# The formal theory of monads II

### Stephen Lack and Ross Street

This appeared in Journal of Pure and
Applied Algebra 175:243-265, 2002.
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.

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Steve Lack
Last modified: Fri Aug 30 09:12:03 EST 2002