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Abstract inner automorphisms can be used to promote any category into a 2-category, and we study two-dimensional limits and colimits in the resulting 2-categories. Existing connected colimits and limits in the starting category become two-dimensional colimits and limits under fairly general conditions. Under the same conditions, colimits in the underlying category can be used to build many notable two-dimensional colimits such as coequifiers and coinserters. In contrast, disconnected colimits or genuinely 2-categorical limits such as inserters and equifiers and cotensors cannot exist unless no nontrivial abstract inner automorphisms exist and the resulting 2-category is locally discrete. We also study briefly when an ordinary functor can be extended to a 2-functor between the resulting 2-categories.
@article{TAC_2024_42_a1,
author = {Pieter Hofstra and Martti Karvonen},
title = {Inner automorphisms as 2-cells},
journal = {Theory and applications of categories},
pages = {19--40},
publisher = {mathdoc},
volume = {42},
year = {2024},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2024_42_a1/}
}
Pieter Hofstra; Martti Karvonen. Inner automorphisms as 2-cells. Theory and applications of categories, Hofstra Festschrift, Tome 42 (2024), pp. 19-40. http://geodesic.mathdoc.fr/item/TAC_2024_42_a1/