Related papers: The van Est Map on Geometric Stacks

The van Est map is a map from Lie groupoid cohomology (with respect to a sheaf taking values in a representation) to Lie algebroid cohomology. We generalize the van Est map to allow for more general sheaves, namely to sheaves of sections…

The classical Van Est theory relates the smooth cohomology of Lie groups with the cohomology of the associated Lie algebra, or its relative versions. Some aspects of this theory generalize to Lie groupoids and their Lie algebroids. In this…

VB-groupoids define a special class of Lie groupoids which carry a compatible linear structure. In this paper, we show that their differentiable cohomology admits a refinement by considering the complex of cochains which are k-homogeneous…

The Van Est homomorphism for a Lie groupoid $G \rightrightarrows M$, as introduced by Weinstein-Xu, is a cochain map from the complex $C^\infty(BG)$ of groupoid cochains to the Chevalley-Eilenberg complex $C(A)$ of the Lie algebroid $A$ of…

In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. This immediately implies a version of Haefliger's conjecture…

This paper belongs to a series devoted to the study of the cohomology of classifying spaces. Generalizing the Weil algebra of a Lie algebra and Kalkman's BRST model, here we introduce the Weil algebra $W(A)$ associated to any Lie algebroid…

We extend the stable motivic homotopy category of Voevodsky to the class of scalloped algebraic stacks, and show that it admits the formalism of Grothendieck's six operations. Objects in this category represent generalized cohomology…

Motivated by our attempt to understand characteristic classes of Lie groupoids and geometric structures, we are brought back to the fundamentals of the cohomology theories of Lie groupoids and algebroids. One element that was missing in the…

Morita equivalence classes of Lie groupoids serve as models for differentiable stacks, which are higher spaces in differential geometry, generalizing manifolds and orbifolds. Representations up to homotopy of Lie groupoids provide a higher…

Weighted Lie algebroids were recently introduced as Lie algebroids equipped with an additional compatible non-negative grading, and represent a wide generalisation of the notion of a VB -algebroid. There is a close relation between two term…

We define the "localized index" of longitudinal elliptic operators on Lie groupoids associated to Lie algebroid cohomology classes. We derive a topological expression for these numbers using the algebraic index theorem for Poisson manifolds…

We study vector bundles over Lie groupoids, known as VB-groupoids, and their induced geometric objects over differentiable stacks. We establish a fundamental theorem that characterizes VB-Morita maps in terms of fiber and basic data, and…

Motivated by the fact that the Hopf-cyclic (co)homologies of function algebras over Lie groups and universal enveloping algebras over Lie algebras capture the Lie group and Lie algebra (co)homologies, we hereby upgrade the classical van Est…

We upgrade the classical operation of \textit{isomonodromic deformations} along a path $\gamma$ to a functor $\mathbb{P}_{\gamma}$ between categories of flat connections with logarithmic singularities along a divisor $D$, which itself…

In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and…

Let E be an H-space acting on a based space X. Then we refer to ev: E -> X, the map obtained by acting on the base point of X, as a ``generalized evaluation map." We establish several fundamental results about the rational homotopy…

We propose a notion of 1-homotopy for generalized maps. This notion generalizes those of natural transformation and ordinary homotopy for functors. The 1-homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As…

This thesis deals with deformations of VB-algebroids and VB-groupoids. They can be considered as vector bundles in the categories of Lie algebroids and groupoids and encompass several classical objects, including Lie algebra and Lie group…

This is the second part of a series of papers devoted to develop Homotopical Algebraic Geometry. We start by defining and studying generalizations of standard notions of linear and commutative algebra in an abstract monoidal model category,…

We introduce and study module structures on both the dgla of multiplicative vector fields and the graded algebra of functions on Lie groupoids. We show that there is an associated structure of a graded Lie-Rinehart algebra on the vector…