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(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 $\mathcal{G}$)?

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  • $\begingroup$ Could you say what it means for a stack to be a gerbe "under a sheaf"? I'm not familiar with that terminology. $\endgroup$ – Jonathan Beardsley Feb 3 '14 at 19:40
  • $\begingroup$ Hm... maybe you did already. $\endgroup$ – Jonathan Beardsley Feb 3 '14 at 19:43
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    $\begingroup$ I'll just say this - at least some of the ideas you're interested in have been approached by homotopy theorists, like Jacob Lurie, and by category theorists. Unfortunately, this seems to be primarily done in the language of infinity categories/infinity topoi (of which the category of chain complexes of sheaves of Abelian groups on some site is an example). However, I'm pretty certain there must be an affirmative answer to your question. One place to start might be ncatlab.org/nlab/show/infinity-gerbe. $\endgroup$ – Jonathan Beardsley Feb 3 '14 at 19:52
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    $\begingroup$ What's really happening in the usual case of cohomology is that you're looking for maps from $X_0$ to various deloopings of $\mathcal{G}_{X_0/S}$ (a sheaf of abelian groups). Now, however, you want to deloop a sheaf valued in chain complexes, which can be, and is done, and look at morphisms from $X_0$ into that. $\endgroup$ – Jonathan Beardsley Feb 3 '14 at 19:59
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A related question is answered in https://arxiv.org/abs/1101.4069.

edit: My paper https://arxiv.org/abs/1712.01384 takes a very similar approach. It deforms modules instead of algebras, so it's simpler but less relevant.

I see no reason to think of this as a gerbe but as a torsor for a group stack. I believe this perspective is due to Grothendieck's book "Categories cofibrees additives et complexe contagent relatif" and a solid modern treatment through a Gromov-Witten lens is established in https://arxiv.org/abs/1111.4200

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