In the paper by Weinstein and Xu: Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), there are two versions of Lie groupoid cohomology. The same differential $\delta$ can be in fact applied to globally smooth cochains or only to cochains which are globally continous and smooth in a neighbourhood of the diagonal.

If $\Gamma$ denotes the total space of the groupoid and $\Gamma^{(n)}$ the set of composable $n$-tuples then one can consider:

$C^n(\Gamma:\mathbb R)={\cal C}^\infty(\Gamma^{(n)},\mathbb R)$,

$C^n_{es}(\Gamma:\mathbb R)=\{\sigma:\Gamma^{(n)}\to\mathbb R, \sigma\quad \mathrm{smooth}\quad \mathrm{around}\quad \Delta^{(n)}\}$

and of course one could consider just continuous cochains

$C^n_0(\Gamma:\mathbb R)={\cal C}(\Gamma^{(n)},\mathbb R)$

In the same paper it is proven that the first two cohomologies are different by giving an example in which globally smooth 2-cohomology is 0 while $H^2_{es}(\Gamma;\mathbb R)$ is non zero (in fact in the example coefficients are in $\mathbb S^1$ but this should make no big difference).

Does anyone knows:

If also continuous cohomology differs from the two previous ones

Other examples in which the cohomologies are different and/or equal

General conditions under which the cohomologies are known to be equal.