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.
Keywords: Inner automorphisms, crossed modules, limits and colimits
2020 MSC: 18A30, 18G45, 18N10
Theory and Applications of Categories, Vol. 42, 2024, No. 2, pp 19-40.
Published 2024-06-19.
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