Fosco Loregian
A n-simplex is a polytope with n vertices and a single arrow forall i ≤ j
There is a category Δ modeling simplices:
finite nonempty totally ordered sets + monotone maps
face
degen
Every category is a suitable presheaf on Δ ([Δ°, Set] = simplicial sets):
More formally, there is a functor i : Δ → Cat
sending a category to its nerve.
General result: Cat is an orthogonal class in simplicial sets
Every n-tuple of arrows eager to be composed has indeed a unique composite.
An ∞-category : an object injective wrt the same class of maps
Every n-tuple of arrows eager to be composed has a composite, unique up to a contractible space of choices.
Boom! you can redo category theory from scratch
Turns out some parts of Mathematics are easier if stated in these terms:
Wanna know more? Check kerodon.net!
A stable ∞-category is an ∞-category
The homotopy category of a stable ∞-cat is always triangulated
The correspondence sending an abelian A into its derived category has a nice and clear universal property
Stable, rational, p-adic, ... homotopy theory are pieces of commutative algebra of ∞-categories
A t-structure on D triangulated is a pair of triangulated subcategories of D such that every object X lies in a sequence
("the part w/ negative homotopy groups", "the part w/ positive homotopy groups")
[FL14] : On stable ∞-cats a t-structure is a factorization system (E,M)
(Algebraic) geometry has been reduced to (categorical) algebra (once again). Yay!
Check. Done in my Ph.D.
is an interesting set [FL-PhD]*
every prime p defines
Blakers-Massey in char p is a thm about factorization systems
Real numbers? Eeew. go p-adic!
has an interest in chromatic homotopy
*Still unpublished!
Fact: Bord(n) is the free (∞,n)-symmoncat on the point
monoidal functors Z : Bord(n) → Vect are completely classified and used in QFT
Morse theory is the theory of suitable factorization systems on Bord(n)
{critical points of a Morse function}
{critical values of a certain "slicing" J : R → FS(Bord(n))}
Too much ∞-categories make us dull boys.
Let's stop to dimension two.
A derivator is a strict 2-functor
satisfying sheafy conditions. They form the 2-category Der.
They subsume most of ∞-category theory (in a suitable sense, they are more general)
[LV17] : A t-structure on a stable derivator is still a certain kind of factorization system*
*And FS are still algebras for the "squaring" 2-monad ( _ )² : A ↦ A² (see [KT93])
[Lor18] : reflective subderivators correspond to reflective factorization systems, and to algebras for idempotent monads*
*(the "formal theory of monads" still holds in Der, a monad T : D → D is just defined objectwise)
To what extent category theory exports to the 2-category Der?
Well...
Uncle Ross to the rescue!
So: we have adjunctions, monads, factorization systems...
There is a Yoneda structure on the 2-category of derivators
If this is true
Preparatory conjecture: there is a pseudo-YS on Der (all universal properties are pseudo-ified)
Fernando, are you here?
Problem: It's all fun(ctors) and games until your grant finishes...
Solution: do applied mathematics. Make good money.
Save the world from climate change.
Avoid (or catalyze) next financial crisis.
I've always wondered, how would a machine do mathematics?
For category theory, I think the answer is in a functional language with dependent types.
class Profunctor p where
dimap :: (a -> b) -> (c -> d) -> p b c -> p a d
dimap f g = lmap f . rmap g
* :: Profunctor b c -> Profunctor a b -> Profunctor a c
(p * q) = \x z -> exist y . ( p x y , q y z )
Open Haskell's Data.Profunctor library
Welp, that's exactly profunctor composition!
A suitable kind of lens (a datatype to access nested records in a datatype) is described as a Tambara module
(profunctors on a monoidal domain A, compatible with an action of A on Prof(A,A))
Changing the action you obtain a whole lot of different lenses, to the point that there is a conjectural bijection between the two classes
Cool.
But we want more: generalize to other kinds of profunctors
A Petri net is a certain kind of multidigraph.
A few things done better with category theory