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.