hi. i think that we said that we'd try meeting on sunday. i just want to try to mention here some topics that i hope we get to talk about. actually, there are so many topics that i want to talk about, that there's little hope that we'll actually be able to talk about all of them in one day (even if you don't have topics of your own to bring up as well, which seems unlikely).
among the topics that i eventually want to talk about are both:
1:
ways of trying to solve some of the problems that we've been working on so far, such as how to prove some improved version of your theorem about the universal property of the quasicoherent sheaves over P^n as a symmetric monoidal cocomplete k-linear category. this includes trying to answer all of the questions that you've been sending to me that i haven't been able to answer yet.
but then also:
2:
topics that i'm interested in but which we haven't gotten around to talking about yet because of focusing up till now on the problems mentioned above (and also because of my general slowness). that is, i have optimistic dreams that i'd like to tell you about, about developing this work into an ambitious program; whereas if i wait to tell you about these dreams until after the foundation results are carefully established, then i might have to wait forever. (for example many of the ideas that alex and i have worked on are very undeveloped so far; i'm much better at formulating approximate conjectures and sketching the big picture of a program than at carefully proving theorems.)
the division into #1 and #2 above is imprecise, in that some topics lie in the overlap of them both. anyway, i'll try to list here some topics, mainly belonging in #2.
first, there are many additional examples of ag theories that i'd like to talk about:
the theory of an epimorphism from the unit object 1 to the direct sum 1 + 1.
the theory of an object.
the theory of a dualizable object.
the theory of an n-dimensional object.
the theory of a lie algebra object. (you mentioned "algebraic theories" the other day; i wasn't sure whether you were deliberately suggesting that the study of them is similar in many ways to the study of what i've been calling "algebraic-geometric theories". in fact, there's even a significant overlap between these two kinds of theories, as exemplified by this example of "the theory of a lie algebra object". in any case, i'm definitely suggesting a parallelism here, that both of these are examples of what i call a "doctrine", about which more later.)
the theory of an n-dimensional lie algebra object.
the theory of a flagged n-dimensional object.
the theory of a flag on the direct sum of n copies of the unit object.
the theory of a curve of genus g. (this example needs a lot of work to make sense, but it's a very interesting example!)
the theory of right-exact functors from a given finitely cocomplete k-linear category. (in other words, "the free symmetric monoidal finitely cocomplete k-linear category on a given finitely cocomplete k-linear category".)
and so forth; there are many other examples...
second, i'd like to discuss the general idea of what i call "doctrines". roughly, a doctrine is a groupoid-enriched category which is "locally finitely presentable" in an approperiate groupoid-enriched sense. ag theories form a doctrine; other examples include the doctrine of algebraic theories, the doctrine of coherent toposes, and so forth.
one particular idea about doctrines that i'd like to discuss is the idea that, for example, the "big zariski topos" of a sufficiently nice scheme x (or of something somewhat more general, such as an algebraic stack of some kind) can be thought of as "the same theory" for which x is the moduli space, but expressed in a different doctrine (namely the doctrine of coherent ringed toposes, instead of the doctrine of ag theories). i haven't succeeded in getting this to actually work yet, but i'm optimistic that the basic idea makes good sense.
third, i'd like to discuss the idea of studying universal properties of symmetric monoidal homotopy-cocomplete differential graded k-linear categories, including especially the ones arising by taking chain complexes of objects from a symmetric monoidal cocomplete k-linear category. very vaguely, i think that this is a "cohomological" or "higher" analog of what we're doing...
well, i guess that i'll stop here for now... there's other topics that i meant to include but i won't get around to mentioning them all here...
i guess that this message came out pretty disorganized, so maybe it won't be too readable for you, but maybe i can at least use it as a reminder for myself about some topics that i might like to eventually bring up...
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