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What is the universal property of the 2-category of monads?

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Stephen Lack and Adrian Miranda

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.

http://www.tac.mta.ca/tac/volumes/42/1/42-01.pdf

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