Equivariance in higher geometry (Q626114)

The authors develop the theory of presheaves in bicategories on Lie groupoids, extending the results of \textit{J. Heinloth} [Lecture notes from the seminars ``Number theory'', ``Algebraic geometry'' and ``Twisted cohomology theories'' held at the University of Göttingen, Göttingen, Germany, 2004, 1--32 (2005; Zbl 1098.14501)] and \textit{D. Metzler} [``Topological and smooth stacks'', \url{arXiv:math/0306176}] for sheaves in categories on smooth manifolds. It is demonstrated that any presheaf \(\mathcal{X}\) in bicategories on the category of manifolds can be naturally extended to a presheaf in bicategories on the category of Lie groupoids. The first main result of the paper is that for a Morita equivalence \(\Gamma\to\Lambda\) of Lie groupoids and a stack \(\mathcal{X}\) on Lie groupoids, the functor \(\mathcal{X}(\Lambda)\to\mathcal{X}(\Gamma)\) given by pullback is an equivalence of bicategories; similarly, if \(\mathcal{X}\) is a prestack, then the resulting functor is fully faithful. This is used to generalize the plus construction; to a \(2\)-prestack \(\mathcal{X}\) on manifolds, the authors associate a \(2\)-stack \(\mathcal{X}^+\) such that the canonical embedding \(\mathcal{X}(M) \to \mathcal{X}^+(M)\) is fully faithful for each manifold \(M\). This construction is applied to bundle gerbes with connection, yielding a local construction of gerbes and the definition of equivariant gerbes, as well as to \(2\)-vector bundles modeled on the \(2\)-vector spaces of \textit{M. M. Kapranov} and \textit{V. A. Voevodsky} [Proc. Symp. Pure Math. 56, Pt.~2, 177--259 (1994; Zbl 0809.18006)], obtaining a stack of \(2\)-vector bundles. As well, the authors introduce the notion of a Jandl gerbe, generalizing gerbes with a Jandl structure defined by \textit{U. Schreiber}, \textit{C. Schweigert} and \textit{K. Waldorf} [Commun. Math. Phys. 274, No.~1, 31--64 (2007; Zbl 1148.53057)], similarly demonstrating that Jandl gerbes form a stack and using them to study holonomy for unoriented surfaces. As another main result, the authors develop a description of equivariant descent.