Monads with arities and their associated theories

TL;DR

This paper generalizes the correspondence between monads and algebraic theories to categories with arities, demonstrating a canonical dense generator for algebras and applying it to groupoids and involutive graphs.

Contribution

It introduces a framework extending finitary monad-theory correspondence to categories with arities, including new arity characterizations for the free groupoid monad.

Findings

The category of algebras for a monad with arities has a canonical dense generator.

Extended the monad-theory correspondence to categories with arities.

Characterized arities for the free groupoid monad and recovered known nerve theorems.

Abstract

After a review of the concept of "monad with arities" we show that the category of algebras for such a monad has a canonical dense generator. This is used to extend the correspondence between finitary monads on sets and Lawvere's algebraic theories to a general correspondence between monads and theories for a given category with arities. As application we determine arities for the free groupoid monad on involutive graphs and recover the symmetric simplicial nerve characterisation of groupoids.