# Twisting an object P by an H-Torsor I

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

## The Statement

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

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?

## Some Details / A sketch of the proof

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.

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 $$w(\sigma)_{\alpha}:= h_{\alpha}\psi_{\alpha}$$.
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 $H^1(U, H)$!