Tag - Higher category theory
In this short talk, I will introduce the notion of n-morphisms between two A∞-algebras. These higher morphisms are such that 0-morphisms correspond to standard A∞-morphisms and 1-morphisms correspond to A∞-homotopies. Their combinatorics are then encoded by new families of polytopes, which I call the n-multiplihedra and which generalize the standard multiplihedra. Elaborating on works by Abouzaid and Mescher, I will then explain how this higher algebra of A∞-algebras naturally arises in the context of Morse theory, using moduli spaces of perturbed Morse gradient trees.
The talk will be on the algebraic structure present in both parts of the title. This algebraic story is most pronounced for E2-algebras in the category of 2-vector spaces (also known as braided fusion categories). Condensation gives rise to an equivalence relation on such E2-algebras (Witt equivalence) with sets of equivalence classes exhibiting nice properties, e.g. being abelian groups (Witt groups). The Witt group of braided fusion categories is a countably generated abelian group with the torsion part annihilated by 32.
Derivator theory, initiated by Grothendieck and Heller in the '90s to correct the shortcomings of triangulated categories, motivated a lot of research regarding the foundation of (∞,1)-category theory, and its applications to algebraic geometry/topology.
For a 2-category theorist, a (pre)derivator is a familiar object - (a suitably co/complete) prestack on the category cat of small categories - and yet still little is known about the formal properties of the 2-category PDer. The present talk is motivated by the belief that time is ripe for a more conceptual look into the foundations of derivator theory, and that far from being a mere exercise in style, such a conceptualization yields many practical advantages.
After briefly outlining the essentials of "formal category theory'' (2-categories can be used to organize the theory of "categories with structure" just as category theory organizes the theory of "sets with structure"), I will report on a conjecture regarding the possibility to provide a "yoneda structure" or a "proarrow equipment" to the 2-category of pre/derivators. Under suitable assumptions, these are equivalent ways to equip PDer with a calculus of Kan extensions, and building on prior work of Di Liberti and myself, this allows to speak about "locally presentable" and "accessible" objects (showing that Adamek-Rosický and Renaudin's definitions eventually coincide); the overall goal is to provide a suitable form of special/general adjoint functor theorem for a morphism of prederivators (such a theorem would simplify a lot the life of the average algebraic geometer).
Factorization systems (both weak and strong) are commonly defined as consisting of two classes of maps satisfying a certain orthogonality relation and a factorization axiom. The standard definition of algebraic weak factorization system, involving comonads and monads, is rather different. The goal of this talk will be to describe an equivalent definition of algebraic weak factorization system emphasising orthogonality and factorization.
We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of ∞-categories, and exploit this to prove many classical theorems about monads in the ∞-categorical setting. Among other things, we prove that the category of algebras for an accessible monad on a locally presentable ∞ category is locally presentable. We also apply the result to construct examples of ∞-categorical monads from pretheories.
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