The Isbell monad

TL;DR

This paper clarifies the structure of the Isbell envelope by framing it as a pseudomonad on the 2-category of categories, and introduces a new concept called cylinder factorisation systems.

Contribution

It characterizes the Isbell monad as a pseudomonad and introduces cylinder factorisation systems as its pseudoalgebras, extending existing factorisation concepts.

Findings

The Isbell envelope forms a pseudomonad on the 2-category of categories.

Pseudoalgebras of the Isbell monad are categories with cylinder factorisation systems.

Cylinder factorisation systems extend Freyd and Kelly's factorisation systems to cocones and cones.

Abstract

In 1966, John Isbell introduced a construction on categories which he termed the "couple category" but which has since come to be known as the Isbell envelope. The Isbell envelope, which combines the ideas of contravariant and covariant presheaves, has found applications in category theory, logic, and differential geometry. We clarify its meaning by exhibiting the assignation sending a locally small category to its Isbell envelope as the action on objects of a pseudomonad on the 2-category of locally small categories; this is the Isbell monad of the title. We characterise the pseudoalgebras of the Isbell monad as categories equipped with a cylinder factorisation system; this notion, which appears to be new, is an extension of Freyd and Kelly's notion of factorisation system from orthogonal classes of arrows to orthogonal classes of cocones and cones.