Wednesday, October 19, 2011
toposes of toric quasicoherent sheaves (octoberfest version)
?? my topic today is officially (??) motivated by an ambitious big philosophical and extremely _general_ (?? perhaps write on blackboard ...) research program (?? or perhaps multiple such programs promoted by different people ... ??? ....) to develop algebraic geometry as a part of categorical logic ( ???? ..... ?? write on blackboard with subset notation ....) ..... but in order to have any chance in a short talk of actually getting to my topic, i have to begin at completely the other end, at the _particular_ end of things, and then if there's any time left over towards the end of my talk, i may be able to say a little bit about the big picture, about the ambitious and very general research programs that this (talk?) is supposed to be a part of ...
so i'm going to start with some pure topos theory, and then only towards the end of the talk will i maybe have a chance to say how it relates to algebraic geometry, and then even less of a chance to try to describe my big picture of how algebraic geometry forms a part of categorical logic ...
however, i do feel compelled to issue a warning right now about the title of my talk ... i'm not sure exactly why i chose this title; i wonder whether i may have been trying to play a trick on someone ... because my title mentions toposes, and then some algebraic geometry stuff, and then sheaves, and for people with sufficient background it's probably sensible to suspect that a talk combining those three ingredients is going to combine them in a familiar way .... whereas in fact i'm _not_ combining them in that familiar way .... so for example i'll be discussing certain toposes whose objects can be interpreted as sheaves, but the way in which the objects qualify as sheaves will be _almost_ unrelated to the way in which they form a topos ... and i'll be using those toposes, which in fact will be grothendieck toposes, for purposes of doing algebraic geometry, but the way in which i use them will be _almost_ unrelated to the way in which grothendieck used toposes in algebraic geometry ...
so here goes with a little bit of pure topos theory ...
when i was first learning topos theory, i found it to be a useful first approximation, or a useful crutch, to think of all grothendieck toposes as being presheaf toposes, or at least as being very similar to presheaf toposes ... in somewhat the same way that when you're first learning about boolean algebras, it may be useful to think of all boolean algebras as being power-sets, or at least very similar to power sets ... but eventually you want to throw away your crutch and understand how there can be grothendieck toposes which _aren't_ presheaf toposes (or boolean algebras which aren't power-sets ...) ... so a perhaps interesting question is : what's the simplest example that you can give of a grothendieck topos that's not a pre-sheaf topos? ... now this isn't a very precise question because i'm not proposing any formal way to measure the simplicity of an example ... but we can still entertain the question in a purely informal way, and i have an answer that i'm going to suggest ... but i'd be interested to hear anyone else's suggestions as well, either right now or later ... ?? so does anyone have any suggestions offhand, as to a nice simple example of a grothendieck topos that's not a pre-sheaf topos?
(?? in case anyone suggests it ... to me, sheaves over (say ...) the real line is a very _obvious_ example, but not a very _simple_ example ... ?? ...)
so let me tell you my example ....
?? but the real interest of this example, i think, is that it actually shows up in nature (so to speak) ... playing a significant role in algebraic geometry, which i'll try to describe right now ...
quasi-affine toric variety that's not affine .... ???? .....
??? hmm, that way of stating it seems to suggest that maybe we should actually even use that as our way of introducing the example, as opposed to giving the toric variety interpretation as an afterthought .... ??? .....
?? maybe even do the switch towards ag / toric varieties while still introducing question, as opposed to while giving answer ... ?? sort of re-interpretation of question... almost ...
??? maybe some slight rewriting required in places above, then ??? .... ?? algebraic geometry as not postponed so much / pure topos theory as not prolonged so much ... ??? maybe mention pure topos theory nature only of _question_, not of answer ... ???? ........ ???? ..... ?? family of examples rather than single example ?? ... ?? place / way to remark upon pleasant surprise of additional intrinsic interest of example(s) ... ??? ...
?? but then also ... ??? other nice sources of non-affine toric varieties .... for example projective instead of quasi-affine .... ????? ...... ?? hmmm, maybe use "non-affine" instead of "quasi-affine but not affine" as the main family ... ?? then mention "quasi-affine but not affine" and "projective" (?? with "... but not affine" almost redundant ?? ....) as prominent sub-families ??? .... ??? projective toric varieties as giving very interesting simple example, but quasi-affine as even simpler (if perhaps less "interesting" ??? ...) .... ????? .....
??hmm, this rough general approach as maybe suggesting ways of introducing some more of the bigger ideas ... ??? ... toric convolution as extra ingredient of tannakian stew in toric case .... ??? .... ?? ....
??... giving opening for somewhat more explicit comment about ... analogy between using "sheaf" to mean "object of topos" and using "quasicoherent sheaf" to mean "object of tensor category" ... ??? where "tensor category" presumably needs to be made somewhat more precise here .... ???? ......
??? something about simplest toric example of "quasi-affine but not affine" as perhaps even simplest not-necessarily-toric example ... ?? illustrating general phenomenon of simplest examples often being toric ??? ......
?? so ... still need further re-organizing here ... ??? .....
