Second-order algebraic theories

Abstract

Second-order universal algebra and second-order equational logic respectively provide a model theory and a formal deductive system for languages with variable binding and parameterised metavariables. This work completes the foundations of the subject from the viewpoint of categorical algebra. Specifically, this thesis introduces the notion of second-order algebraic theory and develops its basic theory. Two categorical equivalences are established: at the syntactic level, that of second-order equational presentations and second-order algebraic theories; at the semantic level, that of second-order algebras and second-order functorial models. The development includes a mathematical definition of syntactic translation between second-order equational presentations. This gives the first formalisation of notions such as encodings and transforms in the context of languages with variable binding.