Ultracategories via Kan extensions of relative monads

TL;DR

This paper explores how ultracategories can be constructed via Kan extensions of relative monads, providing a new perspective on their algebraic structure and the process of ultracompletion.

Contribution

It demonstrates that under certain conditions, the left oplax Kan extension of a relative 2-monad produces a pseudomonad with the same colax algebras, applied specifically to ultracategories.

Findings

Ultracategories can be obtained through Kan extensions of relative monads.

The ultracompletion pseudomonad is recovered using this approach.

The method unifies the treatment of various structured categories.

Abstract

Many structured categories of interest are most naturally described as algebras for a relative monad, but turn out nonetheless to be algebras for an ordinary monad. We show that, under suitable hypotheses, the left oplax Kan extension of a relative 2-monad on categories yields a pseudomonad having the same category of colax algebras. In particular, we apply this to the study of ultracategories to recover the 'ultracompletion' pseudomonad.