?? i guess that theorem should be stated ... ?? something like "subtopos of presheaf topos on commutative monoid x" = "quasi-affine toric variety in affine toric variety spec(k[x])" .... ?? where should worry a bit both about how that's stated (?? localization / open subvariety on toric variety side .... ???? ....) and whether it's actually _true_ (?? open vs more general subtopos on topos side ... ?? ....) ... ???? .... ?? actually similar concerns ... ???? .... one side of the equivalence vs the other ... open subobject (?? ...) vs more general .... ??????? .......
so i'm going to start with some pure topos theory, and then only towards the end of the talk will i maybe have a chance to say how it relates to algebraic geometry, and then even less of a chance to try to describe my big picture of how algebraic geometry forms a part of categorical logic ...
however, i do feel compelled to issue a warning right now about the title of my talk ... i'm not sure exactly why i chose this title; i wonder whether i may have been trying to play a trick on someone ... because my title mentions toposes, and then some algebraic geometry stuff, and then sheaves, and for people with sufficient background it's probably sensible to suspect that a talk combining those three ingredients is going to combine them in a familiar way .... whereas in fact i'm _not_ combining them in that familiar way .... so for example i'll be discussing certain toposes whose objects can be interpreted as sheaves, but the way in which the objects qualify as sheaves will be _almost_ unrelated to the way in which they form a topos ... and i'll be using those toposes, which in fact will be grothendieck toposes, for purposes of doing algebraic geometry, but the way in which i use them will be _almost_ unrelated to the way in which grothendieck used toposes in algebraic geometry ...
so here goes with a little bit of pure topos theory ...
when i was first learning topos theory, i found it to be a useful first approximation, or a useful crutch, to think of all grothendieck toposes as being presheaf toposes, or at least as being very similar to presheaf toposes ... in somewhat the same way that when you're first learning about boolean algebras, it may be useful to think of all boolean algebras as being power-sets, or at least very similar to power sets ... but eventually you want to throw away your crutch and understand how there can be grothendieck toposes which _aren't_ presheaf toposes (or boolean algebras which aren't power-sets ...) ... so a perhaps interesting question is : what's the simplest example that you can give of a grothendieck topos that's not a pre-sheaf topos? ... now this isn't a very precise question because i'm not proposing any formal way to measure the simplicity of an example ... but we can still entertain the question in a purely informal way, and i have an answer that i'm going to suggest ... but i'd be interested to hear anyone else's suggestions as well, either right now or later ... ?? so does anyone have any suggestions offhand, as to a nice simple example of a grothendieck topos that's not a pre-sheaf topos?
(?? in case anyone suggests it ... to me, sheaves over (say ...) the real line is a very _obvious_ example, but not a very _simple_ example ... ?? ...)
so let me tell you my example ....
?? but the real interest of this example, i think, is that it actually shows up in nature (so to speak) ... playing a significant role in algebraic geometry, which i'll try to describe right now ...
quasi-affine toric variety that's not affine .... ???? .....
??? hmm, that way of stating it seems to suggest that maybe we should actually even use that as our way of introducing the example, as opposed to giving the toric variety interpretation as an afterthought .... ??? .....
?? maybe even do the switch towards ag / toric varieties while still introducing question, as opposed to while giving answer ... ?? sort of re-interpretation of question... almost ...
??? maybe some slight rewriting required in places above, then ??? .... ?? algebraic geometry as not postponed so much / pure topos theory as not prolonged so much ... ??? maybe mention pure topos theory nature only of _question_, not of answer ... ???? ........ ???? ..... ?? family of examples rather than single example ?? ... ?? place / way to remark upon pleasant surprise of additional intrinsic interest of example(s) ... ??? ...
?? but then also ... ??? other nice sources of non-affine toric varieties .... for example projective instead of quasi-affine .... ????? ...... ?? hmmm, maybe use "non-affine" instead of "quasi-affine but not affine" as the main family ... ?? then mention "quasi-affine but not affine" and "projective" (?? with "... but not affine" almost redundant ?? ....) as prominent sub-families ??? .... ??? projective toric varieties as giving very interesting simple example, but quasi-affine as even simpler (if perhaps less "interesting" ??? ...) .... ????? .....
??hmm, this rough general approach as maybe suggesting ways of introducing some more of the bigger ideas ... ??? ... toric convolution as extra ingredient of tannakian stew in toric case .... ??? .... ?? ....
??... giving opening for somewhat more explicit comment about ... analogy between using "sheaf" to mean "object of topos" and using "quasicoherent sheaf" to mean "object of tensor category" ... ??? where "tensor category" presumably needs to be made somewhat more precise here .... ???? ......
??? something about simplest toric example of "quasi-affine but not affine" as perhaps even simplest not-necessarily-toric example ... ?? illustrating general phenomenon of simplest examples often being toric ??? ......
?? so ... still need further re-organizing here ... ??? .....
?? i guess that theorem should be stated ... ?? something like "subtopos of presheaf topos on commutative monoid x" = "quasi-affine toric variety in affine toric variety spec(k[x])" .... ?? where should worry a bit both about how that's stated (?? localization / open subvariety on toric variety side .... ???? ....) and whether it's actually _true_ (?? open vs more general subtopos on topos side ... ?? ....) ... ???? .... ?? actually similar concerns ... ???? .... one side of the equivalence vs the other ... open subobject (?? ...) vs more general .... ??????? .......
?? hmm ... in the visualization talk i had a bit of trouble with "non-individuality of the poles of the axis/skewer" in the gimbal/axis picture of the real algebraic b2 building ... ?? maybe the best approach is to emphasize points as configurations of a mechanical system in contrast to lines as mere subspaces of that configuration space ... thus the gimbal is officially part of the mechanical system whereas the skewer isn't; it's just there (?? "auxiliary" ?? ...) to remind you where the spin axis is .... so that "explains" why pulling the skewer out and putting it back in backwards doesn't change the line .... ??? ....
Sunday, October 16, 2011
Saturday, October 15, 2011
i asked tony licata about where to learn about "geometric mackay correspondence" and about "quiver varieties" ...
suggested nakajima in both cases ... "hilbert schemes of points on surfaces" (??...), and original papers from the 1990's ... but also suggested that some notes by ginzburg on quiver varieties might be helpful ...
