On enriched terms and 2-categorical universal algebra

TL;DR

This paper introduces a new concept of enriched terms in 2-categories, generalizing previous notions, and establishes foundational results including free structures and a Birkhoff variety theorem in this enriched 2-categorical setting.

Contribution

It develops a generalized notion of enriched terms and interpretability, extending to 2-categories, and provides explicit constructions and a Birkhoff theorem in this context.

Findings

Defined recursively generated enriched terms for 2-categories

Established a notion of term-interpretability for enriched structures

Proved a 2-dimensional Birkhoff variety theorem

Abstract

We introduce a new notion of recursively generated enriched term which generalizes the one studied in joint work with Rosick\'y. These new terms come together with a notion of term-interpretability, which recovers the same type of interpretability that has been considered for enrichment over posets, metric spaces, and $\omega$-complete posets. As an application of this, we specialize to the 2-categorical case by considering 2-dimensional terms and 2-dimensional equational theories. In this context we also give an explicit description of free structures and prove a 2-dimensional Birkhoff variety theorem.