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This PhD thesis deals with some new models of intensional type theory and theUnivalence Axiom introduced by Vladimir Voevodsky. Our work takes place in theframework of the definitions of type-theoretic fibration categories (the notionof model under consideration in this thesis) and universe in a type-theoreticfibration category, definitions due to Michael Shulman. The goal of this thesisconsists mainly in the exploration of the stability of the univalence axiom, inparticular in the following sense: being given a type-theoretic fibrationcategory C equipped with a univalent universe U, we are eager to endow thefunctor category [D,C], where D is a small category, with the structure of atype-theoretic fibration category plus a univalent universe.

On lifting univalence to the equivariant setting