I am reading Brylinski's Loop Spaces, Characteristic Classes, and Geometric Quantization.


The Statement
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Let $C$ be a gerbe on a space $X$ with "abelian" band $H$, $f: Y \to X$ a local homeomorphism (this context uses local homeomorphisms in place of open subset sets and covers of open sets). Given an object $P$ of $C(Y)$, and an $H$ torsor $I$ over $Y$, there exists an object $Q$ of $C(Y)$ such that $I \overset{\sim}{\to} \underline{\text{Isom}}(P,Q)$ as $H$-torsors. The object $Q$ is called "the object obtained from $P$ by twisting by the H-torsor $I$".


My Question
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> 
> I can get a local ismorphism $I \overset{\sim}{\to} \underline{\text{Isom}}(P,Q)$ but how can I see it is, in fact, global?
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> 
> 


Some Details / A sketch of the proof
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So you take a surjective local homeomorphism $g: Z \to Y$ so that there is a global section of $g^{-1}I$ (i.e. take a fine-enough open cover so that you have sections of $I$). Then pulling back this section $s$ under the maps $p\_1, p\_2: Z \times\_Y Z \to Z$ we get two different sections of a sheaf which is a torsor under the pullback of $H$. So you get this section of the pullback of $H$ and since the pullback of $P$ is also a torsor under this sheaf, you have an automorphism of the pullback of $P$. This automorphism could also be thought of as an isomorphism $u: p\_1^{-1}g^{-1}P \overset{\sim}{\to} p\_2^{-1}g^{-1}P$ which satisfies the cocycle condition. Then by the descent property which this gerbe satisfies, we have an object $Q$ of $C(Y)$ with $\psi: g^{-1}Q \overset{\sim}{\to} g^{-1}P$.  

So we started with a section of the pullback of $I$ (i.e a local section of $I$) and ended with an isomorphism between $P$ and $Q$'s pullbacks (i.e. a local isomorphism).


**What information can I use to show that this local isomorphism is in fact an isomorphism? Or is Brylinski simply abusing language?**


EDIT (2/22/13): I'm fairly certain I just need to show that on double intersections these local isomorphisms differ by an inner automorphism, but since my band is abelian this would say they glue together. I just haven't wrapped my head around this idea in the context of local homeomorphisms ans pullback diagrams quite yet.



Twisting an object P by an H-Torsor I
I am going to put as much detail as possible in this answer without writing any diagrams (too many diagrams!).


In the Sketch of the Proof as outlined above, I was using local homeomorphisms instead of open subsets and open covers. I will switch to open (sets/covers) in this solution for simplicity and variety.


Let $P$ be an objecto of $C(U)$, where $U$ is an open subset of $X$, let $I$ be an $H$-torsor over $U$, and let $(U\_{\alpha})$ be an open cover of $U$ (refined enough such that whenever there is a local property being used we need not refine any more). Then we can, by assumption of the cover, pick a section $s\_{\alpha} \in I\_{\alpha}$ for each $\alpha$. On $U\_{\alpha \beta}$, define $\mu\_{\alpha \beta} \in H(U\_{\alpha \beta})$ by $\mu\_{\alpha \beta}s\_{\beta} = s\_{\alpha}$, using the torsor there. Then since $C$ is a gerbe we have a unique way ( unique by commutativity of the band) to locally identify our torsor element with an automorphism $\mu\_{\alpha \beta} : P\_{\alpha \beta} \to P\_{\alpha \beta} $ satisfying the cocycle condition (up to natural transformations on double and triple intersections; I will avoid these natural transformations in this answer at all costs. Having diagrams is the only way for me to feel comfortable explaining them).


By the descent property that $C$ satisfies, since we have objects $P\_{\alpha}$ and isomorphisms $\mu\_{\alpha \beta}: P\_{\beta}|\_{U\_{\alpha \beta}} \to P\_{\alpha}|\_{U\_{\alpha \beta}}$ (so here I combine the natural transformations between restricting once to the double intersection and restricting two times to the double intersection, along with my automorphism $\mu$ on the double intersection to get an isomorphism between my two $P$'s. ), then I have an object $Q$, with isomoprhisms $\psi\_{\alpha}: P\_{\alpha} \to Q\_{\alpha}$ such that (up to natural transformations on $U\_{\alpha \beta}$) $$\psi\_{\alpha} = \mu\_{\alpha \beta} \psi\_{\beta}$$
**Claim**: We have an isomorphism of $H$-torsors $w: I \to \underline{\text{Isom}}(P,Q)$.
Let $ \sigma \in I(V)$ and define $V\_{\alpha} := V \cap U\_{\alpha}$. For each $V\_{\alpha}$, define $h\_{\alpha} \in H(V\_{\alpha})$ by $h\_{\alpha}s\_{\alpha} = \sigma\_{\alpha}$. Then we can think of $h\_{\alpha}$ as an automorphism of $P\_{\alpha}$ (now thinking of $P\_{\alpha}:= P\_{V\_{\alpha}}$) so we define our corresponding isomorphism $w(\sigma): P\_V \to Q\_V$ by $$w(\sigma)\_{\alpha}:= h\_{\alpha}\psi\_{\alpha}$$. 


It is straightforward to check that this map is well-defined, a morphism of torsors, and thus an isomorphism. 


*Note: the bit about the band being abelian helps with a different choice of $Q$ resulting in a uniquely isomorphic situation; thus allowing a bijective correspondence with the isomorphism classes of objects in $C(U)$ and $H^1(U, H)$!*


