Okay, but what’s the point of looking at monoids in the category of endofunctors?
Recall that if is a monoid in a monoidal category , then we can talk about a module over : it’s an object equipped with an action map
satisfying some axioms. But this is clearly not the most general notion of action of a monoid we can think of. For example, we know what it means for an ordinary monoid , in , to act on the objects of any category whatsoever (we just want a monoid homomorphism ), and that notion of monoid action isn’t subsumed by this definition.
Here’s something more general that comes closer. It’s not necessary that and live in the same category. Instead, can live in a monoidal category while lives in a category equipped with the structure of a module category over . In particular this means that there is an action functor
with some extra structure satisfying some axioms. This allows us to make sense of as an object in and hence to make sense of an action map as before, and even to state the usual axioms. (Another instance of the microcosm principle.)
Now, fix . What is the most general monoidal category over which is a module category? Of course, it’s the monoidal category of endofunctors of . Hence the most general kind of monoid that can act on an object in is a monoid in , or equivalently a monad.
In fact, since an action of a monoidal category on can be described as a monoidal functor , any action of a monoid on an object in the sense described above naturally factors through an action of a monad.
Example. Any cocomplete category is naturally a module category over with the action given by
More precisely, in this situation we say that is tensored over (which is in some sense dual to being enriched over ). This lets us describe what it means for a monoid in to act on an object .
Example. Any symmetric monoidal cocomplete category (this includes the hypothesis that the monoidal operation distributes over colimits) is naturally a module category over the monoidal category (see this blog post) of species (equipped with the composition product ) with the action given by
where denotes, as above, , and denotes the quotient of this by the diagonal action of .
This lets us describe what it means for a monoid in to act on an object . And a monoid in is precisely an operad.