nLab orbital ∞-category

Redirected from "orbital $\infty$-category".
Note: orbital ∞-category and orbital ∞-category both redirect for "orbital $\infty$-category".
Contents

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

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

Definition

An \infty-category 𝒯\mathcal{T} is orbital if 𝔽 𝒯\mathbb{F}_{\mathcal{T}} admits finite (co)products.

Definition

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

Basic properties

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

Span(𝔽 𝒯)A eff(𝔽 𝒯), \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.

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.

References

Many of the original papers on this topic:

On disjunctive (∞,1)-categories

On the slice filtration for inductive orbital categories

Last revised on May 11, 2024 at 23:44:56. See the history of this page for a list of all contributions to it.