nLab G-∞-category

Idea

Let $\mathcal{S}_G$ be the (∞,1)-category of G-spaces, and let $\mathcal{O}_G$ be the orbit category. Elmendorf's theorem provides an equivalence $\mathcal{S}_G \simeq \mathrm{Fun}(\mathcal{O}_G^{\op},\mathcal{S})$. The starting point of Parametrized Higher Category Theory and Higher Algebra is to define $G$-$\infty$-categories so that they satisfy Elmendorf's theorem.

Definition

Definition

The ∞-category of small $G$-$\infty$-categories is

$$\mathrm{Cat}_{G,\infty} \coloneqq \mathrm{Fun}(\mathcal{O}_G^{\op}, \mathrm{Cat}_\infty).$$

More generally, often $\mathcal{T}$ will be an orbital ∞-category; in any case, we make the analogous definition.

Definition

If $\mathcal{T}$ is an (∞,1)-category, then the (∞,1)-category of small $\mathcal{T}$-$\infty$-categories is

$$\mathrm{Cat}_{\mathcal{T},\infty} \coloneqq \mathrm{Fun}(\mathcal{T}^{\op}, \mathrm{Cat}_\infty).$$

Related concepts

• G-space

• equivariant homotopy theory

• Parametrized Higher Category Theory and Higher Algebra, orbital ∞-category, equivariant symmetric monoidal category

References

• 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)