nLab
groupoid infinity-action

Contents

Idea

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

Definition

Definition

For 𝒢 Grpd (H)\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H}) a groupoid object, PHP \in \mathbf{H} any object equipped with a morphism a:P𝒢 0a \colon P \to \mathcal{G}_0 to the object of objects of 𝒢\mathcal{G}, a 𝒢 \mathcal{G}_\bullet-groupoid ∞-action on XX with anchor aa is a groupoid (X//𝒢) (X//\mathcal{G})_\bullet over 𝒢 \mathcal{G}_\bullet of the form

X×𝒢 0𝒢 2 𝒢 2 X×𝒢 0𝒢 1 𝒢 1 X a 𝒢 0, \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 aa with the leftmost 0-face map 𝒢 ({0}{0,,n})\mathcal{G}_{(\{0\} \hookrightarrow \{0, \cdots, n\})} and the horizontal morphisms are the corresponding projections on the second factor.

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

Example

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

Last revised on November 5, 2014 at 09:05:54. See the history of this page for a list of all contributions to it.