KZ-pseudomonads and Kan injectivity

Ivan Di Liberti, Gabriele Lobbia, and Lurdes Sousa

We introduce the notion of Kan injectivity in 2-categories and study its properties. For an adequate 2-category K, we show that every set of morphisms H induces a KZ-pseudomonad on K whose 2-category of pseudoalgebras is the locally full sub-2-category of all objects (left) Kan injective with respect to 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.

Keywords: 2-category, Kan injectivity, KZ-pseudomonad, small object argument

2020 MSC: 18D70, 18D65, 18N10, 18N15

Theory and Applications of Categories, Vol. 40, 2024, No. 16, pp 430-478.

Published 2024-05-24.

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