Wednesday, June 30, 2010

discussion with baez today

1 algebroid of monos as equivalent to algebroid of epis?? because both equivalent to algebroid of short exact sequences??

2 ??situations where category of projective modules is finitely cocomplete but with its own colimits?? or something like that??

3 doctrine of cauchy complete algebroids?? some work of trimble with baez...

4 walking epi symmetric monoidal finitely cocomplete algebroid... an object in the nth grade of the syntactic algebroid as something like "a filtered vector space with n+1 stages, with an action of n! on the total space, such that the jth stage is closed under the corresponding young subgroup" ?? ??or something like that...

5 classification of indecomposable reps of borel lie algebra ... how this relates to #4 above and/or to quiver representations ...

Monday, June 28, 2010

doctrines 6

on the one hand, doctrines are to (weakly) groupoid-enriched categories what "finite limits theories" (aka "lex theories") are to ordinary categories. this lets us apply to the study of doctrines ideas coming from the study of finite limits theories, notably "gabriel-ulmer duality" and the method of constructing finite limits theories by "sketches". this makes doctrines fairly easy to work with.

on the other hand, the concept of doctrine arises naturally in trying to understand the foundations of algebraic geometry (the reasons for going beyond affine varieties to projective varieties and schemes, the importance of line bundles and vector bundles and coherent sheaves, classifying toposes and moduli stacks, and so forth), so doctrines aren't just easy to work with; there's also something worthwhile to be accomplished by working with them.

(in fact for us the necessity of working with and understanding doctrines much preceded the realization that doctrines can be seen as the groupoid-based analog of finite limits theories, and that ideas from the study of finite limits theories come ready-made to apply to the study of doctrines.)

first we'll set up a "dictionary" connecting concepts from the study of finite limits theories with concepts from the study of doctrines; then we'll set up a "catalog" of doctrines including some toy examples but also examples of actual relevance to algebraic geometry.

a "finite limits theory" t is a small category (sometimes called "the syntactic category of t" or "the category of formulas of t") with all finite limits. a "doctrine" d is a small groupoid-enriched category (sometimes called "the syntactic 2-category of d" or "the 2-category of theories of d") with all finite homotopy limits.

for a finite limits theory t, the category of finite-limit-preserving functors from the syntactic category of t to the category of sets is called "the category of models of t". it's a large category of a certain kind, from which the syntactic category of t can be recovered by certain means. for a doctrine d, the groupoid-enriched category of finite-homotopy-limit-preserving groupoid-enriched functors from the syntactic 2-category of d to the groupoid-enriched category of groupoids is called "the groupoid-enriched category of environments for d". it's a large 2-category of a certain kind, from which the syntactic 2-category of d can be recovered by certain means.


hmmm, what about the idea of trying to use "infinity-doctrines" and have the (infinity,1)-category of infinity-doctrines be the (infinity,1)-category of infinity-environments of an infinity-doctrine .... ?????or something like that .... ?????....

??is it really true that the concept of "finite homotopy limits" is a/the good concept in the same way that the concept of "finite limits" is?? .... or something like that...

??so the way that gabriel-ulmer duality (of the "classical" kind?? or something like that) works is that _no_ formula is a model, nor vice versa ... ??no wait a minute, i'm very confused here now.... i guess that i was thinking of something like how no small category of formulas is a large category of models (or something like that...) ... so some sort of level slip or something...

i suppose that i should also consider the finite limits theory of "categories with all finite limits" ... or something like that...

Sunday, June 27, 2010

how does the relationship between the module category of an associative algebra and the algebraic representation category of its multiplicative algebraic monoid and/or group relate to the relationship between the finitely cocomplete algebroid doctrine and the symmetric monoidal finitely cocomplete algebroid doctrine?

Saturday, June 26, 2010

so let's try to explicitly describe the syntactic category of "the algebraico-geometric theory of one epimorphism"...

what about something like "homological algebra" and "derived functors" and so forth for finitely cocomplete algebroids that are far from being abelian?? ...or something like that...

also, consider the forgetful algebro-geometric morphism from the moduli stack of epimorphisms to the moduli stack of morphisms ...

