Here we prove the existence of sheaf cohomology theory in arbitrary o-minimal structures.

By recent work on some conjectures of Pillay, each definably compact group G in a saturated o-minimal expansion of an ordered field has a normal "infinitesimal subgroup" G00 such that the quotient G/G00, equipped with the "logic topology", is a compact (real) Lie group. Our first result is that the functor G → G/G00 sends exact sequences of definably compact groups into exact sequences of Lie groups. We then study the connections between the Lie group G/G00 and the o-minimal spectrum (...) G̃ of G. We prove that G/G00 is a topological quotient of G̃. We thus obtain a natural homomorphism ψ* from the cohomology of G/G00 to the (Čech-)cohomology of G̃. We show that if G00 satisfies a suitable contractibility conjecture then $\widetilde{G^{00}}$ is acyclic in Čech cohomology and ψ* is an isomorphism. Finally we prove the conjecture in some special cases. (shrink)