nLab fiberwise core

Definition

Definition

Let $\varphi:\mathcal{C} \rightarrow \mathcal{B}$ be a cocartesian fibration. Then, the fiberwise core of $\mathcal{C}$ is the cocartesian fibration

$$\mathrm{St} \left( \mathcal{B} \xrightarrow{\mathrm{Un} \varphi} \mathbf{Cat}_\infty \xrightarrow{(-)^{\simeq}} \mathcal{S} \hookrightarrow \mathbf{Cat}_{\infty} \right),$$

where $\mathrm{St}$ and $\mathrm{Un}$ denote the straightening and unstraightening functors and $(-)^{\simeq}$ denotes the core.

If $\mathcal{B} = \mathcal{O}_G^{\mathrm{op}}$ is the orbit category of a finite group, then this is naturally a G-space via Elmendorf's theorem, and we also refer to this G-space as the fiberwise core of $\mathcal{C}$.

Related concepts

• G-∞-category