??? hmm, so what about possibility that i might actually already understand this "geometric mackay correspondence" (?? ...) stuff better than i realized ?? .... "sophisticated fiber" vs this "resolution" stuff .... ??? ..... ?? "blow-up" and or generalized such .... ???? ...... ???? .....
??? hmmm ... ??? blow-up (?? generalized ?? ...) of subvariety of infinitesimal variety ?? ..... ?? non-functoriality of resolution .... ???? toric case ???? ......data ... symmetry-breaking ... kontsevitch .... ???? ..... ?? irreducible components of singular fiber .... ???? ...... ?? (??dramatic ?? ...) non-flatness of blow-up ... ??? ....
suggested nakajima in both cases ... "hilbert schemes of points on surfaces" (??...), and original papers from the 1990's ... but also suggested that some notes by ginzburg on quiver varieties might be helpful ...
??? hmm, so what about possibility that i might actually already understand this "geometric mackay correspondence" (?? ...) stuff better than i realized ?? .... "sophisticated fiber" vs this "resolution" stuff .... ??? ..... ?? "blow-up" and or generalized such .... ???? ...... ???? .....
??? hmmm ... ??? blow-up (?? generalized ?? ...) of subvariety of infinitesimal variety ?? ..... ?? non-functoriality of resolution .... ???? toric case ???? ......data ... symmetry-breaking ... kontsevitch .... ???? ..... ?? irreducible components of singular fiber .... ???? ...... ?? (??dramatic ?? ...) non-flatness of blow-up ... ??? ....
Wednesday, October 12, 2011
proj geom = dimensional analysis 9 (outline)
1 preamble ... imperfection .... ???? ....
2 dimensional analysis example ...
3 audience torture quiz : how to translate into projective algebraic geometry ? .... 2 key ingredients in particular ...
4 quiz answer
5 how to understand quiz answer .... two formal approaches ...graded commutative algebra vs (much more interestingly...) "dimensional category" .... sort-of equivalence between the two approaches ...
6 what's so interesting about lawvere's approach (the dimensional category approach), namely how it makes projective algebraic geometry (?? and in fact algebraic geometry more generally) into a branch (?? ...) of "categorical logic" .... in sense of study of structured categories .... moduli stack of models .... ???? ..... ???? "state of affairs" .... ?? .... "theory with free parameters" .... ??? ..... ?? inspired by idea of "physical theory" as well as of "logical theory" .... ?? .... ????? tannakian program ..... ???? ..... ?? message to "categorical logicians" in general .... ???? .....
7 interpretations of tannakian program ....
???? graded commutative algebra and "torus-equivariant affine algebraic geometry" ??? ....
2 dimensional analysis example ...
3 audience torture quiz : how to translate into projective algebraic geometry ? .... 2 key ingredients in particular ...
4 quiz answer
5 how to understand quiz answer .... two formal approaches ...graded commutative algebra vs (much more interestingly...) "dimensional category" .... sort-of equivalence between the two approaches ...
6 what's so interesting about lawvere's approach (the dimensional category approach), namely how it makes projective algebraic geometry (?? and in fact algebraic geometry more generally) into a branch (?? ...) of "categorical logic" .... in sense of study of structured categories .... moduli stack of models .... ???? ..... ???? "state of affairs" .... ?? .... "theory with free parameters" .... ??? ..... ?? inspired by idea of "physical theory" as well as of "logical theory" .... ?? .... ????? tannakian program ..... ???? ..... ?? message to "categorical logicians" in general .... ???? .....
7 interpretations of tannakian program ....
???? graded commutative algebra and "torus-equivariant affine algebraic geometry" ??? ....
Tuesday, October 11, 2011
proj geom = dimensional analysis 8
the title of my talk is ... (pointing to it on whiteboard) a lie ... or at least, an exaggeration ... in fact, neither side of this equation is precisely enough defined to be exactly equal to itself, let alone to anything else. for example, projective geometry comes in at least two different flavors, "synthetic" and "analytic", and both of these flavors are important in the study of group representations as it happens, and they're related to each other in a funny way ... although analytic projective geometry is perhaps more commonly known these days as _algebraic_ projective geometry (ot at least, those are roughly the same thing ...), and it's that form of projective geometry, namely projective _algebraic _ geometry, that i'm going to be connecting to (/ equating with ??) dimensional analysis in this talk...
but all of this vagueness and fuzziness is pretty typical when you're ... living in this universe, or more particularly when you're trying to set up one of these secret dictionaries, or "cryptomorphisms" as they're sometimes called, that translate between two different branches of mathematics or two different ways of thinking. typically there'll be some words in each of the two languages that sound funny when you translate them into the other language, which might make you doubt the validity of the dictionary... but that's actually supposed to be one of the benefits of such a dictionary, that it forces you to stretch your conceptual view of each side of the equation, trying to make the picture fit together ....
in any case, before worrying about some of the more nitpicking ways in which this equation might be false, i'm going to start by explaining the ways in which it's _true_ ...
[??? emphasizing idea of category of graded vector spaces as category of group representations ..... ??? when we get to that ... ???? .....]
[?? as to how "crypto" (hidden ... ?? ...) the dictionary here really is, well, it really did take me by surprise ... though in retrospect does seem really obvious in some ways .... ?? both study of "homogeneous quantities", which can be taken to mean "quantities that transform in a particularly nice simple way under rescaling transformations" .... ??? which hints at relationship of this stuff to conference theme of representation theory .... ??? _abelian_ case ....... ????? ??? save this "retrospectively obvious" stuff for right after audience opinions ... ???? ... "i described this relationship as a "cryptomorphism", hidden, secret .... but in retrospect it seems really obvious to me now ..... homogenous quantities ....re-scaling .... line-bundle = one-dimensional object ..." .... ????? .....]
