MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 2 down vote accepted

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 $H^1(U, H)$!

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.