This is the webpage for Seminar ∞, a seminar on higher category theory and its applications run during the fall 2016 semester at Ohio State.
spacetime coordinates: Tuesdays at 1pm in MA 105, beginning Sept. 6. We'll generally run for roughly 60-90 minutes, depending on a few factors.
core topic: fundamentals of ∞-category theory (and its relation with other methods of abstract homotopy theory, especially model categories)
statement of purpose: There's a serious disconnect between how ∞-categories are usually presented and how they're actually used (and actually useful). My central aim is to provide a working familiarity with both aspects of the theory.
prerequisites: Basic category theory (categories, functors, natural transformations, adjunctions) and basic homotopy theory (homotopy and co/homology groups, co/fiber sequences, weak homotopy equivalences). A bit of homological algebra could be helpful for motivation, but shouldn't really be necessary.
possible subsequent topics:
- stable ∞-categories (which completely repair the various defects of triangulated categories)
- fundamentals of derived algebraic geometry
- "Spec(F1)" via categorified DAG (i.e. chromatic homotopy theory); topological modular forms
- factorization homology via the ∞-category of manifolds: homology theories for manifolds, nonabelian Poincaré duality, Poincaré/Koszul duality and its interpretation via deformation theory
- ∞-categories generally: geometric representation theory (cf. e.g. Gaitsgory--Rozenblyum); Tanaka's spectral Fukaya category
- DAG: derived intersections (e.g. RCrit, the quantum (i.e. derived) Euler--Lagrange solutions); stacks; the co/tangent complex formalism; shifted symplectic structures and shifted Poisson structures
- Spec(F1) and CHT: recent work of Barthels--Heard--Valenzuela on Grothendieck duality
- factorization homology: knot homology theories; the bordism hypothesis, QFTs, and duality phenomena (e.g. PKD); Costello--Gwilliam's algebraic approach to Feynman integrals
Here is a guide which will hopefully be helpful for transitioning to student talks in the spring semester.