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.

Specifically, in many examples, $\mathcal{C}_{n}$ happens to be an $(\infty,n)$-category; these are the co-connective categorical spectra. In this case, each stage $\mathcal{C}_n$ is exhibited as a monoidal $(\infty,n)$-category and, by iteration, as a symmetric monoidal $(\infty,n)$-category.

If these symmetric monoidal $(\infty,n)$-categories are presentable (so that they have a good supply of colimits playing the role of addition) and the monoidal structure preserves these colimits (this playing the role of linearity), then Scholze 2026 Def. 1.18 speaks of StRings (formally dual to “Gestalten”), a grand categorification of commutative rings (and their formally dual affine schemes).

Stagewise forming Picard $\infty$-groupoids of these symmetric monoidal $(\infty,n)$-categories yields an ordinary spectrum of $\infty$-groupoids, which may be thought of as the $\infty$-group of units of the st-ring:

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

Examples

Example

The $(\infty,n)$-categories of complex super $n$-vector spaces form a categorical spectrum whose Picard spectrum (2) is the Anderson dual $I_{\mathbb{C}^\times}$ of the sphere spectrum.

(Due to David Reutter and Theo Johnson-Freyd, upcoming.)

Related concepts

• combinatorial spectrum

• (∞,Z)-category

References

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]

See also:

• Félix Loubaton: Theory and models of $(\infty,\omega)$-categories [arXiv:2307.11931]

On the generalization of Whitehead-generalized homology to coefficients being categorical spectra:

• Hadrian Heine: Homology of higher categories [arXiv:2505.22640]

On presentable categorical spectra as $\infty$-categorical commutative rings (“StRings”), formally dual to Gestalten (cf. duality between algebra and geometry):

• Peter Scholze; Def. 1.18 in: Geometry and Higher Category Theory, lecture notes (2026) [webpage, pdf]