Factorization homology I: Higher categories

Factorization Homology of Enriched ∞-categories

For an arbitrary symmetric monoidal∞-category V, we define the factorization homology of V-enriched∞-categories over (possibly stratified) 1-manifolds and study its basic properties. In the case that V is cartesian symmetric monoidal, by considering the circle and its self-covering maps we obtain a notion of unstable topological cyclic homology, which we endow with an unstable cyclotomic trace ...

Higher Operads, Higher Categories

AN INTRODUCTION TO HIGHER CLUSTER CATEGORIES

In this survey, we give an overview over some aspects of the set of tilting objects in an $m-$cluster category, with focus on those properties which are valid for all $m geq 1$. We focus on the following three combinatorial aspects: modeling the set of tilting objects using arcs in certain polygons, the generalized assicahedra of Fomin and Reading, and colored quiver mutation.

Interpolation Categories for Homology Theories

For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by using truncated versions of resolu...

Factorization Homology of Topological Manifolds

Factorization homology theories of topological manifolds, after Beilinson, Drinfeld and Lurie, are homology-type theories for topological n-manifolds whose coefficient systems are n-disk algebras or n-disk stacks. In this work we prove a precise formulation of this idea, giving an axiomatic characterization of factorization homology with coefficients in n-disk algebras in terms of a generalizat...