The nerve theorem for relative monads

Nathanael Arkor ; Dylan McDermott

Theory and applications of categories, Lawvere Festschrift, Tome 43 (2025), pp. 403-454.

Résumé

A fundamental result in the theory of monads is the characterisation of the category of algebras for a monad in terms of a pullback of the category of presheaves on the category of free algebras: intuitively, this expresses that every algebra is a colimit of free algebras. We establish an analogous result for enriched relative monads with dense roots, and explain how it generalises the nerve theorems for monads with arities and nervous monads. As an application, we derive sufficient conditions for the existence of algebraic colimits of relative monads. More generally, we establish such a characterisation of the category of algebras in the context of an exact virtual equipment. In doing so, we are led to study the relationship between a j-relative monad T and its associated loose-monad E(j,T), and consequently show that the opalgebra object and the algebra object for T may be constructed from certain double categorical limits and colimits associated to E(j,T).

Détail

Publié le : 2025-05-11

Classification : 18D70, 18D65, 18C15, 18C20, 18A40, 18D60, 18C10, 18D20, 18N10
Keywords: Relative monad, relative adjunction, monadicity, virtual double category, virtual equipment, formal category theory, enriched category theory