The formal theory of monads II

We give an explicit description of the free completion of a 2-category under the Eilenberg–Moore construction, and show that this has the same underlying category as the 2-category of monads in . 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 as the 2-category of monads rather than . We also introduce the wreathsin ; these are the objects of , 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.