?? "so we have 2 key ingredients here:
1 _dimensions_, which are the boxes in the table here ...
2 _quantities_ that live in these dimensions, which are the variables that appear in those boxes ... (???? .....)
and my claim is that we can translate what we're doing here from the language of dimensional analysis into the language of projective algebraic geometry ....
so let me whether anyone in the audience has any opinions about how these 2 key ingredients should get translated .... ????? .....
but all of this vagueness and fuzziness is pretty typical when you're ... living in this universe, or more particularly when you're trying to set up one of these secret dictionaries, or "cryptomorphisms" as they're sometimes called, that translate between two different branches of mathematics or two different ways of thinking. typically there'll be some words in each of the two languages that sound funny when you translate them into the other language, which might make you doubt the validity of the dictionary... but that's actually supposed to be one of the benefits of such a dictionary, that it forces you to stretch your conceptual view of each side of the equation, trying to make the picture fit together ....
in any case, before worrying about some of the more nitpicking ways in which this equation might be false, i'm going to start by explaining the ways in which it's _true_ ...
[??? emphasizing idea of category of graded vector spaces as category of group representations ..... ??? when we get to that ... ???? .....]
[?? as to how "crypto" (hidden ... ?? ...) the dictionary here really is, well, it really did take me by surprise ... though in retrospect does seem really obvious in some ways .... ?? both study of "homogeneous quantities", which can be taken to mean "quantities that transform in a particularly nice simple way under rescaling transformations" .... ??? which hints at relationship of this stuff to conference theme of representation theory .... ??? _abelian_ case ....... ????? ??? save this "retrospectively obvious" stuff for right after audience opinions ... ???? ... "i described this relationship as a "cryptomorphism", hidden, secret .... but in retrospect it seems really obvious to me now ..... homogenous quantities ....re-scaling .... line-bundle = one-dimensional object ..." .... ????? .....]
?? "so we have 2 key ingredients here:
1 _dimensions_, which are the boxes in the table here ...
2 _quantities_ that live in these dimensions, which are the variables that appear in those boxes ... (???? .....)
and my claim is that we can translate what we're doing here from the language of dimensional analysis into the language of projective algebraic geometry ....
so let me whether anyone in the audience has any opinions about how these 2 key ingredients should get translated .... ????? .....
???? graded modules of graded commutative algebra as forming some sort of "hopf category" with comultiplication corresponding to .... ????? ......
??? some duality confusion ... ??? .... ??? ...
??? try to straighten out ... ??? ....
????? any relationship to "combined doctrine" and associated compatibility condition ..... ??????? ...... ????? ..... ??? hmmm, a problem with this idea is that it seems we should expect hopf category structure here to be rather delicate, disappearing for example as soon as you pass from graded modules to some sort of constrained such ...... ????? .....
???? categorified comultiplication and tensor product of categorified modules .... ???? ....
??? some duality confusion ... ??? .... ??? ...
??? try to straighten out ... ??? ....
????? any relationship to "combined doctrine" and associated compatibility condition ..... ??????? ...... ????? ..... ??? hmmm, a problem with this idea is that it seems we should expect hopf category structure here to be rather delicate, disappearing for example as soon as you pass from graded modules to some sort of constrained such ...... ????? .....
???? categorified comultiplication and tensor product of categorified modules .... ???? ....
Monday, October 10, 2011
proj geom = dimensional analysis 7
?? seems like one key to pulling things together here is .... ??? basic dictionary entries get introduced right aruond time lawvere's approach and competing approach are introduced ....
?? speed of light joke around here ???? ..... section vs morphism ... algebra vs category .... ???? ....
pointing to tableau on whiteboard : "part of what's going on here is that we're introducing an alleged sort of mental/conceptual hygiene; separating quantities of different kinds into separate boxes .... ?? thereby allegedly avoiding certain kinds of mental/conceptual confusion ..." ..... ?????? ??? maybe stuff about related ideas ..... structured programming and so forth .... ???? "typed lambda calculus" ???? .....
?? hmm, maybe torture quiz right after introducing dimensional analysis example .... "we're supposed (according to me) to get a cryptomorphism to projective algebraic geometry .... ?? any opinion as to how that might go ?? .... what might correspond to the dimensions, and to the quantities that live in them ??? ....." ....
??? "i promised you a dictionary ... well this is basically it .... rather short .... you can add lots more entries to it if you want, but all the rest are basically determined by these two" .... ???? ....
??? rant about retrospective obviousness of correspondence here ... terminologies "dimension" vs "line-bundle" .... !!! "line" as "one-dimensional object", aka "dimension" !!!!!! (??? this rant as probably somewhat separate from rant about invertibility under tensor product as hallmark of "one-dimensionality", coming up slightly later ???? ....)
"dimensional analysis is essentially the study of dimensions and quantities that live in those dimensions, projective algebraic geometry is essentially the study of line bundles and sections that live on those bundles, and the two subjects are secretly isomorphic or "cryptomorphic" under this (pointing to 2-entry dictionary on whiteboard) correspondence" ....
(if you doubt the characterization of pag as "all about line bundles and their sections", then that's an additional propaganda point that i can try to rant about if you like .... ???? .....)
"at this point there are two main ways to mathematically formalize [?? what we're doing here ?? ... / ?? this correspondence ... ?? ...] :
1 organize the sections of all the line bundles over a projective variety into a _graded commutative algebra_, with the _grades_ being the line bundles ....
