The Algebra of Categorical Spectra

Categorical spectra are spectrum objects in pointed $(\mathrm{\infty },\mathrm{\infty })$-categories: sequences $(X_n)$ equipped with equivalences $X_n\simeq \mathrm{\Omega }{X}_{n+1}$. This thesis develops foundations for categorical spectra and constructs their tensor product, the stabilized analogue of the lax Gray tensor product of $(\mathrm{\infty },\mathrm{\infty })$-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.