defn - Symtr(X, C), category of symmetries of an object

Let $C$ be a symmetric monidal ""$\infty$-Category, $X$ and object.

Idea: $Symtr(X)$ is the $\infty$-category whose objects is data of an algebra $A$ and a left action of $A$ on $X$ in $C$.

Defn

The category of symmetries of $X\in C$ is the (total space of) the cocartesian fibration

$Symtr(X,C)\to Al{g}^{(1)}(C)$ Classifying the functor

$$LMo{d}_{(-)}(C){\times }_C\{x\}:Al{g}^{(1)}(C)\to Ca{t}_{\infty }$$