2 .... lawvere approach .... _obects_ being the line bundles .... ???? .....
...............
??speed-of-light joke here somewhere ... ??? but wasn't there something else i wanted here too???? ...
oh yes!!! ..... bit about .... sections of line bundles ..... ???? " ... lowbrow approach involving lowbrow geometric quotienting by re-scaling action on vector space .... before the quotienting (=?= introduction of grading ... g = a ... ??? ....) had functions on a space, but what are they now only functions on some space lying over the space you're interested in .... and if you think about this and pursue it far enough, you can see that what's happening with these former functions (from affine picture ...) is that they're now (in projective geometry) achieving the status of ("twisted functions" or) sections of line bundles ..... ???"
now what makes [this second approach / lawvere's category approach / ... ??? ...] so interesting is that because of the way in which it uses _categories_ it reveals pag as being a part(/aspect ??? ....) of the vast program of _categorical logic_ (?? or "category-theoretic logic") of which lawvere is one of the principal founders ....
?? and in fact that's really my central message here, that projective algebraic geometry can be seen as a part of categorical logic (?? and in fact algebraic geometry more generally can be seen as a part of categorical logic) ... and this message is directed mainly at categorical logicians themselves, to tell them how they can use what they already know in order to understand algebraic geometry and to contribute to its development .... in particular, if you're a categorical logician and you're interested in learning algebraic geometry, then there's a royal road for you to take in order to get there from where you already are ... instead of having to trudge along the commoners's (??) road, which may involve swallowing an awful lot of preliminary introductory material .... and this royal road is pretty different from what algebraic geometers might tell you themselves about how category theory is applied to algebraic geometry ....
(?? "turf wars" ... ?? .... category theory as very naively about getting past hierarchies to more interestingly reciprocal webs of interconnectedness, and that's part of what's going on here .... silly to argue over whose pastime most fundamentally subsumes whose; my message for a particular audience about something they understand as subsuming something else doesn't preclude other ways of looking at it ..... mutual subsumption and so forth .... ???? ...)
the basic idea of category-theoretic logic is (, very roughly,) to study categories with some kind of extra structure and to think of these structured categories as _theories_ of a sort, and to think of (????? "categorical logic" vs "categorical algbera" here ???????? .......) a functor preserving the structure as a _model_ of the domain theory, or as an _interpretation_ of the domain theory inside the codomain theory ....
(?? somewhere .... secondary central message (= not quite central message) directed particularly at attendees of a conference on "category-theoretic methods in representation theory" or whatever .... : that it's important to study categories of representations alongside certain other kinds of categories .... "unification" aspect of tannakian philosophy .... stack concept unifying groups (pure stackiness) with spaces (completely un-stacky stacks) .... ??? .... for example studying interpretation taking g-reps to quasicoherent sheaves over scheme x, as ess g-torsor over x ....... ????? .....)
????!!! get to stuff about .... moduli stack of models ..... as more or less "proj" construction here .... ??? ..... gauge theory and general relativity as prototypical examples of physical theories where "state of affairs" may have automorphism, aka moduli stack of models may be stacky (?? property:structure:stuff::axiom:predicate:type .... ???? ...) .... ???? ..... ??? whole idea of lawvere's "theories" (for example algebraic theories, toposes ....) as inspired (????) both by "logical theories" and by "physical theories" ...... ???? ......
?? speed of light joke around here ???? ..... section vs morphism ... algebra vs category .... ???? ....
pointing to tableau on whiteboard : "part of what's going on here is that we're introducing an alleged sort of mental/conceptual hygiene; separating quantities of different kinds into separate boxes .... ?? thereby allegedly avoiding certain kinds of mental/conceptual confusion ..." ..... ?????? ??? maybe stuff about related ideas ..... structured programming and so forth .... ???? "typed lambda calculus" ???? .....
?? hmm, maybe torture quiz right after introducing dimensional analysis example .... "we're supposed (according to me) to get a cryptomorphism to projective algebraic geometry .... ?? any opinion as to how that might go ?? .... what might correspond to the dimensions, and to the quantities that live in them ??? ....." ....
??? "i promised you a dictionary ... well this is basically it .... rather short .... you can add lots more entries to it if you want, but all the rest are basically determined by these two" .... ???? ....
??? rant about retrospective obviousness of correspondence here ... terminologies "dimension" vs "line-bundle" .... !!! "line" as "one-dimensional object", aka "dimension" !!!!!! (??? this rant as probably somewhat separate from rant about invertibility under tensor product as hallmark of "one-dimensionality", coming up slightly later ???? ....)
"dimensional analysis is essentially the study of dimensions and quantities that live in those dimensions, projective algebraic geometry is essentially the study of line bundles and sections that live on those bundles, and the two subjects are secretly isomorphic or "cryptomorphic" under this (pointing to 2-entry dictionary on whiteboard) correspondence" ....
(if you doubt the characterization of pag as "all about line bundles and their sections", then that's an additional propaganda point that i can try to rant about if you like .... ???? .....)
"at this point there are two main ways to mathematically formalize [?? what we're doing here ?? ... / ?? this correspondence ... ?? ...] :
1 organize the sections of all the line bundles over a projective variety into a _graded commutative algebra_, with the _grades_ being the line bundles ....
2 .... lawvere approach .... _obects_ being the line bundles .... ???? .....
...............
