What is a monad in a $2$-category?

Question

The wikipedia article on monads somewhat mysteriously notes that

Monads can be defined in any 2-category ${\mathfrak C}$. The monads defined above are for ${\mathfrak C}$ = Cat.

where Cat is the 2-category where objects are categories, morphisms are functors and 2-morphisms are natural transformations.

What does a monad in a general 2-category look like? My first attempt at defining it would be

If ${\mathfrak C}$ is a 2-category, a monad on an object $C\in {\mathfrak C}$ consists of an endomorphism $T:C\to C$ and a pair of 2-morphisms, $\eta:{\rm id}_C\to T$ and $\mu : T\circ T \to T$ with some coherence conditions.

I can't quite formulate the coherence conditions - it seems to hinge on knowing what $F\phi$ means when $F$ is a morphism and $\phi$ is a 2-morphism. I can understand the notation when we're talking about Cat, but not in the general case.

So my questions are:

1. What does the definition of a monad in a general 2-category look like?
2. Are there examples of 2-categories ${\mathfrak C}$, other than Cat, where monads on ${\mathfrak C}$ correspond to something we already know about?

Answer 1

Your attempt is correct. The coherence conditions are as expected. In a $2$-category, when $f$ is a $1$-morphism and $\eta$ is a $2$-morphism, then $f\eta$ abbreviates $\mathrm{id}_f \circ \eta$. For questions of this type you can almost always consult the nlab, in this case the article on monads states the definition you are looking for.

Concrete examples:

• Monoidal monads are monads in the $2$-category of monoidal categories.

• In the $2$-category of rings and bimodules a monad is an algebra over a ring.

• See also other examples in the nlab artice. You can also look at other concrete examples of $2$-categories and see what happens.

Answer 2

One of the many bits of data defining a 2-category is the notion of horizontal composition of 2-cells: given 1-cells $f, f' : X \to Y$ and $g, g' : Y \to Z$, and 2-cells $\alpha : f \Rightarrow f'$, $\beta : g \Rightarrow g'$, there is then a 2-cell $\beta \circ \alpha : g \circ f \Rightarrow g' \circ f'$. In $\mathfrak{Cat}$, we have the explicit formula $$\beta \circ \alpha = \beta f' \bullet g \alpha = g' \alpha \bullet \beta f$$ and this is sometimes called the Godement product. The composition of natural transformations in $\mathfrak{Cat}$ that people are familiar with is instead a special case of vertical composition (which I denote by $\bullet$, as above).

In a general 2-category, one defines the 2-cell $g \alpha : g f \Rightarrow g f'$ to be the horizontal composite $\textrm{id}_g \circ \alpha$. Once you know this, then the usual definition of monad carries over straightforwardly... at least in a strict 2-category.

In a weak 2-category (i.e. bicategory in the sense of Bénabou), because we need not have $(h \circ g) \circ f \stackrel{?}{=} h \circ (g \circ f)$ for 1-cells, one has to be a little bit more careful with the definition of monad. Of course, there is a natural 2-isomorphism from one composite to the other, and these satisfy coherence axioms similar to those for a monoidal category. Since we are only interested in defining monads, however, one can cheat a little using the following observation:

Proposition. If $X$ is an object in a bicategory $\mathfrak{C}$, then the hom-category $\mathfrak{C}(X, X)$ is naturally a monoidal category with unit $\textrm{id}_X$ and monoidal product $\circ$. Conversely, every monoidal category gives rise to a one-object bicategory in this way.

A monad on $X$ in $\mathfrak{C}$ then turns out to be the same thing as a monoid in $\mathfrak{C}(X, X)$. The only downside of this approach is that it is not so clear how to define a morphism of monads in $\mathfrak{C}$, but of course that can be done too. Nonetheless, this shows that the notion of monad subsumes that of monoid.