nLab orbital ∞-category

Idea

An orbital $\infty$-category is an (∞,1)-category satisfying a weak axiomatic framework generalizing the properties of the orbit category of a finite group allowing for the construction of the Burnside category.

Definitions

Given $\mathcal{T}$ an (∞,1)-category, we write $\mathbb{F}_{\mathcal{T}} \coloneqq \mathcal{T}^{\amalg}$ for its finite coproduct completion.

Definition

An $\infty$-category $\mathcal{T}$ is orbital if $\mathbb{F}_{\mathcal{T}}$ admits pullbacks.

Definition

An orbital $\infty$-category $\mathcal{T}$ is atomic if every retract in $\mathcal{T}$ is an equivalence.

Basic properties

If $\mathcal{T}$ is orbital, then $\mathbb{F}_{\mathcal{T}}$ is disjunctive, so we may form the Burnside category

$$\mathrm{Span}(\mathbb{F}_{\mathcal{T}}) \coloneqq A^{\mathrm{eff}}(\mathbb{F}_{\mathcal{T}}),$$

the latter denoting the (∞,1)-category of correspondences constructed by Barwick.

Examples

The following are all examples of atomic orbital $\infty$-categories.

• orbit categories of finite groups; more generally, orbit categories of profinite groups (where the stabilizers are required to be open);

• locally finite groups (where the stabilizers are required to be finite);

• any space;

• the cyclonic orbit 2-category (see at cyclotomic spectrum);

• the 2-category of connected finite groupoids and covering maps (i.e. the global orbit category);

• the category of finite sets of cardinality $\leq n$ and surjective functions;

• the topological categories of finite-dimensional inner product spaces (over $\mathbb{R}$ and $\mathbb{C}$) of dimension $\leq n$ and orthogonal projections.

For many cases of these atomic orbital $\infty$-categories there is a conservative (∞,1)-functor to a poset, and so they are EI (∞,1)-categories. Essentially finite EI orbital discrete categories have been considered by Wilson on work concerning the slice filtration, where they are called inductive orbital categories.

Related concepts

• equivariant symmetric monoidal category

• G-∞-category, Parametrized Higher Category Theory and Higher Algebra

References

Many of the original papers on this topic:

• Clark Barwick, Emanuele Dotto, Saul Glasman, Denis Nardin, Jay Shah, Parametrized higher category theory and higher algebra: A general introduction, (arXiv:1608.03654)

• Clark Barwick, Parametrized higher category theory and parameterized higher algebra, video

• Clark Barwick, Emanuele Dotto, Saul Glasman, Denis Nardin, Jay Shah, Parametrized higher category theory and higher algebra: Exposé I – Elements of parametrized higher category theory, (arXiv:1608.03657)

• Jay Shah, Parametrized higher category theory, (arXiv:1809.05892)

• Jay Shah, Parametrized higher category theory II: Universal constructions, (arXiv:2109.11954)

• Denis Nardin, Parametrized higher category theory and higher algebra: Exposé IV - Stability with respect to an orbital ∞-category, (arXiv:1608.07704)

• Denis Nardin, Stability and distributivity over orbital $\infty$-categories, (thesis)

• Saul Glasman, Stratified categories, geometric fixed points and a generalized Arone-Ching theorem, (arxiv:1507.01976)

• Saul Glasman, Goodwillie calculus and Mackey functors, (1507.01976)

On disjunctive (∞,1)-categories

• Clark Barwick, section 4 of Spectral Mackey functors and equivariant algebraic K-theory (I), Adv. Math., 304:646–727 (arXiv:1404.0108)

On the slice filtration for inductive orbital categories

• Dylan Wilson, On categories of slices (arXiv:1711.03472v1)