Is Quillen's bracket a "universal enveloping" something? $\newcommand{\G}{\mathcal{G}}$
In K-theory, there is a construction due to Quillen as follows. Let $(\G, \oplus, 0)$ be a monoidal groupoid. Then the bracket $\langle \G, \G \rangle$, sometimes also denoted as $U\G$, is a new category. Its objects are the same as the objects of $\G$. For $A,B$ such objects, the hom set is:
$$\hom_{U\G}(A,B) = \operatorname{colim}_{X \in \G} \hom_\G(X \oplus A, B).$$
In less abstract terms, a morphism from $A$ to $B$ is an equivalence class of pairs $(X,f)$ where $X \in \G$ and $f : X \oplus A \to B$ is a morphism in $\G$. Two such pairs $(X,f)$ and $(X',f')$ are deemed equivalent if there exists $g : X \to X'$ such that $f' = f \circ (g \oplus \operatorname{id}_A)$.
(For example, if $(\Sigma, \sqcup, \varnothing)$ is the groupoid of finite sets and bijections, then $U\Sigma$ is the category of finite sets and injections, usually denoted $\mathsf{FI}$ or $\Theta$. If $\G$ is the groupoid of finitely-generated free $R$-modules and isomorphisms, then $U\G$ is the category of fg free $R$-modules and "free split injections".)
Under some condition, $\G$ is the underlying groupoid of $U\G$. The notation $U\G$ (that I have found in papers of Randal-Williams–Wahl and Soulié, and it certainly appears elsewhere) definitely suggests that this is the "universal enveloping... something" of $\G$. Is there such an interpretation? Or is this just an artifact of notation?
 A: Here's how I like to think about it.


*

*Let $\mathcal G$ be a monoidal groupoid.

*Let $\mathbb B \mathcal G$ be the delooping of $\mathcal G$: this is a $(2,1)$-category with one object $\bullet$.

*Let $\mathcal X$ be a groupoid that $G$ acts on. 

*Then the action of $\mathcal G$ on $\mathcal X$ may be thought of as a (pseudo)functor $\chi_{\mathcal X} : \mathbb B \mathcal G \to \mathsf{Gpd}$, which sends $\bullet$ to $\mathcal X$.

*Let $\mathbb E_{\mathcal G} \mathcal X \to \mathbb B \mathcal G$ denote the fibration of $(2,1)$-categories classified by $\chi_{\mathcal X}$. One way of thinking of this is that $\mathbb E_{\mathcal G}$ is the lax colimit of the functor $\chi_{\mathcal X}$ in $(\infty,1)$-categories.

Then $\langle \mathcal G, \mathcal X\rangle = \tau_{\leq 1} \mathbb E_{\mathcal G} \mathcal X$.

Here $\tau_{\leq 1}$ is the 1-truncation: it takes homotopy classes levelwise to turn a $(2,1)$-category into a 1-category. As a left adjoint, $\tau_{\leq 1}$ preserves colimits. 

So $\langle \mathcal G, \mathcal X\rangle$ is the lax colimit of the functor $\chi_{\mathcal X}: \mathbb B \mathcal G \to \mathsf{Cat}$. 

In particular, when $\mathcal X = \mathcal G$ with the left translation action, then $\mathbb E_{\mathcal G} \mathcal X$ deserves to be called $\mathbb E \mathcal G$, and we have

$\langle \mathcal G, \mathcal G \rangle = \tau_{\leq 1} \mathbb E \mathcal G$ is the lax colimit of the tautological functor $\mathbb B \mathcal G \to \mathsf{Cat}$.

I'm still not sure conceptually why one wants to apply $\tau_{\leq 1}$, rather than viewing everything as taking place in $\mathsf{Cat}_\infty$. But at some point I did write some notes for myself developing parts of Grayson's Higher Algebraic K-Theory II from this perspective, and it seemed to be not a terrible perspective to take.
