On presheaf submonads of quantale enriched categories

TL;DR

This paper investigates presheaf monads and their submonads within $V$-categories for a quantale $V$, providing characterizations via $V$-distributors and analyzing their algebraic structures.

Contribution

It introduces two new characterizations of presheaf submonads using $V$-distributors and explores the algebraic categories associated with key examples like the formal ball and Lawvere monads.

Findings

Characterizations of presheaf submonads via admissible classes and Beck-Chevalley conditions

Analysis of Eilenberg-Moore categories for specific monads

Insights into the structure of $V$-categories and their monads

Abstract

This paper focus on the presheaf 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. Then we focus on the study of the corresponding Eilenberg-Moore categories of algebras, having as main examples the formal ball monad and the so-called Lawvere monad.