We show that the maximal subgroup of the free profinite semigroup associated by Almeida to an irreducible sofic shift is a free profinite group, generalizing an earlier result of the second author for the case of the full shift (whose corresponding maximal subgroup is the maximal subgroup of the minimal ideal). A corresponding result is proved for certain relatively free profinite semigroups. W...

We study equivariant sheaves over profinite spaces, where the group is also taken to be profinite. resolve a serious deficit in existing theory by constructing good notion of presheaves, with suitable sheafification functor. Using sheafification, we develop general modules ring, give explicit constructions infinite products and introduce an analogue skyscraper sheaves. These results underlie re...

We construct a model structure on simplicial profinite sets such that the homotopy groups carry a natural profinite structure. This yields a rigid profinite completion functor for spaces and pro-spaces. One motivation is the étale homotopy theory of schemes in which higher profinite étale homotopy groups fit well with the étale fundamental group which is always profinite. We show that the profi...