Let $\mathcal{A}$ be an abelian sheaf of groups on $X$. We establish the bijection: $\{\text{$\mathcal{A}$-gerbes on } X\}/\text{equivalence} \leftrightarrow H^2(X, \mathcal{A})$

Step 1: From Gerbes to Cohomology

Given an $\mathcal{A}$-gerbe $\mathcal{G}$ on $X$, choose a covering $\{U_i\}$ such that $\mathcal{G}|_{U_i}$ is neutral (has an object $x_i$).

Since $\mathcal{G}$ is a gerbe, over $U_i$ we have $\mathcal{G}|_{U_i} \simeq \mathcal{B}\mathcal{A}|_{U_i}$. Over overlaps $U_{ij}$, the objects $x_i|_{U_{ij}}$ and $x_j|_{U_{ij}}$ are isomorphic. Choose isomorphisms: $\phi_{ij}: x_i|_{U_{ij}} \xrightarrow{\sim} x_j|_{U_{ij}}$

These are well-defined up to the action of $\mathcal{A}(U_{ij})$.

On triple overlaps $U_{ijk}$, we have three isomorphisms forming a triangle. The failure of this triangle to commute is measured by: $g_{ijk} = \phi_{jk} \circ \phi_{ij} \circ \phi_{ik}^{-1} \in \mathcal{A}(U_{ijk})$

Claim: $g_{ijk}$ is a Čech 2-cocycle.

Over quadruple overlaps $U_{ijkl}$, the cocycle condition: $g_{jkl} - g_{ikl} + g_{ijl} - g_{ijk} = 0$ follows from the coherence of the stack structure. Thus $g \in Z^2(\{U_i\}, \mathcal{A})$.

Independence of Choices: Different choices of $\phi_{ij}$ differ by $h_{ij} \in \mathcal{A}(U_{ij})$, which modifies $g_{ijk}$ by a coboundary: $g'_{ijk} = g_{ijk} + \delta h_{ij}$

Thus the cohomology class $[g] \in H^2(X, \mathcal{A})$ is well-defined.

Step 2: From Cohomology to Gerbes

Given $\alpha = [g_{ijk}] \in H^2(X, \mathcal{A})$ represented by a cocycle over $\{U_i\}$, construct a gerbe $\mathcal{G}_\alpha$:

• Over $U_i$, set $\mathcal{G}_\alpha|_{U_i} = \mathcal{B}\mathcal{A}|_{U_i}$ (the neutral gerbe)
• Over $U_{ij}$, the two trivializations differ by an $\mathcal{A}$-valued 1-cocycle $\phi_{ij}$ (to be determined)
• The cocycle condition on $U_{ijk}$ forces: $\phi_{jk} \circ \phi_{ij} \circ \phi_{ik}^{-1} = g_{ijk}$

We can choose $\phi_{ij}$ satisfying this (since $\mathcal{A}$ is abelian, this is a linear problem). The gerbe $\mathcal{G}_\alpha$ is then defined by gluing using these $\phi_{ij}$.

Verification: We must check:

1. The gluing is associative (uses cocycle condition)
2. Different choices of $\phi_{ij}$ give equivalent gerbes
3. $\mathcal{G}_\alpha$ is indeed a gerbe (locally non-empty and connected)

All of these follow from the construction and the cohomology conditions.

Step 3: The Maps are Inverse

Given a gerbe $\mathcal{G}$, construct its class $[\mathcal{G}] \in H^2(X, \mathcal{A})$, then reconstruct a gerbe $\mathcal{G}_{[\mathcal{G}]}$.

Claim: $\mathcal{G} \simeq \mathcal{G}_{[\mathcal{G}]}$.

Both gerbes trivialize over the same cover $\{U_i\}$ with the same transition data (up to coboundary), so they're equivalent.

Conversely, given $\alpha \in H^2(X, \mathcal{A})$, construct $\mathcal{G}_\alpha$, then extract its class. The cocycle we extract is cohomologous to $\alpha$ by construction, so we recover $\alpha$.

Step 4: Automorphisms

The automorphism group of a gerbe $\mathcal{G}$ with class $\alpha$ consists of automorphisms of the neutral gerbe that respect the gluing. These are precisely $H^1(X, \mathcal{A})$: $\text{Aut}(\mathcal{G}) = H^1(X, \mathcal{A})$

This can be seen by noting that an automorphism is given by choosing $h_i \in \mathcal{A}(U_i)$ on each piece, compatible with the transition data.

Conclusion

We have established: $\{\text{$\mathcal{A}$-gerbes}\}/\simeq \xleftrightarrow{1:1} H^2(X, \mathcal{A})$ with automorphisms of a gerbe classified by $H^1(X, \mathcal{A})$. The neutral gerbe corresponds to $0 \in H^2(X, \mathcal{A})$.