TITLE: Are there other semidirect product/crossed products in other areas
QUESTION [14 upvotes]: Suppose $(O, G, \alpha)$ is a triple where $O$ is some mathematical object, $G$ is a group and $\alpha : G \rightarrow Aut(O)$. Many different areas of mathematics study such triples. However, I only know of a couple of examples of a new object that encodes this system.
1) If $G$ is a group, $N$ a normal subgroup of $G$, $H$ a subgroup of $G$ and $\alpha(h)(n) = hnh^{-1}$, then one can encode $(N, H, \alpha)$ into the (outer) semidirect product $N \rtimes_\alpha H$. 
2) If $O = \mathcal A$, a C$^*$-algebra, $G$ is a locally compact group and $\alpha : G \rightarrow Aut(\mathcal A)$ acts by $*$-automorphisms then $(\mathcal A, G, \alpha)$ is encoded as the crossed product algebra $\mathcal A \rtimes_\alpha G$. In very simple terms, this is accomplished by considering unitary representations of $G$. There is also a related construction in Statistical Mechanics called the covariance algebra.
My question then: are there any other semidirect/crossed product type constructions for such triples $(O, G, \alpha)$?

REPLY [10 votes]: In any higher category $C$, given an object $X$ and an action of a group $G$ on it you can ask for the homotopy quotient $X_{hG}$ of $X$ by the action of $G$, which is defined by a homotopy-coherent version of the universal property of the usual quotient. This recovers the following constructions in the following higher categories:

In the 2-category of groups (thinking of groups as one-object categories), this recovers the semidirect product, and actually a generalization of it describing general group extensions.
In the 2-category of rings (thinking of rings as one-object linear categories), this recovers the crossed product, and again a generalization of it including e.g. twisted group algebras.
In the $\infty$-category of spaces, this recovers the construction described in Mark Grant's answer. 
In the $\infty$-category of chain complexes, this recovers a general form of group cohomology with nontrivial coefficients.