??speed-of-light joke here somewhere ... ??? but wasn't there something else i wanted here too???? ...
oh yes!!! ..... bit about .... sections of line bundles ..... ???? " ... lowbrow approach involving lowbrow geometric quotienting by re-scaling action on vector space .... before the quotienting (=?= introduction of grading ... g = a ... ??? ....) had functions on a space, but what are they now only functions on some space lying over the space you're interested in .... and if you think about this and pursue it far enough, you can see that what's happening with these former functions (from affine picture ...) is that they're now (in projective geometry) achieving the status of ("twisted functions" or) sections of line bundles ..... ???"
now what makes [this second approach / lawvere's category approach / ... ??? ...] so interesting is that because of the way in which it uses _categories_ it reveals pag as being a part(/aspect ??? ....) of the vast program of _categorical logic_ (?? or "category-theoretic logic") of which lawvere is one of the principal founders ....
?? and in fact that's really my central message here, that projective algebraic geometry can be seen as a part of categorical logic (?? and in fact algebraic geometry more generally can be seen as a part of categorical logic) ... and this message is directed mainly at categorical logicians themselves, to tell them how they can use what they already know in order to understand algebraic geometry and to contribute to its development .... in particular, if you're a categorical logician and you're interested in learning algebraic geometry, then there's a royal road for you to take in order to get there from where you already are ... instead of having to trudge along the commoners's (??) road, which may involve swallowing an awful lot of preliminary introductory material .... and this royal road is pretty different from what algebraic geometers might tell you themselves about how category theory is applied to algebraic geometry ....
(?? "turf wars" ... ?? .... category theory as very naively about getting past hierarchies to more interestingly reciprocal webs of interconnectedness, and that's part of what's going on here .... silly to argue over whose pastime most fundamentally subsumes whose; my message for a particular audience about something they understand as subsuming something else doesn't preclude other ways of looking at it ..... mutual subsumption and so forth .... ???? ...)
the basic idea of category-theoretic logic is (, very roughly,) to study categories with some kind of extra structure and to think of these structured categories as _theories_ of a sort, and to think of (????? "categorical logic" vs "categorical algbera" here ???????? .......) a functor preserving the structure as a _model_ of the domain theory, or as an _interpretation_ of the domain theory inside the codomain theory ....
(?? somewhere .... secondary central message (= not quite central message) directed particularly at attendees of a conference on "category-theoretic methods in representation theory" or whatever .... : that it's important to study categories of representations alongside certain other kinds of categories .... "unification" aspect of tannakian philosophy .... stack concept unifying groups (pure stackiness) with spaces (completely un-stacky stacks) .... ??? .... for example studying interpretation taking g-reps to quasicoherent sheaves over scheme x, as ess g-torsor over x ....... ????? .....)
????!!! get to stuff about .... moduli stack of models ..... as more or less "proj" construction here .... ??? ..... gauge theory and general relativity as prototypical examples of physical theories where "state of affairs" may have automorphism, aka moduli stack of models may be stacky (?? property:structure:stuff::axiom:predicate:type .... ???? ...) .... ???? ..... ??? whole idea of lawvere's "theories" (for example algebraic theories, toposes ....) as inspired (????) both by "logical theories" and by "physical theories" ...... ???? ......
Sunday, October 9, 2011
proj geom = dimensional analysis 6 (outline)
1 .... preamble and introduce baez's example ....
2 introduce two ways of formally interpreting the example, namely graded commutative algebra way and (more interestingly) lawvere's way, as "dimensional category" ....
(?? peculiar aspect here ... not really focusing on (?? contrast/relationship between 2 sides of) title equation .... rather, on contrast/relationship between standard approach and lawvere approach ... ?? towards .... understanding projective geometry ???? ..... ???? ...... towards establishing title equation ???? ..... .... some confusion here ... ???? ......)
3 using both ways from 2 above to connect to projective geometry .... _much_ more interestingly using lawvere's way ... ??? "now there are at least two ways to formalize what we're doing here in such a way as to reveal the connection to projective geometry" ...... ?????? ...... "first traditional way with perhaps most obvious direct connection to nominal topic of this conference; but more interestingly, ..." ..... ???? .....
3.1 ...."proj" ... "modding out by scaling group" ..... ???? ..... ????? ........ .... ??? "... try to return to this later ..."
3.2 ..... "taking moduli stack of models of dimensional theory" .... leading to 4 .... ????sa "model as state of affairs" .... "theory with free parameters...." .... ???? .....
4 rant about lawvere's concept of "theory" ... ??? and beck's concept of "doctrine" .... ??? and these as forming one approach ("categorical logic" approach ???? .....) to formulating/understanding/developing "tannakian program" / "geometric function theory" ...... ??????? ..... ???? include stuff (real rant ...) about .... word "theory" .... ideas it's supposed to encompass .... logic(tie in with mention of "categorical logic" above), physics .... ???? .... ??? also include stuff about higher doctrines and vector bundles and quasicoherent sheafs, going beyond line bundles .... ???? .... ???? lawvere's idea of theory as cat .... tied in with stuff, structure, propety = type, predicate, axiom .... ???? ..... ??gauge theories and general relativity as prototypical of physical theories where state of affairs may have automorphism / moduli stack may be genuinely stacky ... ?? ...
???? where in above (?? and/or below??) do we give the two basic dictionary entries (dimension = line bundle, quantity of given dimension = section of given line bundle) ??? ..... ???? should be semi-early, not too late ???? .....
5 ?? return as promised in 3.1 ??? ..... orbit stack .... "unification" approach to tammakian program .... ??? ....
??? somewhere try to work in "walking line-nicely-embedded-in-standard-n-space" .... brandenburg's correction .... ??? ..... ?? tied in with emphasis on "algebraic geometry as part/aspect of categorical logic" .... ???? ....
