For a 2-category K, we consider Street's 2-category Mnd(K) of monads in K, along with Lack and Street's 2-category EM(K) and the identity-on-objects-and-1-cells 2-functor Mnd(K) → EM(K) between them. We show that this 2-functor can be obtained as a "free completion" of the 2-functor 1: K → K. We do this by regarding 2-functors which act as the identity on both objects and 1-cells as categories enriched a cartesian closed category BO whose objects are identity-on-objects functors. We also develop some of the theory of BO-enriched categories.
Keywords: monads, Eilenberg-Moore objects, limit completions, 2-categories, enriched categories
2020 MSC: 18C15, 18C20, 18D20, 18N10, 18A35
Theory and Applications of Categories, Vol. 42, 2024, No. 1, pp 2-18.
Published 2024-06-13.
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