Algebraic theories, monads, and arities

TL;DR

This paper explores how monads with arities generalize finitary monads, enabling the reconstruction of complex algebraic structures like n-categories through Kan extensions, with applications in semantics and higher category theory.

Contribution

It extends the concept of finitary monads to monads with arities, providing a framework for reconstructing various algebraic and categorical structures.

Findings

Monads with arities can be reconstructed using Kan extensions.

The framework applies to algebraic theories and higher categories.

Examples from semantics and higher category theory are provided.

Abstract

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.