Normalization by Evaluation and Algebraic Effects

We examine the interplay between computational effects and higher types. We do this by presenting a normalization by evaluation algorithm for a language with function types as well as computational effects. We use algebraic theories to treat the computational effects in the normalization algorithm in a modular way. Our algorithm is presented in terms of an interpretation in a category of presheaves equipped with partial equivalence relations. The normalization algorithm and its correctness proofs are formalized in dependent type theory (Agda).