??? have we managed above yet to work in all the various approaches to tannakian program that we want to ?? ...
?? maybe part of some sort of cocnclusion .... "algebraic geometry for category theorists" idea ... that is, not to trust algebraic geometers's ideas as to how to apply category theory in algebraic geometry .... ??? ... instead, philosophy of "categorical loguc" .... ???? ....
2 introduce two ways of formally interpreting the example, namely graded commutative algebra way and (more interestingly) lawvere's way, as "dimensional category" ....
(?? peculiar aspect here ... not really focusing on (?? contrast/relationship between 2 sides of) title equation .... rather, on contrast/relationship between standard approach and lawvere approach ... ?? towards .... understanding projective geometry ???? ..... ???? ...... towards establishing title equation ???? ..... .... some confusion here ... ???? ......)
3 using both ways from 2 above to connect to projective geometry .... _much_ more interestingly using lawvere's way ... ??? "now there are at least two ways to formalize what we're doing here in such a way as to reveal the connection to projective geometry" ...... ?????? ...... "first traditional way with perhaps most obvious direct connection to nominal topic of this conference; but more interestingly, ..." ..... ???? .....
3.1 ...."proj" ... "modding out by scaling group" ..... ???? ..... ????? ........ .... ??? "... try to return to this later ..."
3.2 ..... "taking moduli stack of models of dimensional theory" .... leading to 4 .... ????sa "model as state of affairs" .... "theory with free parameters...." .... ???? .....
4 rant about lawvere's concept of "theory" ... ??? and beck's concept of "doctrine" .... ??? and these as forming one approach ("categorical logic" approach ???? .....) to formulating/understanding/developing "tannakian program" / "geometric function theory" ...... ??????? ..... ???? include stuff (real rant ...) about .... word "theory" .... ideas it's supposed to encompass .... logic(tie in with mention of "categorical logic" above), physics .... ???? .... ??? also include stuff about higher doctrines and vector bundles and quasicoherent sheafs, going beyond line bundles .... ???? .... ???? lawvere's idea of theory as cat .... tied in with stuff, structure, propety = type, predicate, axiom .... ???? ..... ??gauge theories and general relativity as prototypical of physical theories where state of affairs may have automorphism / moduli stack may be genuinely stacky ... ?? ...
???? where in above (?? and/or below??) do we give the two basic dictionary entries (dimension = line bundle, quantity of given dimension = section of given line bundle) ??? ..... ???? should be semi-early, not too late ???? .....
5 ?? return as promised in 3.1 ??? ..... orbit stack .... "unification" approach to tammakian program .... ??? ....
??? somewhere try to work in "walking line-nicely-embedded-in-standard-n-space" .... brandenburg's correction .... ??? ..... ?? tied in with emphasis on "algebraic geometry as part/aspect of categorical logic" .... ???? ....
??? have we managed above yet to work in all the various approaches to tannakian program that we want to ?? ...
?? maybe part of some sort of cocnclusion .... "algebraic geometry for category theorists" idea ... that is, not to trust algebraic geometers's ideas as to how to apply category theory in algebraic geometry .... ??? ... instead, philosophy of "categorical loguc" .... ???? ....
?? passage from comm ring to ag theory via taking module ag theory ... ??? as doctrine morphism, or as not ... ??? .....
?? globalization/spectrum adjunction here .... ??? hmmm, i was really expecting this _not_ to be a doctrine morphsm, but maybe it really is ..... ???? "underlying propositional theory of predicate theory" ??????? ....... ?????? .......... ?????????? ..........
?? wait, maybe (??...) some confusion here .... "spectrum" as not landing in _comm ring_ ???? .....
(i think part of what i was originally going to suggest here is that if this wasn't/isn't a doctrine morphism then maybe whatever we do with it might be interesting to try with other non-[doctrine morphism]s ... ?? but maybe also now, if it _is_ a doctrine morphism, interesting to try whatever we do with it to other doctrine morphisms .... ???? ..... ???? "change of colimits" .... ????? .....)
??? colimit vs 2-colimit here ??? ....
_comm ring_ -> _comm ring formal (2,1)-colimit_ -> _ag theory_
???? right adjoints to both of those ??? ..... ??? and to composite ??? ...
??? confusion ??? ....
??? fe ..... ???? consider poorer doctrine = dimensional, richer = ag .... ???? "dimensional spectrum of ag theory" ....... .... ??? .... globalization ..... ???? ..... ... ??? ....
?? globalization/spectrum adjunction here .... ??? hmmm, i was really expecting this _not_ to be a doctrine morphsm, but maybe it really is ..... ???? "underlying propositional theory of predicate theory" ??????? ....... ?????? .......... ?????????? ..........
?? wait, maybe (??...) some confusion here .... "spectrum" as not landing in _comm ring_ ???? .....
(i think part of what i was originally going to suggest here is that if this wasn't/isn't a doctrine morphism then maybe whatever we do with it might be interesting to try with other non-[doctrine morphism]s ... ?? but maybe also now, if it _is_ a doctrine morphism, interesting to try whatever we do with it to other doctrine morphisms .... ???? ..... ???? "change of colimits" .... ????? .....)
??? colimit vs 2-colimit here ??? ....
_comm ring_ -> _comm ring formal (2,1)-colimit_ -> _ag theory_
???? right adjoints to both of those ??? ..... ??? and to composite ??? ...
??? confusion ??? ....
??? fe ..... ???? consider poorer doctrine = dimensional, richer = ag .... ???? "dimensional spectrum of ag theory" ....... .... ??? .... globalization ..... ???? ..... ... ??? ....
