This question is in reference to Gaitsgory's preprint *[Contractibility of the space of rational maps](http://arxiv.org/abs/1108.1741)*. On p. 5 of the preprint, Gaitsgory defines a prestack $\mathscr{Y}$ (say over affine $\mathbb{C}$-schemes) to be *homologically contractible* if the functor $\text{Vect}\longrightarrow\mathfrak{D}(\mathscr{Y})$ taking $V\mapsto V\otimes\omega\_{\mathscr{Y}}$ is fully faithful. Here $\mathfrak{D}(\mathscr{Y})$ denotes the DG category of D-modules on $\mathscr{Y}$. Further down the same page, Gaitsgory mentions that homological contractibility of $\mathscr{Y}$ is equivalent to the condition that $H\_{\bullet}(\mathscr{Y}(\mathbb{C})^{\text{top}},\mathbb{Q})\cong\mathbb{Q}$. What is the argument for the equivalence of these two formulations of homological contractibility? I would be happy to see the argument even in a simple case, like the case where $\mathscr{Y}$ is a smooth complex variety.



Homological contractibility of a prestack
This is proven in some detail in section 3 [Gaitsgory's writeup](http://www.math.harvard.edu/~gaitsgde/GL/Tamagawa.pdf) of his the Atiyah-Bott formula. He starts with the fully faithfulness definition, then proves the equivalance with homological statement at the very end of the section.


