The Algebra of Categorical Spectra

TL;DR

This thesis develops the theory of categorical spectra in higher categories, introduces their tensor product, and applies these concepts to derive the cobordism hypothesis with singularities.

Contribution

It constructs the tensor product for categorical spectra and uses it to analyze stability phenomena and derive the cobordism hypothesis with singularities.

Findings

Defined the tensor product for categorical spectra.

Applied the tensor product to study stability phenomena.

Provided a categorical derivation of the cobordism hypothesis with singularities.

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.