Codensity: Isbell duality, pro-objects, compactness and accessibility

TL;DR

This paper explores the properties of codensity monads induced by certain subcategories, revealing their connections with Isbell duality, pro-objects, and compactness, and discusses their accessibility and a new notion of generically idempotent monads.

Contribution

It establishes links between codensity monads, Isbell duality, pro-objects, and compact spaces, and introduces the concept of generically idempotent monads.

Findings

Codensity monads are generally not accessible.

Connections between codensity monads and Isbell duality are demonstrated.

Introduction of the notion of generically idempotent monads.

Abstract

We study codensity monads $T$ induced by (mostly small, mostly dense) full subcategories $A \subset K$. These monads behave quite similarly, we show some connections with the Isbell duality, pro-finite objects and compact spaces. We prove that they are quite unlikely to be accessible. Finally, we introduce the notion of generically idempotent monad and comment its properties.