KZ-pseudomonads and Kan Injectivity

TL;DR

This paper introduces Kan injectivity in 2-categories and demonstrates how sets of morphisms induce KZ-pseudomonads, linking Kan injectivity to pseudoalgebra structures via a chain construction and small object argument.

Contribution

It develops the concept of Kan injectivity in 2-categories and shows how it leads to KZ-pseudomonads with pseudoalgebras characterized by Kan injectivity.

Findings

Every set of morphisms induces a KZ-pseudomonad in an adequate 2-category.

The 2-category of pseudoalgebras corresponds to objects Kan injective w.r.t. a set of morphisms.

Construction of a (pseudo)chain ensures convergence via a small object argument.

Abstract

We introduce the notion of Kan injectivity in 2-categories and study its properties. For an adequate 2-category $\mathcal{K}$, we show that every set of morphisms $\mathcal{H}$ induces a KZ-pseudomonad on $\mathcal{K}$ whose 2-category of pseudoalgebras is the locally full sub-2-category of all objects (left) Kan injective with respect to $\mathcal{H}$ and morphisms preserving Kan extensions. The main ingredient is the construction of a (pseudo)chain whose appropriate ``convergence" is ensured by a small object argument.