Nominal Lawvere Theories: A category theoretic account of equational theories with names

•A novel class of categories called Nominal Lawvere Theories.•A sound and complete categorial interpretation of Nominal Equational Logic (NEL).•Classifying categories for NEL theories.•A correspondence between NEL theories and Nominal Lawvere Theories.

Names, or object-level variables, are a ubiquitous feature in programming languages and other computational applications. Reasoning with names, and related constructs like binding and freshness, often poses conceptual and technical challenges. Nominal Equational Logic (NEL) is a logic for reasoning about equations in the presence of freshness side conditions. This paper gives a category theoretic account of NEL theories, by analogy with Lawvere's classic correspondence between equational theories and small categories with finite products. This development reveals the abstract structure behind reasoning with equations modulo freshness.