(This is probably a very naive question. My understanding of the cotangent complex is quite vague.)

Let me first recall the picture for **deformations of a smooth morphism**:

If $f:X_0\to S_0$ is a smooth morphism of schemes and $S_0\to S$ is a closed immersion defined by a square zero ideal $I$, the stack (in the Zariski topology) over $X_0$ associating to an open $U_0\subseteq X_0$ the category of smooth lifts $U$ of $U_0$ over $S$ is naturally a *gerbe* under the sheaf $$\mathcal{G}_{X_0/S}:=\mathcal{H}om (\Omega^1_{X_0/S_0}, f^* I).$$ This just means that deformations locally exist and are locally unique, and that the sheaf of automorphisms of a local deformation is canonically isomorphic to $\mathcal{G}_{X_0/S}|_{U_0}$.

It is a formal consequence of this that **(a)** there is an obstruction class $o(X_0/S) \in H^2(X_0, \mathcal{G}_{X_0/S})$ to deforming $X_0$ over $S$, **(b)** if it vanishes then such deformations are a torsor under $H^1(X_0, \mathcal{G}_{X_0/S})$, and **(c)** the group of automorphisms of a deformation is $H^0(X_0, \mathcal{G}_{X_0/S})$.

The above three conclusions hold generally for *any* morphism $f$ if we set $\mathcal{G}_{X_0/S} = R\mathcal{H}om (L_{X_0/S_0}, f^* I)$ where $L_{X_0/S_0}$ is the cotangent complex. Therefore it seems natural to ask whether, as in the smooth case, local deformations form a some sort of "gerbe" under $\mathcal{G}_{X_0/S}$. Of course the problem is that $\mathcal{G}_{X_0/S}$ is not a sheaf of groups but an object of the derived category. So it would have to be some "derived gerbe".

**Question:** Is there a notion of a "derived gerbe" (under an object $\mathcal{G}$ of the derived category of sheaves of abelian groups), such that local deformations of $X_0/S$ as above form a "derived gerbe" under $R\mathcal{H}om (L_{X_0/S_0}, f^* I)$, and such that the analogues (a)-(c) above hold (for any $\mathcal{G}$)**?**