I'm sure this is very well known but I haven't found any references for this searching the internet so hence the question:


> What's the neat abstract framework for obstruction theory for Gerbes in an $\infty$-topos?


This is somewhat imprecise so I'll try to come up with a precise question so that it will be clear what's missing for me. I'll work with $\infty$-gruopoids all the while trying to make arguments which are easily adapted to the general case.

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Let $X$ be a (connected) space and let $E \to X$ be a map whose fibers are connected and 1-truncated, (i.e. BG for some group $G$). This fibration is classified by a map $X \to BAut(BG)$.

Over $BAut(BG)$ we have the universal gerbe 

$$BG \to BAut_*(BG) \to BAut(BG)$$ 

Which pulls back to 

$$BG \to E \to X$$

However $BAut(BG)$ is itself a $K(Z(G),2)$-gerbe over its 1-truncation.

$$K(Z(G),2) \to BAut(BG) \to BOut(BG)$$

By analyzing the maps a bit we find out that $BAut_*(BG) \cong BAut(G)$ is the natural $BInn(G)$ gerbe over $BOut(G)$. So we have a natural map of gerbes $BAut(G) \to BAut(BG)$ over $BOut(G)$.

> Lets fix a "band" $\mathcal{B}: X \to BOut(G)$ and pullback $BAut(G) \to BAut(BG)$ of
> gerbes to get a map of gerbes $\mathcal{I} \to \mathcal{Z}$ over $X$
> where $\mathcal{I}$ is an $BInn(G)$-gerbe and $\mathcal{Z}$ is a
> $K(Z(G),2)$-gerbe. We now have the following interpretations:
> 
> 1. Sections of $\mathcal{Z}$ classify **$BG$-gerbes banded by $\mathcal{B}$**
> 2. Sections of $\mathcal{I}$ classify **$BG$-gerbes banded by $\mathcal{B}$ equipped with a a global section.**
> 3. The natural map $\mathcal{I} \to \mathcal{Z}$ induces on a sections map which **takes a gerbe equipped with a section and forgets the section.**


Now imagine for now on that the map $BAut_*(BG) \to BAut(BG)$ was the homotopy fiber (relatively over $BOut(G)$) of some universal map $BAut(BG) \to K$ for some universal fibration $K \to BOut(BG)$ equipped with a section $s: BOut(G) \to K$ (meaning that $BAut_*(BG) \cong BOut(G) \times_{K} BAut(BG)$).

If that was the case we would have a universal fiber sequence over $BOut(BG)$:

$$BAut_*(BG) \to BAut(BG) \to K$$

Lets pull this back to $X$ via the band $\mathcal{B}$ to get a (relative) fiber sequence:

$$\mathcal{I} \to \mathcal{Z} \to \mathcal{K}$$


Taking homotopy groups (or "homotopy sections") we get a long exact sequence:


$$\dots \to \pi_1(\mathcal{K}) \to \pi_0(\mathcal{I}) \to \pi_0(\mathcal{Z}) \to \pi_0(\mathcal{K}) \to *$$


> **Filling in the interpretations from above we might expect the 
 following sequence exists:**

> $$ \dots \to \{ \text{???} \} \to \{ \text{sections of gerbes banded
 by $\mathcal{B}$ } \} \to \{ \text{gerbes banded by $\mathcal{B}$} \} \to \{ \text{obstructions for sections of gerbes banded by
 $\mathcal{B}$} \} \to *$$

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> **Questions:**
>
>  1. Is there in fact such universal $\mathcal{K}$?
>  2. Given (1) is true and the analysis above is correct, how does one calculate $\pi_0(\mathcal{I})$ given that $\mathcal{I}$ is itself a gerbe? There seems to be a circular trap here...
>  3. How much of this can be cleanly stated and simplified in the language of $\infty$-topoi and how?
>  4. What's a good modern source for this topic?