nLab groupoid infinity-action

Idea

The generalization of ∞-group ∞-actions to $\infty$-actions of groupoid objects in an (∞,1)-category.

Definition

Definition

For $\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ a groupoid object, $P \in \mathbf{H}$ any object equipped with a morphism $a \colon P \to \mathcal{G}_0$ to the object of objects of $\mathcal{G}$, a $\mathcal{G}_\bullet$-groupoid ∞-action on $X$ with anchor $a$ is a groupoid $(X//\mathcal{G})_\bullet$ over $\mathcal{G}_\bullet$ of the form

$$\array{ \vdots && && \vdots \\ \downarrow \downarrow \downarrow \downarrow && && \downarrow \downarrow \downarrow \downarrow \\ X \underset{\mathcal{G}_0}{\times} \mathcal{G}_2 && \to && \mathcal{G}_2 \\ \downarrow \downarrow \downarrow && && \downarrow \downarrow \downarrow \\ X \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 && \to && \mathcal{G}_1 \\ \downarrow \downarrow && && \downarrow \downarrow \\ X && \stackrel{a}{\to} && \mathcal{G}_0 } \,,$$

where the homotopy fiber products on the left are those of the anchor $a$ with the leftmost 0-face map $\mathcal{G}_{(\{0\} \hookrightarrow \{0, \cdots, n\})}$ and the horizontal morphisms are the corresponding projections on the second factor.

We call $(X//\mathcal{G})_\bullet$ also the action groupoid of the action of $\mathcal{G}_\bullet$ on $(X,a)$ and call its realization $X \to (X//\mathcal{G})$ the homotopy quotient of the action.

Example

For $\mathcal{G}_\bullet = (\mathbf{B}G)_\bullet$ the delooping of a group object, def. reduces to the definition of an ∞-action of the ∞-group $G$.

Related concepts

• groupoid representation

• ∞-groupoid-principal ∞-bundle

• ∞-groupoid bibundle

References

For the more classical definition of an action of a Lie groupoid on a smooth manifold see e.g. Section 1.6 in

• Kirill Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press, 2005 (doi:10.1017/CBO9781107325883)