TITLE: What is known about the category of monads on Set?
QUESTION [24 upvotes]: Monads on the category Set of sets and functions are somehow fundamental objects of category theory, and moreover they have important applications to computer science.  We know of a good number of monads on Set, but they all appear (at least to me) as isolated examples (other than three big classes of them I'll discuss below).  I'm wondering what is known about the category of monads on Set; call it $Mnd$.  Below, I'll refer to a monad $(M,\eta,\mu)$ in $Mnd$ simply by its functor, $M$.
Clearly, the category $Mnd$ is not small because for any set $E$, there is a monad $X\mapsto X\amalg E$ and another monad $X\mapsto X^E$.  It would be nice to "classify" monads so as to see these as two, rather than as a large number, of examples.  Are there ways to "classify" objects in $Mnd$ to notice more broad patterns?  Here's another class of them for example: those coming from algebraic theories (e.g. the free group monad, the free ring monad, etc.)
For what types of diagrams does $Mnd$ have limits or colimits?  Clearly $Mnd$ has an initial object (the identity monad) and a final object ($X\mapsto\{\star\}$).
Suppose given a monad $M\in Mnd$, and suppose we know (1) the set $M(\emptyset),$ (2) the set $M(\{\star\})$, and (3) the function $M(\emptyset\to\{\star\})$.  Can we say anything else about $M$?  Can we limit the three possible answers to (1),(2),(3) above for monads? 
Are any other interesting facts known about $Mnd$?

REPLY [23 votes]: The category of all monads on Set (often this is called $Mnd$, and $Mon$ means the category of monoids) is not just large, but has large hom-sets. 
But if you restrict it to the full subcategory $Mnd_f$ of all finitary monads, then you get a beautiful category: it is not just complete and cocomplete but locally finitely presentable.
It is even monadic over the category $Set^N$ of families of sets indexed by the natural numbers. 
The finitary monads are the ones that correspond to finitary (one-sorted) algebraic theories, and so include the monads for monoids, commutative monoids, groups, rings, Lie algebras (over a given field), and so on. More formally, a monad is finitary when its endofunctor part $T:Set\to Set$ preserves filtered colimits, or equivalently when $T$ is the left Kan extension of its restriction to finite sets. 
An example of a monad which is not finitary is the ultrafilter monad, whose algberas are compact Hausdorff spaces. 
Instead of finitary monads, you can take monads of rank $\alpha$, for some regular cardinal $\alpha$ - these are the ones whose endofunctors preserve $\alpha$-filtered colimits, and which can be described in terms of $\alpha$-small operations and equations. The full subcategory $Mnd_\alpha$ of $Mnd$ consisting of the monads of rank $\alpha$ is also locally finitely presentable. For example $Mnd_f$ is just the case where $\alpha=\aleph_0$.
The inclusion $Mnd_\alpha\to Mnd$ does preserve colimits (in fact it has a right adjoint), and so $Mnd$ does have colimits of diagrams of monads with bounded rank. 
An example of a monad which does not have a rank is the powerset monad, whose algebras are the complete semilattices.