Let $\Lambda \stackrel{F}{\to} \Omega \stackrel{G}{\leftarrow} \Gamma$ be a diagram of groupoids and functors and $\Gamma \times\_\Omega \Lambda$ the homotopy pullback. We will regard all these groupoids as spaces and compute the cohomology with coefficients in some field.


There should be a map
$$ C^\*(\Gamma) \stackrel{\mathbb{L}}{\otimes}\_{C^\*(\Omega)} C^\*(\Lambda) \to C^\* (\Gamma \times\_\Omega \Lambda)$$
from the derived tensor product to the cohomology of the homotopy pullback.


Is this map an equivalence?



Cohomology of a homotopy pullback of groupoids
No, consider the multiplication by $2$ map on the integers. The homotopy fiber is two points. But the derived tensor product (for $\mathbb Q$ coefficients) is $\mathbb Q \otimes^{L}\_{\mathbb Q[x]} \mathbb Q[x] = \mathbb Q$. 


