The van Est Map on Geometric Stacks

We generalize the van Est map and isomorphism theorem in three ways, and we discuss conjectured connections with homotopy theory, including a proposal of a category which unifies differentiable stacks, Lie algebroids and homotopy theory. In Part 2 of this thesis we generalize the van Est map from a comparison map between Lie groupoid cohomology and Lie algebroid cohomology to a (more conceptual) comparison map between the cohomology of a stack $\mathcal{G}$ and the cohomology of a simple foliation $\mathcal{H}\to\mathcal{G}$. In Part 1 we generalize the functions that we can take cohomology of in the context of the van Est map. Instead of using functions valued in representations, we can use functions valued in modules, eg. we can use $S^1$-valued functions and $\mathbb{Z}$-valued functions. Finally, everything we do works in both the smooth and holomorphic categories. These generalizations allow us to derive results, including classical ones, that could not be obtained with the usual van Est map, and they give a new method of computing cohomology. Part 3 of this thesis involves a proposal of a definition of Morita equivalences in the categories of Lie algebroids and LA-groupoids. What we find is that our proposed category of LA-groupoids unifies differentiable stacks, Lie algebroids and homotopy theory. In particular, there are morphisms between Lie algebroids and Lie groupoids and we show that such objects can be Morita equivalent. This gives another interpretation of the van Est map. Using this category, we argue that a classifying space for $G$ is just a space which is homotopy equivalent it. We discuss higher generalized morphisms (given by gerbes) and we conjecture the existence of a smooth version of Grothendieck's homotopy hypothesis.