Eilenberg-Moore category

The Eilenberg-Moore Category: a short introduction

Part III Seminar Talk, 28 November 2007, 4.30-5.15pm, CMS Meeting Room 15.

An adjunction between two categories called C and D is a way of 'comparing' the two categories by means of two functors: one functor (F) going out of C, and another one (G) going into C.

A good example of an adjunction is the one between Sets and Groups: out of sets we have the Free functor, sending a set A to the free group on the elements of A, and to make this into an adjunction, we just go back from Groups to Sets by "forgetting" about the group structure of a given group B, so that GB is the underlying set of B.

For our purposes, we don't mind what the category D is, and just focus on the composite GF, which goes out of C and then back into C again (this is called an endofunctor on C). We wonder what kind of 'structure' GF imposes on C. (For example, what is GFA in the Free adjunction between Sets and Groups as described above?)

The kind of structure an adjunction imposes on C is called a monad. So any adjunction between any two categories C and D gives rise to a monad on the category we start from, C.

As often in mathematics, we'd like to find a converse to the statement 'every adjunction gives rise to a monad': suppose you find some monad T lying around somewhere, but you can't remember where that monad ever came from. Is it possible to construct an adjunction which gives rise to this monad?

The answer is: yes! An example of a construction of an adjunction which does this for you, is the adjunction with the Eilenberg-Moore category, on which I'll focus in this talk. I will describe the construction of the Eilenberg-Moore category and the associated adjunction, and then show that this adjunction does what we want: it gives rise to the monad T.

Actually, there exists a whole category of adjunctions giving rise to your personal favourite monad, and the Eilenberg-Moore category is a very special one in this category: it's terminal. At the end of my talk I'll hint on this fact, and the associated idea of monadicity of an adjunction.

Last modified: 21 November, 2007.