Saturday, October 8, 2011
?? is it just some middle perversity for which poincare duality and so forth survive to the non-singular case ?? .... ?? "complementary perversities" here maybe ???? ..... ??? upper and lower middle as complementary ??? ....
?? hmmm, "Intersection homology groups of complementary dimension and complementary perversity are dually paired." (wpa) ..... ???? .....
?? "The (lower) middle perversity m is defined by m(k) = integer part of (k − 2)/2. Its complement is the upper middle perversity, with values the integer part of (k − 1)/2. If the perversity is not specified, then one usually means the lower middle perversity. If a space can be stratified with all strata of even dimension (for example, any complex variety) then the intersection homology groups are independent of the values of the perversity on odd integers, so the upper and lower middle perversities are equivalent." ... ??? don't remember how close i came to thinking it outloud, but i was vaguely imagining something like that happening ...
?? hmm, at the moment not seeing any perversity-dependence in wpa definition of "perverse sheaf" .... ??? .... ?? or stratification-dependence, even .... ??? ....
?? only one google hit for "middle perverse sheaf", none for "lower middle perverse sheaf" ... ?? ....
?? maybe that wpa is being sloppy ... ??? stratification as not pre-specified ... ??? while maybe mentioned core status requires pre-specification ?? .... ... ???? .....
?? hmmm, "Intersection homology groups of complementary dimension and complementary perversity are dually paired." (wpa) ..... ???? .....
?? "The (lower) middle perversity m is defined by m(k) = integer part of (k − 2)/2. Its complement is the upper middle perversity, with values the integer part of (k − 1)/2. If the perversity is not specified, then one usually means the lower middle perversity. If a space can be stratified with all strata of even dimension (for example, any complex variety) then the intersection homology groups are independent of the values of the perversity on odd integers, so the upper and lower middle perversities are equivalent." ... ??? don't remember how close i came to thinking it outloud, but i was vaguely imagining something like that happening ...
?? hmm, at the moment not seeing any perversity-dependence in wpa definition of "perverse sheaf" .... ??? .... ?? or stratification-dependence, even .... ??? ....
?? only one google hit for "middle perverse sheaf", none for "lower middle perverse sheaf" ... ?? ....
?? maybe that wpa is being sloppy ... ??? stratification as not pre-specified ... ??? while maybe mentioned core status requires pre-specification ?? .... ... ???? .....
"When MacPherson and I first started thinking about intersection homology, we realized that there was a number that measured the "badness" of a cycle with respect to a stratum. This number had the property that when you (transversally) intersected two cycles, their "badness" would add. The best situation occurs for cocycles, in which case that number was zero, and the intersection of two cocycles was again a cocycle. The worst situation was for ordinary homology, in which case that number could be as large as the codimension of the stratum. In that case, the intersection of two cycles could even fail to be a cycle. After a while it became clear that we needed a name for this number and we tried "degeneracy", "gap", etc., but nothing seemed to fit. It seemed that the bad cycles were being "obstinate", but "obstinateness" did not sound reasonable. Finally we said, "let's just call it the perversity, and we'll find a better word later". We tried again later, with no success. (We did not realize that in some languages the word is profane.) When we first went to talk with Dennis Sullivan and John Morgan about these ideas, we were calling the resulting groups "perverse homology", but Sullivan suggested the alternative, "intersection homology", which seemed fine with us. This was 1974-75. Later, when it was discovered that, for any perversity, there is an abelian category of sheaves, whose simple objects are the intersection cohomology sheaves (with that perversity) of closures of strata, Deligne coined the term "faisceaux pervers"."
?? hmm, having trouble getting some of this to parse ... ???
?? "The worst situation was for ordinary homology, in which case that number could be as large as the codimension of the stratum." ... ?? as maybe promising ?? ... ?? might fit with idea of "top perversity = normal" ...?? hmm, having trouble getting some of this to parse ... ???
(??? grothendieck bit about "bad object vs bad category" ?? ....)
Friday, October 7, 2011
?? intersection homology for given perversity as homology of certain canonical perverse sheaf for it ?? ...
?? coincidence ?? .... same number of perversities for n-manifold as arrow-flippings on quiver corresponding to pgl(n) ???? ..... ????? possibility (??upper/lower ???) perversity as maybe giving "best version of hall algebra" ??? .... ???? .....
?? coincidence ?? .... same number of perversities for n-manifold as arrow-flippings on quiver corresponding to pgl(n) ???? ..... ????? possibility (??upper/lower ???) perversity as maybe giving "best version of hall algebra" ??? .... ???? .....
?? relationship of intersection homology to relative homology ... ??? ....
?? perversity as device for systematically getting alternative core of arbitrary abelian category ???? ..... ???? .... ??? too strong to be true ?? .... ???? reliance on "geometric structure" beyond mere chain complex ... ???? ..... ??? ......
?? inclusion of higher stratum, vs of down-segment of lower stratums .... ???? ....
??? perversity as ... sort of moore-postnikov factorization operation ??? ...... ??? .....
??again (?? ...), any particular perversity acting as "identity operation" from any of these viewpoints ?? ....
quivers .... flag varieties .... ???? ....
?? perversity as device for systematically getting alternative core of arbitrary abelian category ???? ..... ???? .... ??? too strong to be true ?? .... ???? reliance on "geometric structure" beyond mere chain complex ... ???? ..... ??? ......
?? inclusion of higher stratum, vs of down-segment of lower stratums .... ???? ....
??? perversity as ... sort of moore-postnikov factorization operation ??? ...... ??? .....
??again (?? ...), any particular perversity acting as "identity operation" from any of these viewpoints ?? ....
quivers .... flag varieties .... ???? ....
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