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](http://en.wikipedia.org/wiki/Hochschild_homology)?



How exactly is Hochschild homology a monad homology?
[Tyler, are you sure about this?](https://mathoverflow.net/questions/3078/how-exactly-is-hochschild-homology-a-monad-homology/3085#3085) I thought the bar construction comes from the adjunction between R-modules and k-modules for R a given k-algebra (i.e. relative Tor). Besides, what you say only makes sense if we're taking coefficients in R itself and not a general bimodule M.


If memory serves correctly, starting with a k-algebra R and looking at a simplicial resolution for it via the adjunction k-modules -- k-algebras leads to *cyclic* homology as in the paper of Feigin-Tsygan.


That wiki page also looks off to me: the Loday construction is for the Hochschild homology and decomposition for *commutative* algebras, and this isn't made very clear in the wiki.


**quick edit:** There's a good if terse discussion of the bar construction via monads in Weibel Chapter 8 (p.283). I suspect you could also extract the desired information out of the much more general machinery in [Jon Beck's thesis](http://www.tac.mta.ca/tac/reprints/articles/2/tr2abs.html), modulo some possible struggle with notation.


**encore une fois:** Consider the adjunction between k-mod and R-mod. If M is an object of R-mod then the simplicial construction provided by the adjunction looks like this


M <--- R\otimes M <--- R \otimes R \otimes M <----- etc


where I've not been able to draw in all the face maps, but hopefully you get what I mean. Now by taking the alternating sum of face maps in each degree, we get a split exact sequence of R-module maps


M <--- R\otimes M <--- R \otimes R \otimes M <----- etc


which is a resolution in the classical sense of M in R-mod-R by R-mod projectives - I'm assuming k is a field for sake of convenience. (So you can use it to calculate Tor^R if you wish.)


Now take M=R and note that we have a resolution of R by R^e-projectives. Apply Hom{R^e}(\_\_, X) where X is your coefficient module, and you get *precisely* the Hochschild chain complex as in the original papers.


Of course, we didn't have to take sums of face maps before applying the Hom functor. So, if we start with R regarded as an object of R-mod, the canonical simplicial construction (for M=R) would give us a contractible simplicial object in R-mod with M=R at the bottom, this object would in fact live in R^e-mod, and so is eligible to be hit with HomR^e(\_\_,X). If we do this, we get a simplicial object in k-mod, and said object *should* be the one described in the wiki article, corresponding to the Hochschild chain complex.


