TITLE: How exactly is Hochschild homology a monad homology?
QUESTION [14 upvotes]: Many texts which praise the generality of the bar construction associated to a monad, say that Hochschild homology is an example of this. 
What exactly is in this case the underlying endofunctor of the monad, on which category is it an endofunctor, what are the monad structure maps and, most important (since I think my confusion lies here), why do then the face maps look as on the wikipedia page?

REPLY [4 votes]: Assuming that the base ring $k$ is a field (or that the algebra is projective over $k$), a functor whose monad computes Hochschild homology is the one which maps $k$-modules $M$ to the $A$-bimodule $A\otimes M\otimes A$, which is adjoint to the forgetful functor in the other direction.
The bar complex corresponding to this adjunction is not exactly the one used by Hochschild to define his cohomology, but it is easily seen to give naturally isomorphic results, since it more or less evidently constructs gives projective resolutions of bimodules (not only of $A$, as is the case with the monad coming from one-sided extension of scalars)