En-algebras in m-categories

Abstract

The study of EE_n-algebras in higher categories has attracted growing interests, both from various categorification programs in mathematics as well as the study of higher dimensional topological orders in physics. However, the complexity of these structures increases rapidly with the category level. In this thesis, we prove a connectivity bound for maps of infinity-operads of the form AA_{k_1} otimes \cdots otimes AA_{k_n} -> EE_n, and as a consequence, give an inductive way to construct EE_n-algebras in m-categories. To prove this result, we first develop a theory of arity restricted unital infinity-operads. Given k >= 1, we define unital k-restricted infinity-operads, which are variants of infinity-operads which have only (= k)-arity morphisms, as complete Segal presheaves on closed k-dendroidal trees, which are closed trees built from corollas with valences = k. Furthermore, we prove that the restriction functors from unital infinity-operads to unital k-restricted infinity-operads admit fully faithful left and right adjoints by showing that the left and right Kan extensions preserve complete Segal objects. Varying k, the left and right adjoints give a filtration and a co-filtration for any unital infinity-operad by k-restricted infinity-operads, generalizing the AA_k filtration for EE_1. Second, We prove a version of Eckmann-Hilton argument that takes into account both connectivity and arity of infinity-operads. Along the way, we prove a technical Blakers-Massey type statement for algebras of coherent infinity-operads.