Dear All, As usual, there have been plenty of people with comments about history. There was also a second part to the question:
Btw, I frequently encounter monads in a categories of functors which are not endofunctors. An example are finite dimensional vectorspaces which can be constructed via a monoid in the category of functors FinSet -> Set, here I is the embedding and (x) can be constructed from the left kan extension and composition. The unit is given by the Kronecker delta and join can be constructed from Matrix multiplication. Should one call these beasts monads as well? Is there a good reference for this type of construction?
The category of functors from FinSet to Set is equivalent to the category of endofunctors of Set which preserve filtered colimits: such endofunctors are usually called finitary. Thus a monoid in [FinSet,Set] with respect to this tensor product is the same thing as a monad on Set whose endofunctor part is finitary: this is called a finitary monad. These finitary monads on Set are equivalent to Lawvere theories and so in turn to (finitary, single-sorted) varieties. Finitary monads can also be considered on other base categories than Set, especially on locally finitely presentable ones. It is true that vector spaces are the algebras for a finitary monad on Set. There is no need to restrict to finite-dimensional vector spaces; in fact it is not true that there is a monad on Set whose algebras are the finite-dimensional vector spaces. Steve.