Second-Order Algebraic Theories

TL;DR

This paper develops the foundational categorical algebra for second-order universal algebra and equational logic, providing a formal framework for languages with variable binding and parameterized metavariables.

Contribution

It introduces second-order algebraic theories and establishes categorical equivalences, formalizing syntactic translations and encodings in variable binding languages.

Findings

Established categorical equivalences between syntactic and semantic notions.

Provided a formal definition of syntactic translation for second-order presentations.

First formalization of encodings and transforms in languages with variable binding.

Abstract

Fiore and Hur recently introduced a conservative extension of universal algebra and equational logic from first to second order. 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, the paper 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. Our 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.