Friday, June 25, 2010

dimensional analysis = projective algebraic geometry

it's no big surprise that dimensional analysis is related to projective algebraic geometry, although you could have spent some time thinking about either or both of these subjects without particularly noticing it. both concern the study of "homogeneous" quantities, meaning quantities that transform under rescaling in certain simple ways. what might be more surprising though is that clarifying this relationship is linked with a unified development of good category-theoretic foundations for algebraic geometry and physics and logic based on the concept of "theory of a doctrine" occurring in the work of beck and lawvere.

doctrines 5

alex hoffnung asked me today about why i think that for purposes of algebraic geometry we should be developing a concept of "doctrine" based on groupoids instead of (as jon beck apparently originally envisioned) on categories, so i want to try to answer that to some extent here.

my first reaction is that one of the big reasons for this is that one of the key examples of a doctrine, namely the doctrine of "dimensional theories", probably needs to be groupoid-based rather than category-based because the objects of a dimensional theory, being "line objects", are invertible under tensor product, and the "inverse object" functor is contravariant, thus not fitting with the most straightforward version of the idea of category-based doctrines.
one thing that i've gotten confused about in the past is about how groupoid objects in for example the category of locales can correspond to toposes whose model categories are actual categories rather than mere groupoids. this is coming up again in some things that i'm working on now, so i should try to work out soem of the ideas and examples here...

categorified gabriel-ulmer duality and meta-sketching of doctrines

since a doctrine (in my current usage) is a categorified version of a finite limits theory, and since finite limits theories can be "sketched", doctrines can also be sketched, in a slightly categorified way that i'll refer to for now as "meta-sketching". moreover it seems instructive to actually meta-sketch a bunch of examples of doctrines, some toy examples as well as some examples closer to being actually relevant.

Thursday, June 24, 2010

let's consider homotopy-limit-preserving functors from _simplicial set_^op to _simplicial set_...

for example, when does a strictly representable functor have this property?

??is there a "stackification" issue here?? ...or something like that...
baez reminded me today (actually yesterday i guess) about some ideas about dircihlet functions and "string theory" and harmonic oscillator and "partition functions" ... at one point i thought that i understood some little bits of this stuff...

baez tried to tell me about how this was supposed to relate to some semi-mysterious comments of connes's about the generating function "x/(1-exp(x))", or something like that ...

attaching map for bruhat cell 2

can we understand the attaching process here in terms of something like "artin-wraith (co-)glueing for ag theories"? we've had some difficulties with similar ideas before, but ... ??...

and... ??is there something here about the "anchor" being not a quotient of the projective space of the attached cell but rather a limiting case of it?? ??or something like that?? ??something about how this relates to stuff that we've run into recently about "irreducible varieties converging towards reducible ones", or something like that?? ...

Wednesday, June 23, 2010

lawvere

something i've wondered about, about whether lawvere explicitly meant to draw certain connections between ideas that i remember him talking about...

i have a pretty isolated and vague memory of lawvere talking once about a certain category-theoretic interpretation of "dimensional analysis" that he'd developed. i think that this was in the middle of a course of lectures or a seminar on other stuff, and was presented as somewhat of a digression. (i think that i remember him writing it down on the side blackboard as though apart from the main action on the front blackboard.) i think that he also presented it somewhat laconically as an idea that he'd been somewhat excited about without managing to get too many other people excited about it.

i pretty much forgot the details of his idea, but later came to suspect that it was more or less the same as an idea that i rediscovered semi-recently, which is that a symmetric monoidal algebroid with all objects invertible and all braidings trivial is on the one hand equivalent to a sort of graded commutative algebra, while on the other hand can also be interpreted as a "dimensional theory", meaning a theory (in a certain sense) built according to the rules of dimensional analysis. the objects of the algebroid correspond to the grades of the graded commutative algebra and also to the "dimensions" in which the quantities of the theory live.

moreover, the usual technique (what algebraic geometers sometimes abbreviate as "proj") for extracting a projective variety from a certain kind of graded commutative algebra can be applied in this setting (in a slightly generalized way) to show that dimensional analysis is secretly essentially equivalent to slightly generalized projective algebraic geometry. in retrospect this is "obvious", in that both are about the study of "homogeneous quantities" and of covariance of quantities with respect to rescaling. it is similarly "obvious" that the concept of "dimension" in dimensional analysis manifests itself in algebraic geometry as the concept of "line bundle" (which is an abstract way of looking at a projective embedding); after all a "line bundle" is really just a _line_ from an "internal" viewpoint, and a "line" is a pure 1-dimensional object, aka a "dimension".

despite this "obviousness" i don't remember hearing people (not even lawvere)
talk about this. maybe it's one of those things that's so "obvious" that people don't talk about it, but i generally favor talking about such things. or maybe people do talk about it but i just haven't been listening carefully enough.

