Ultracategories as colax algebras for a pseudo-monad on CAT

TL;DR

This paper demonstrates that ultracategories, as defined by Lurie, can be characterized as normal colax algebras for a specific pseudo-monad on the category of categories, linking ultrafunctors to algebra morphisms.

Contribution

It provides a new algebraic perspective on ultracategories by relating them to colax algebras for a pseudo-monad, clarifying the structure of ultrafunctors.

Findings

Ultracategories are normal colax algebras for a pseudo-monad on CAT.

Ultrafunctors correspond to lax/colax algebra morphisms.

The approach offers a new algebraic framework for ultracategories.

Abstract

We show a result inspired by a conjecture by Shulman claiming that ultracategories as defined by Lurie are normal colax algebras for a certain pseudo-monad on the category of categories CAT. Such definition allows us to regard left and right ultrafunctors as defined by Lurie as instances of lax/colax algebras morphisms