  Information Geometry generalizes to infinite dimension by modeling the
tangent space of the relevant manifold of probability densities with
exponential Orlicz spaces. We review here several properties of the exponential
manifold on a suitable set $\mathcal E$ of mutually absolutely continuous
densities. We study in particular the fine properties of the Kullback-Liebler
divergence in this context. We also show that this setting is well-suited for
the study of the spatially homogeneous Boltzmann equation if $\mathcal E$ is a
set of positive densities with finite relative entropy with respect to the
Maxwell density. More precisely, we analyse the Boltzmann operator in the
geometric setting from the point of its Maxwell's weak form as a composition of
elementary operations in the exponential manifold, namely tensor product,
conditioning, marginalization and we prove in a geometric way the basic facts
i.e., the H-theorem. We also illustrate the robustness of our method by
discussing, besides the Kullback-Leibler divergence, also the property of
Hyv\"arinen divergence. This requires to generalise our approach to
Orlicz-Sobolev spaces to include derivatives.%
