This is a page to record ideas that I have learned in discussions with James Dolan over the last several months, or possibly years. I have many, many pages of notes, which I would like to organize and develop here.
We consider the algebraic stack of “bi-flagged $n$-dimensional vector spaces” and develop the idea that coherent sheaves over this stack can be understood as “categorified Hecke operators”. Similar ideas have been discussed by various authors (Ben-Zvi/Francis/Nadler, Baez/Hoffnung/Walker, Lurie, Schreiber) though in somewhat different language. Here, we attempt to take seriously the philosophy of treating “coherent sheaves” first; that is, instead of first describing an algebraic stack and then calculating its category of coherent sheaves, we argue that it is generally more illuminating to do it the other way around. In particular, we specify the algebraic stack only by “presenting” the category of coherent sheaves over it as a “theory” of the “doctrine” of finitely co-complete symmetric monoidal algebroids, with the stack itself appearing as an afterthought as the moduli stack of “models” of the theory.
The “doctrine” philosophy (including its relationship to the philosophy of “categorified function theory”) and how we plan to apply it to concrete problems about categorified hecke operators.
How to use the “doctrine” philosophy to extract an incidence geometry from a torsor of a simple algebraic group. (some use of topos theory aka “doctrine of geometric logic”.)
Specific examples of sesquicoherent sheaves (categorified hecke operators).
Relationships to more mainstream work involving perverse sheaves, the category O, and so forth.
The “Galois-Schubert correspondence” classifying nice subgroups of simple algebraic groups.
Some of this should overlap quite a bit with Baez’s personal web page on Doctrines.
Definition A doctrine is a $(2,1)$-category with finite homotopy limits.
This is, of course, a very general definition. The point is to think of this as a categorification of the notion of finite limits theory or essentially algebraic theory, and to see through examples that this is somehow a useful idea.
Example - finite limits theory of a monomorphism
We can, in some sense, define this finite limits theory by the universal property of the “walking monomorphism”:
We want to think of this arrow as a sketch of our theory. In particular, it will be a “limit-sketch”, since we are specifying a finite limits theory. Then we know that we should expect the categories of models to be a locally presentable category. (Note: it may or may not be very useful to be very careful in writing down this sketch, so for now this is a bit sketchy.)
Since we are expecting our sketch to “generate” our theory, we want to use the sketch to write down set-builder formulas for other objects in the theory. (NEED TO DO THIS NEXT)
The reason to try to understand finite limits theories by sketching is that we can try to understand a categorified version of Gabriel-Ulmer duality for degenerate doctrines by learning how to categorify our sketching tools to the setting of doctrines.
Example - Toy Doctrines, the theory of one-object