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Register a new userOn two notions of a Gerbe over a stack
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Let $mathcal{G}$ be a Lie groupoid. The category $Bmathcal{G}$ of principal $mathcal{G}$-bundles defines a differentiable stack. On the other hand, given a differentiable stack $mathcal{D}$, there exists a Lie groupoid $mathcal{H}$ such that $Bmathcal{H}$ is isomorphic to $mathcal{D}$. Define a gerbe over a stack as a morphism of stacks $Fcolon mathcal{D}rightarrow mathcal{C}$, such that $F$ and the diagonal map $Delta_Fcolon mathcal{D}rightarrow mathcal{D}times_{mathcal{C}}mathcal{D}$ are epimorphisms. This paper explores the relationship between a gerbe defined above and a Morita equivalence class of a Lie groupoid extension.
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Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact etale Lie groupoid $M$, we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated projective modules over an Azumaya algebra on $M$. This result can be seen as an equivariant Serre-Swan theorem for twisted vector bundles.
We construct and study general connections on Lie groupoids and differentiable stacks as well as on principal bundles over them using Atiyah sequences associated to transversal tangential distributions.
A Lie 2-group $G$ is a category internal to the category of Lie groups. Consequently it is a monoidal category and a Lie groupoid. The Lie groupoid structure on $G$ gives rise to the Lie 2-algebra $mathbb{X}(G)$ of multiplicative vector fields, see (Berwick-Evans -- Lerman). The monoidal structure on $G$ gives rise to a left action of the 2-group $G$ on the Lie groupoid $G$, hence to an action of $G$ on the Lie 2-algebra $mathbb{X}(G)$. As a result we get the Lie 2-algebra $mathbb{X}(G)^G$ of left-invariant multiplicative vector fields. On the other hand there is a well-known construction that associates a Lie 2-algebra $mathfrak{g}$ to a Lie 2-group $G$: apply the functor $mathsf{Lie}: mathsf{Lie Groups} to mathsf{Lie Algebras}$ to the structure maps of the category $G$. We show that the Lie 2-algebra $mathfrak{g}$ is isomorphic to the Lie 2-algebra $mathbb{X}(G)^G$ of left invariant multiplicative vector fields.
Let $mathbb{X}=[X_1rightrightarrows X_0]$ be a Lie groupoid equipped with a connection, given by a smooth distribution $mathcal{H} subset T X_1$ transversal to the fibers of the source map. Under the assumption that the distribution $mathcal{H}$ is integrable, we define a version of de Rham cohomology for the pair $(mathbb{X}, mathcal{H})$, and we study connections on principal $G$-bundles over $(mathbb{X}, mathcal{H})$ in terms of the associated Atiyah sequence of vector bundles. We also discuss associated constructions for differentiable stacks. Finally, we develop the corresponding Chern-Weil theory and describe characteristic classes of principal $G$-bundles over a pair $(mathbb{X}, mathcal{H})$.
We compute the analytic torsion of a cone over a sphere of dimension 1, 2, and 3, and we conjecture a general formula for the cone over an odd dimensional sphere.
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