Algebraic theories, monads, and arities

Monads are of interest both in semantics and in higher dimensional algebra. It turns out that the idea behind usual notion finitary monads (whose values on all sets can be computed from their values on finite sets) extends to a more general class of monads called monads with arities, so that not only algebraic theories can be computed from a proper set of arities, but also more general structures like n-categories, the computing process being realized using Kan extensions. This Master thesis compiles the required material in order to understand this question of arities and reconstruction of monads, and tries to give some examples of relevant interest from both semantics and higher category theory. A discussion on the promising field of operads is then provided as appendix.

💡 Research Summary

The thesis “Algebraic theories, monads, and arities” develops a unifying categorical framework that extends the classical notion of finitary monads to a broader class called “monads with arities”. In the traditional setting, a finitary monad on Set is completely determined by its values on finite sets; the values on arbitrary sets are obtained by a left Kan extension from the finite‑set subcategory. The author abstracts the role of “finite sets” into the concept of an arity: a small dense subcategory A of a base category C. A monad T has arities A if the restriction T|A exists and the full monad can be recovered as the left Kan extension Lan_A (T|A). An additional “arity‑preserving” condition guarantees that the unit and multiplication of T are also preserved under this extension.

The work proceeds in four main parts. The first chapter reviews monads, Lawvere theories, and the finitary case, emphasizing the Kan‑extension viewpoint. The second chapter formalizes arities, introducing density and presentability requirements on A, and proves that under these hypotheses the left Kan extension yields a genuine monad on C. The third chapter applies the theory to three distinct domains.

1. Algebraic theories – When A is the subcategory of finite sets, the construction reproduces the classical equivalence between finitary monads and Lawvere theories, confirming that the new definition genuinely generalizes the old one.

2. Higher‑dimensional category theory – By taking A to be the globular shapes (0‑globe, 1‑globe, …, n‑globe), the author builds the free n‑category monad. The values of the monad on arbitrary globular sets are obtained by left Kan extending from the globes, providing a clean categorical derivation of the computad construction and showing that the same machinery works uniformly for all n.

3. Programming semantics – Effects such as state, exceptions, or nondeterminism are modeled by signatures that serve as arities. The corresponding effect monads are then reconstructed via Kan extension, which yields a systematic treatment of effect combination and interpretation. This demonstrates that the arity‑monad framework can serve as a semantic backbone for effectful programming languages.

The final chapter, presented as an appendix, sketches a connection with operads. Operads encode multi‑input operations; the author observes that a multi‑arity subcategory can be regarded as the collection of operadic input shapes. By feeding an operad’s action into the Kan‑extension process one obtains an “operadic monad”, suggesting a pathway to integrate operadic algebra with monadic effects and higher‑dimensional structures.

Overall, the thesis establishes that choosing an appropriate arity subcategory allows one to reconstruct a wide variety of algebraic and higher‑categorical structures from a small amount of data. The left Kan extension acts as the computational engine, while the arity‑preserving condition ensures that the monadic laws survive the extension. This perspective not only unifies existing results (finitary monads, Lawvere theories, free n‑categories) but also opens new research directions, notably the systematic study of operadic monads and their applications to semantics and higher algebra.