  In this document, we aim to gather various results related to a
compositional/categorical approach to rigorous Statistical Mechanics. Rigorous
Statistical Mechanics is centered on the mathematical study of statistical
systems. Central concepts in this field have a natural expression in terms of
diagrams in a category that couples measurable maps and Markov kernels. We
showed that statistical systems are particular representations of partially
ordered sets (posets), that we call A-specifications, and expressed their
phases, i.e., Gibbs measures, as invariants of these representations. It opens
the way to the use of homological algebra to compute phases of statistical
systems. Two central results of rigorous Statistical Mechanics are, firstly,
the characterization of extreme Gibbs measures as it relates to the zero-one
law for extreme Gibbs measures, and, secondly, their variational principle
which states that for translation invariant Hamiltonians, Gibbs measures are
the minima of the Gibbs free energy. We showed how the characterization of
extreme Gibbs measures extends to A-specifications; we proposed an Entropy
functional for A-specifications and gave a message-passing algorithm, that
generalized the belief propagation algorithm of graphical models, to find
critical points of the associated variational free energy.