anyway, one of the things that i wonder about lawvere is whether he explicitly intended to connect dimensional analysis with projective algebraic geometry in this way. but furthermore, i wonder whether his use of the word "theory" in certain contexts is explicitly intended to emphasize this kind of connection.

thus my own usage of "dimensional theory" to describe a certain kind of graded commutative algebra or equivalently a certain kind of symmetric monoidal algebroid is itself intended as a sort of pun, in that a dimensional theory is a "theory" in what i perceive of as two ways (though i wonder whether lawvere himself perceives it as just a single way). first, it's a "theory" in a sort of lowbrow old-fashioned "physical" sense; it contains some quantities (such as perhaps "x-component of momentum of particle 1" or "y-component of angular momentum of particle 2") living in various dimensions (such as "momentum" or "angular momentum") and obeying some homogeneous (aka "dimensionally consistent") algebraic constraints (such as "conservation of momentum" or "conservation of angular momentum"). but second, it's a "theory" in the sense developed (according to lawvere) by jon beck and later explored in more detail by lawvere, of being a category equipped with some sort of extra structure amounting to a so-called "doctrine"; in this case the doctrine of "symmetric monoidal algebroids with all objects invertible and all braidings trivial".

thus i wonder whether this was lawvere's (and/or even beck's) intent all along- that when we hear of a "theory" we're supposed to think about it in both of these ways.

(perhaps the concept of "logical theory" is a third strand here, again problematic to decide how entangled it is with these other two.)

another synonym for "line bundle" is "invertible sheaf". invertible sheaves take their place among the "coherent sheaves", which in turn live in a ringed topos. the puns turning on the meaning of "theory" become only sharper in the context of these enrichenings.

anyway i tried writing to lawvere about this, to try to find out some of the history of this, about how explicitly he intended various connections to be drawn, but i haven't heard back from him. maybe he's written about this somewhere without my having read it or understood it properly.
discussion with baez today... "splitting principle" ... sa weak pullback of stacks / weak pushout of theorys ... putting a flag on particular object in the theory...

sa... whether 1) "structure augmentation" here gives an embedding on "k-theory", or something like that... but also, whether 2) the given object splits in k-theory under the augmentation... sa taking the augmentation to be a flag ct a splitting... taking "k-theory" to be direct sum k-theory ct exact sequence k-theory ... sa which of 1), 2) fails in these 4 cases... or something like that...

??sa theory of lambda rings as "structure theory of k-theory classifying space/spectrum" in some sense??

??baez suggested sa "tall-wraith monoid" here...

sa "multiplicative characteristic class" as map (of spectrums or of spaces os) from classifying space for vector bundles to eilenberg-maclane spectrum of some commutative ring, taking direct sum of vector bundles to multiplication, in some sense...

??sa multiplicative characteristic class as determined by values on line bundles... sa splitting principle... sa using "splitting principle" (os...) to get multiplicative characteristic class from formal power series in x... sa splitting vector bundle into line bundles and then... ??? ... sa "formal power series in first chern class" ... ??sa "first chern class as "_only_" characteristic
class for line bundles...

??sa "cohomology of cp^infinity as close to formal power serieses" ... os...

??sa "symmetric function, suppressing mention of variables because it's symmetric in them..."
os... ??....

??sa "chern-weil" ... invariant polynomials... [classifying space for 1d vector bundles]^n -> classifying space for nd vector bundles ... sa strict equalizer of weyl group action at cohomology level ... os, asf os... sa case of general simple lie gp, os... asf os...

...didn't get around to discussing galois theory aspects (??sa "adams operations" asf...) today...
so why does the space of isotropic linear 1d subspaces of a symplectic vector space form a contact manifold? is this an interesting way of thinking about it?
something about idea of "multiplicative characteristic class" or something?? something about... map from classifying space for something like vector bundles to classifying space for something like cohomology classes with coefficients in something pretty simple... or something... something about something like ring structure on classifying space for vector bundles... something about how map gets along with ring structure here... ??or something like that??...

doctrines 4

consider "the theory of an epimorphism" in the doctrine of finitely co-complete algebroids. give a nice explicit description of this theory in terms of a serre subcategory, or something like that...

