Study of various categories gravitating around $(\phi ,\mathrm{\Gamma })$-modules

Functors involved in Fontaine equivalences decompose as extension of scalars and taking of invariants between full subcategories of modules over a topological ring equipped with semi-linear continuous action of a topological monoid. We give a general framework for these categories and the functors between them. We define the categories of étale projective $\mathcal{S}$-modules over $R$ to englobe categories that will correspond by Fontaine-type equivalences to finite free representations of a group. We study their preservation by base change, taking of invariants by a normal submonoid of $\mathcal{S}$ and coinduction to a bigger monoid. We define and study categories corresponding to finite type continuous representations over ${\mathbb{Z}}_p$ through the notions of finite projective $(r,\mu )$-dévissage and of topological étale $\mathcal{S}$-modules over $R$.