Sites whose topoi are the smooth representations of locally profinite groups

We define a class of sites such that the associated topos is equivalent to the category of smooth sets (representations) of some locally prodiscrete monoids (to be defined). Examples of locally prodiscrete monoids include profinite groups and finite adele valued points of algebraic groups. This is a generalization of the fact that the topos associated with the \'etale site of a scheme is equivalent to the category of sets with continuous action by the \'etale fundamental group. We then define a subclass of sites such that the topos is equivalent to the category of discrete sets with a continuous action of a locally profinite group.