Tuesday, June 22, 2010

attaching map for bruhat cell

let's consider the example of the big bruhat cell for the flag variety of pgl(3). this is the image of the map

str upper tri -> str upper unipotent -> cosets(lower tri)

where the first arrow is exponentiation and the second is... almost obvious, but we're going to have to be somewhat careful here...

and then we're going to be interested in limiting behavior...

let me try again...

we're interested in the decomposition of g/k wrt a certain "observer" subgp h (secretly the stabilizer borel of the favorite total flag). we have certain representatives in g/k of the components, i guess.

actually, how are we going to get those representatives in our example?? what we actually seem to have so far is the conjugate stabilizer subalgebras of those representatives...

i guess actually it should be ok; officially the weyl group is defined as special cosets of the cartan... or something like that... hmm, still need to work this out...

doctrines 3

consider "the finite limits theory of a set", and try to interpret it as a doctrine. contrast this to "the doctrine of a groupoid"...

also, consider taking models of a finite limits theory in the finite limits environment of (say for example) abelian groups rather than sets, as an analog of taking theories of a doctrine in the meta-environment of categories rather than groupoids...

Monday, June 21, 2010

adams operations and galois stacks

this could be completely the wrong idea, but...

baez has been telling me about the idea of taking the "grothendieck lambda ring" of an algebraico-geometric theory, and then interpreting its "adams operations" as some sort of action of the absolute abelian galois group (or something like that). and i'm wondering now whether in the case where the algebraico-geometric theory is something like the modules of a number ring (or maybe the coherent sheaves over the associated galois stack, or something like that ...???...), you get some hopefully semi-obvious action sort of embodying "artin reciprocity" ... ???or something like that...

hmm, so what _about_ the adams operations on the character ring of a finite group, and on the lambda ring of coherent sheafs over a galois stack?? _is_ this the right idea, or is there still some bad level slip here?? or something...

galois-schubert correspondence

galois stacks

sesquicoherent sheaves

champ vs gerbe

doctrines 2

some background on why i'm trying to (re?)invent a concept of "doctrine"...

in some of my semi-recent attempts to understand algebraic geometry, i found that an important aspect of what's going on is that the objects that algebraic geometers study tend to form themselves into 2-categories rather than 1-categories as you might naively expect.

(it could be that further study would show that n-categories for n>2 (or other sorts of higher categorical structures) are even more salient than 2-categories here, but 2 is at least a step on the way to n, and it could be in some sense the important step.)

thus naively you might expect that algebraic geometry is roughly the study of the objects in the category of projective varieties (and/or in a few related categories such as the category of "schemes"). to a projective variety x however is associated the category x# of so-called "coherent sheaves" over x, and it turns out that x# knows everything important there is to know about x, and that focusing on x# gives a purer and more illuminating picture of what's going on geometrically and conceptually than focusing on x does. then when you start axiomatizing what kind of object x# itself is, you realize that it's some kind of category-with-extra-structure and that this kind of categories-with-extra-structure will most usefully form a 2-category rather than a 1-category.

one reason

motives

questions

crackpot matrixes

hasse-weil zeta function of elliptic curve

euclidean coxeter complex

stackification

moduli stack of stable genus g curves

Sunday, June 20, 2010

doctrines

definition: a "doctrine" is a small (weakly) groupoid-enriched category with all finite homotopy-limits.

first of all, to what extent does this definition really make sense? and what about an alternative version using simplicially-enriched categories instead of groupoid-enriched? (does this reveal any problems with the given definition?)

what about interpreting an ordinary finite-limits theory as a doctrine according to this definition?
for example the finite-limits theory of a monoid. does this idea contradict some other idea that we had about categorifying the theorem that says that "the finite-limits theory is formed by the opposite of the finitely-presented models" in a certain context??

Saturday, June 19, 2010

invariant contact distribution on flag manifold

i've been working with john huerta for a while on a project to understand how the simple lie algebra "g2" is related to the geometry of a rolling ball. this idea has probably been pretty much completely worked out by other people already, but as usual that doesn't stop me from trying to understand it in my own way, sometimes consulting what other people have done and sometimes not. also this research isn't very narrowly focused and instead digresses off in other directions; the benefit of learning all sorts of things during such digressions is perhaps one of the excuses for taking such a long time working on what should probably be a simple project not especially close to the research frontier.

i forget exactly how it happened, but john and i were reading some paper that talked about an "invariant 2,3,5-distribution" on one of the "grassmanian" homogeneous spaces of g2 , and also about invariant distributions associated with graded nilpotent lie algebras, and for some reason i decided to stare at the "root system" of g2:

[still working on finding a good way to post hand-drawn pictures here]

and see whether i could see anything in this picture that meshed with what the paper was talking about.