Continuous and smooth Lie groupoid cohomology 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.
 A: Sorry for answering so late, I just was pointed to this question. Before answering  your questions in the case of a Lie group (i.e., one object) let me point out that the difference between $\mathbb{R}$-coefficients and $\mathbb{S}^1$-coefficients it not negligible but is somehow the whole heart of the story.


*

*The cohomology of the continuous cochains is equal to the cohomology of the smooth cochains. Moreover, there is yet another complex $C^n_{ec}$ of cochains that are continuous around $\Delta^{(n)}$. Then $H^n_{ec}$ equals $H^n_{es}$ (all this for coefficients in $\mathbb{R}$, $\mathbb{S}^1$ or more general tori).

*The universal cover $\mathbb{Z}\to\mathbb{R}\to\mathbb{S}^1$ is described by a cocycle in $Z^2_{es}$ that cannot be smooth (otherwise $ \mathbb{R}\to\mathbb{S}^1$ would have a smooth global section), see also 3.

*$H^n_{es}=H^n$ (in your notation form above) if either the Lie group $G$ or the coefficients are contractible (in particular the difference between $\mathbb{R}$
and $\mathbb{S}^1$ plays a significant role).
A reference for the equality of smooth and continuous cohomology is Hochschild, Mostow, "Cohomology of Lie groups" and for the more general case this paper, together with F. Wagemann.
I expect that the case of general Lie groupoids can be treated similarly, for instance I would expect that $H^n=H^n_{es}$ if the Lie groupoid $\Gamma$ is topologically trivial (i.e., its simplicial manifold $N\Gamma$ is contractible). However, the generalisation of our approach is not entirely straight forward.
