The Algebra of Categorical Spectra

Abstract

Categorical spectra are spectrum objects in pointed $(\infty ,\infty )$-categories: sequences $(X_n)$ equipped with equivalences $X_n\simeq \Omega X_{n+1}$. This thesis develops foundations for categorical spectra and constructs their tensor product, the stabilized analogue of the lax Gray tensor product of $(\infty ,\infty )$-categories. We use this tensor product to study stability phenomena, expressed as the coincidence of certain finite weighted colimits and limits. As an application, we give a precise categorical derivation of the cobordism hypothesis with singularities from the ordinary cobordism hypothesis, making rigorous a sketch of Lurie.

Revised version of the author's thesis, originally submitted in July 2024. The main results remain unchanged, but several proofs have been improved and typos are fixed. https://jscholarship.library.jhu.edu/handle/1774.2/70013 v2: removed incorrect claims about closure of strong Steiner complexes under partial duality (Prop 2.4.17 and Remark A.2.5)