Van Est isomorphism for homogeneous cochains

Abstract

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 on the linear fiber. Our main result is a Van Est theorem for such cochains. We also work out two applications to the general theory of representations of Lie groupoids and algebroids. The case k=1 yields a Van Est map for representations up to homotopy on 2-term graded vector bundles. Arbitrary k-homogeneous cochains on suitable VB-groupoids lead to a novel Van Est theorem for differential forms on Lie groupoids with values in a representation

On the Lie 2-algebra of sections of an LA-groupoid

“…where the last equality follows from the second condition of a representation up to homotopy (6). On the other hand, we have that…”

Differential forms with values in VB-groupoids and its Morita invariance

“…For q = 0, this is exactly the complex calculating the cohomology of G with values in the ruth E = C[1] ⊕ E, H • (G, E). It is also important to note that, for E = 0, the complex reduces to (Ω q (B • G, t * C), δ C ); this complex was introduced in [11] in the context of van Est isomorphisms. We shall now investigate the invariance of the cohomology H • (C •,q (V)) under Morita equivalence.…”

“…, U j p ) ∈ T (g1,...,gp) B p G. Let us now define C p ext (G) as the space of functions B p G → R which are multi-linear with respect to the decomposition (5.14) and skew-symmetric on the first q-components. From [11,Prop. 4.7], we know that C • ext (G) is a subcomplex of the differentiable cochain complex of G (in fact, there is a chain map P ext : C • (G) → C • ext (G) which is a projection!).…”

“…. Now, it is straightforward to see that F preserves the C • ext complexes and the existence of a chain projection P ext : C • → C • ext (see [11]) implies that F induces an isomorphism between the ext-chain cohomologies. Now, the proof follows from the following diagram…”

Deformation cohomology of Lie algebroids and Morita equivalence

“…vector bundle objects in the category of Lie groupoids, is defined and discussed in [8], where the authors prove the Morita invariance of the VB-groupoid cohomology. Finally, VB-groupoid cohomology is related to VB-algebroid cohomology by a Van-Est type map [3].…”