We give an explicit description of the free completion EM(K) of a 2-category K under the Eilenberg-Moore construction, and show that this has the same underlying category as the 2-category Mnd(K) of monads in K. We then demonstrate that much of the formal theory of monads can be deduced using only the universal property of this completion, provided that one is willing to work with EM(K) as the 2-category of monads rather than Mnd(K). We also introduce the wreaths in K; these are the objects of EM(EM(K)), and are to be thought of as generalized distributive laws. We study these wreaths, and give examples to show how they arise in a variety of contexts. © 2002 Elsevier Science B.V. All rights reserved.
Lack, S., & Street, R. (2002). The formal theory of monads II. Journal of Pure and Applied Algebra, 175(1–3), 243–265. https://doi.org/10.1016/S0022-4049(02)00137-8