Coherence of strict equalities in dependent type theories

We study the coherence and conservativity of extensions of dependent type theories by additional strict equalities. By considering notions of congruences and quotients of models of type theory, we reconstruct Hofmann's proof of the conservativity of Extensional Type Theory over Intensional Type Theory. We generalize these methods to type theories without the Uniqueness of Identity Proofs principle, such as variants of Homotopy Type Theory, by introducing a notion of higher congruence over models of type theory. Our definition of higher congruence is inspired by Brunerie's type-theoretic definition of weak $\infty$-groupoid. For a large class of type theories, we reduce the problem of the conservativity of equational extensions to more tractable acyclicity conditions.