KZ-pseudomonads and Kan Injectivity
Ivan Di Liberti, Gabriele Lobbia, Lurdes Sousa
TL;DR
This paper develops a 2-categorical analogue of orthogonal subcategory/reflective-subcategory theory via Kan injectivity, connecting left Kan extensions to lax-idempotent pseudomonads (KZ-pseudomonads). By constructing a transfinite Kan-injective pseudochain built from coequinserters, it produces a left adjoint to the inclusion of Kan-injective objects, and demonstrates that the resulting pseudoalgebras coincide with Kan-injective objects, i.e., the Eilenberg–Moore 2-category is equivalent to a Kan-injective subcategory. The main result shows that, under smallness and bicolimit hypotheses, the inclusion of a Kan-injective sub-2-category is the right part of a KZ-adjunction, providing KZ-monadicity and a structural bridge to left Kan injectivity. The latter sections integrate Lex-colimits and distributive laws, showing how Kan injectivity characterizes lex-exactness, and linking to Garner–Lack’s theory of lex-colimits, with concrete characterizations in terms of $D$-maps and $\, ext{Phi}_l$-algebras. Collectively, the work offers a comprehensive framework for presenting and studying 2-categories via Kan-injectivity and KZ-doctrines, with broad implications for exactness, monadicity, and lex-theoretic constructions.
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.