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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.

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1 Answer 1

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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 $H^1(U, H)$!

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