On presheaf submonads of quantale-enriched categories

Abstract

This paper focuses on the presheaf monad, or the free cocompletion monad, and its submonads on the realm of V-categories, for a quantale V. First we present two characterisations of presheaf submonads, both using V-distributors: one based on admissible classes of V-distributors, and other using Beck-Chevalley conditions on V-distributors. Further we prove that lax idempotency for 2-monads on V-Cat can be characterized via such a Beck-Chevalley condition. Then we focus on the study of the Eilenberg-Moore categories of algebras for our monads, having as main examples the formal ball monad and the Lawvere-Cauchy completion monad.