nLab coefficient system

Definition

Let $G$ be a group, $\mathcal{O}_G$ its orbit category, and $\mathcal{C}$ an ∞-category (or, in particular, a 1-category).

Definition

A G-coefficient system valued in $\mathcal{C}$ is a functor $\mathcal{O}_G^{op} \rightarrow \mathcal{C}$.

Examples

1. A version of Elmendorf's theorem states that the $\infty$-category of G-spaces is equivalent to $G$-coefficient systems in the $\infty$-category $\mathcal{S}$ of spaces.

2. G-∞-categories are simply coefficient systems of ∞-categories.

3. Coefficient systems of sets and Abelian groups play a prominent role in equivariant homotopy theory; for instance, the equivariant homotopy groups $\pi_n^{\bullet}(-)$ form a coefficient system of sets, for all $n$, which may be upgraded to groups when $n \geq 1$ and to abelian groups when $n \geq 2$.

4. Coefficient systems of spectra are the stabilization of G-spaces.

5. Mackey functors have underlying coefficient systems, given by pullback along the embedding $\mathcal{O}_G^{\op} \hookrightarrow \mathbb{F}_G^{\op} \hookrightarrow \mathrm{Span}(\mathbb{F}_G)$, the latter inclusion being the wide subcategory of backwards maps.