we understand to some extent the important role played by the doctrine of symmetric monoidal finitely cocomplete algebroids in algebraic geometry... but on the other hand, this leaves the role played by finite limits still somewhat mysterious... thus it raises foundational (or something...) questions like: why are people all the time using abelian categories as a tool in algebraic geometry, rather than mere finitely cocomplete algebroids? ...
so roughly i want to try to present an answer to this sort of question... though i'm not sure yet exactly how good an answer it is, and the answer is still somewhat fuzzy...
very roughly, my tentative answer is that when you pass from the doctrine of symmetric monoidal finitely cocomplete algebroids to the infinity-doctrine (or something...) of differential graded such (or something like that...) (now why would you want to do such a thing?? ... well, i'll try to get to that...), the fundamental opposition between limits and colimits is ameliorated by a reconciliation... first of all, in this higher context ordinary limits and colimits get replaced by homotopy limits and homotopy colimits, but moreover, it turns out that the difference between homotopy limits and homotopy colimits evaporates... in a certain sense... and in a certain context... well, there's a lot of qualifications needed here...
the story that i'm trying to tell john here is one of those stories whose telling seems to require an irreducible amount of suspense... the story evaporates if you tell it all at once, because the tension to be resolved in a later chapter needs to be built up in an earlier chapter... perhaps this indicates that the story-teller's own understanding is still lacking, or they'd be able to convey their understanding more directly, without detouring through intermediate stages of partial understanding... certainly my own understanding is lacking in this case... but sometimes that's the way it is, and you need to tell the story in stages like that... i remember william zame, when he wanted to try to tell us such a story in a first-year graduate analysis course, would sometimes, short on time or maybe just patience, try to compress it into a single paragraph or maybe just a single sentence, like "at first you think such-and-such... but then later you realize some-other-such-and-such..."... that's the sort of situation i'm in here, short on time or maybe patience... someone's patience, at least...
the intermediate stage of understanding that i think i need to detour through here, and that i'm suspecting john hasn't had a chance to see much of yet, is the stage where you actually take the doctrine of abelian categories seriously as a doctrine, more specifically as (more or less) a lawvere-style doctrine, a monad (though "weak"...) on the (weak...) 2-category of algebroids... or something like that... apparently it was worked out by peter freyd a very long time ago that this monad can be understood as a composite of the monad for finitely complete algebroids and the one for finitely cocomplete algebroids, with a barr-beck braiding (aka "distributivity law") between the two monads... that is, a monad is a type of monoid, and the tensor product (in this case functor composition) of two monoids isn't generally a monoid unless you have a sufficiently nice "braiding morphism" (aka "yang-baxter operator", although there's no yang-baxter constraint required until three objects are involved instead of just two) between the two monoid objects, either a "nonce" braiding or one coming from a "global" braiding (making the monoidal category in which the monoid objects live braided monoidal). a nonce braiding making two monoid objects tensorable is called a "barr-beck distributivity law", and that's what we have in this case...
ideally, i would like to go into lots of detail about this... exactly how the braiding works (to the extent that i understand it so far... the apparent adjointness between cokernel and kernel, and/or the "commutativity" between them...), and going through some nice simple examples such as "the walking short exact sequence abelian category" and how it relates to "the walking epi finitely cocomplete algebroid"... all directed towards building up the central tension, the opposition between limits and colimits in algebroids... but also planting the seed of the eventual reconciliation of the opposition, which is the striking overlap between limits and colimits in algebroids, namely the way in which finite direct sums qualify as both... i also want to discuss freyd's analogy between abelian categories and toposes (or perhaps better, "coherent geometric theories" or something like that), with a similar barr-beck braiding between limits and colimits, and how this relates to flat modules and "the abelian analog of diaconescu's theorem"...
No comments:
Post a Comment