???what about something about "connection information" ?????? .... ????......
??maybe just something about "in-/efficiency" here??
?? ... more generally (...), what about something about ... ??stone space of t-structures on some stuff ... ???or something ... ??...
???hmm, so maybe the subquotients of a set really do form a stone space??? ??or something???... ....solutions of a system of boolean equations ... ??...
???something about ... ??"continuous map from stone space x to stone space of subquotients of s, with union of all the subs equal to s and coarsest mutual refinement of all the quotients equal to actual equality on s" ??? .... ???o something ???....
???but what about some "inefficiency" here ????.....
hmmmm...... ????.....
for example something about if the continuous map is constant ... ???....
??hmm, so what about something about ... ???stone space (or something...) of subquotients (or something) of a set x ???? .... ???... ???hmm, what _about_ something about "orthogonality of partitions" (or something...) here??? .... ???what about something about "connection information" ?????? .... ????......
??maybe just something about "in-/efficiency" here??
?? ... more generally (...), what about something about ... ??stone space of t-structures on some stuff ... ???or something ... ??...
???hmm, so maybe the subquotients of a set really do form a stone space??? ??or something???... ....solutions of a system of boolean equations ... ??...
???something about ... ??"continuous map from stone space x to stone space of subquotients of s, with union of all the subs equal to s and coarsest mutual refinement of all the quotients equal to actual equality on s" ??? .... ???o something ???....
???but what about some "inefficiency" here ????.....
hmmmm...... ????.....
for example something about if the continuous map is constant ... ???....
Monday, February 28, 2011
??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 ... ???....
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 ???.....
??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 ???.....
??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" ... ?? ...
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 _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???
??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???
??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 ..... ???????.......
???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 ..... ???????.......
Sunday, February 27, 2011
??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???...
????...
???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.
???.....
??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.
???.....
Subscribe to:
Posts (Atom)