nLab categorical spectrum

Idea

The notion of categorical spectra is the full categorification of that of (Omega-) spectra of spaces/ $\infty$ -groupoids, hence the generalization of spectra from $(\infty,0)$ -categories to $(\infty,\infty)$ -categories:

A categorical spectrum (Stefanich 2021) is an $\mathbb{N}$ -indexed set of pointed $(\infty,\infty)$ -categories $(\mathcal{C}_n, X_n)_{n \in \mathbb{N}}$ such that each stage is equivalent to the endomorphism object of the next stage:

(1)

\begin{array}{ccc} \mathcal{C}_n &\overset{\sim}{\longrightarrow}& End_{\mathcal{C}_{n+1}}\big(X_{n+1}\big) \\ X_n &\mapsto& id_{X_{n+1}} \mathrlap{\,.} \end{array}

This may be thought of as stagewise revealing a higher category structure with k-morphism possibly also in negative degrees: The k-morphisms of $\mathcal{C}_{n}$ are the $k-n$ -morphisms of the categorical spectrum. Therefore one also speaks of $(\infty,\mathbb{Z})$ -categories (Kern 2024, following Lessard 2019, 2022).

Since endomorphism objects $End(-)$ as in (1) are canonically monoid objects, the stages of a categorical spectrum carry the structure of monoid objects internal to $(\infty,\infty)$ -categories and, by iteration, in fact of coherently commutative monoid objects.

Concretely, in many examples, $\mathcal{C}_{n}$ happens to be an $(\infty,n)$ -category. In this case, each stage $\mathcal{C}_n$ in a categorical spectrum is exhibited as a monoidal $(\infty,n)$ -category and, by iteration, as a symmetric monoidal $(\infty,n)$ -category.

(-)^\times \;\colon\; CatSpectra \longrightarrow Spectra \mathrlap{\,.}

The $(\infty,n)$ -categories of complex super $(n-1)$ -vector spaces form a categorical spectrum whose Picard spectrum is the Anderson dual $I_{\mathbb{C}^\times}$ of the sphere spectrum.

Precursor discussion referring only to strict $\omega$ -categories:

Paul Lessard: Spectra as Locally Finite $\mathbb{Z}$ -Groupoids (2019) [video:YT]
Paul Lessard: $\mathbf{Z}$ -Categories I [arXiv:2206.00849]

Original discussion in the generality of $(\infty,\infty)$ -categories:

Germán Stefanich: Categorical spectra, §8.6 & §13 in: Higher Quasicoherent Sheaves, PhD thesis, UC Berkeley (2021) [escholarship:19h1f1tv, pdf]
Naruki Masuda: The Algebra of Categorical Spectra, PhD thesis, Johns Hopkins University (2024) [handle:1774.2/70013, arXiv:2605.03114]
David Kern: Categorical spectra as pointed $(\infty,\mathbb{Z})$ -categories [arXiv:2410.02578]