This is partly in response to Reid, but also intended as general clarification.
As I understand it, Peter's original question was:
-- here is the Hochschild chain complex for an algebra A and bimodule M, as defined in Hochschild's original papers;
-- it is the chain complex associated to a certain simplicial object as defined on the Wikipedia page;
-- one is told that this object comes from the bar construction (or standard resolution) associated to some monad;
-- where/what is the monad?
The last one seems to be Reid's underlying point/question. Tyler says you can get it, up to a dimension-shift, from the adjunction between k-modules and k-algebras (at least when A=M). My earlier recollection was that this more naturally leads to cyclic homology a.k.a. additive K-theory as defined by Feigin and Tsygan, but I have yet to check this against a copy of their paper. (The point is that in characteristic zero, the cyclic homology of a free tensor algebra on a given k-module, coincides with the cyclic homology of the ground field, so one can take free resolutions of a given k-algebra and then use spectral sequence arguments.) On reflecting a bit more, because the Hochschild homology of a free (=tensor) algebra is confined to degrees 0 and 1, perhaps one can also obtain Hn(A,M) as Tyler suggests, by taking the free algebra resolution of A (in the category of k-algebras) and then hitting the resulting simplicial object with a suitable functor - but this seems trickier than in the commutative case (Andre-Quillen) and I can't get hold of a copy of Quillen's paper at the moment.
Alors. As I understand it, following Weibel's book (and the papers of Barr & Beck et al), the simplicial object (in the category of k-modules) that yields the Hochschild chain complex, arises by applying a certain Hom-functor (namely ArmHomA(cdot,X) ) to another simplicial object, say beta(A), in the category of A-bimodules.
Now beta(A) is not contractible in the category of A-bimodules, in general, and doesn't come from a (co)monad on that category. However, beta(A) can be identified with another simplicial object F(A), which lives in the category of A-modules.
What is F(A)?
Well, take a step back and consider the adjunction between k-modules and A-modules (maybe you need k to be a field at this point, maybe not). That gives rise to a bar construction in A-mod, namely for any given M in A-mod one obtains a simplicial object F(M) which is given in each degree by
F−1(M)=Mquad,quadFn(M)=MotimesAotimesn+1rmforngeq0.
Note that this is contractible in A-mod by the general machinery of the bar resolution associated to a monad. There was nothing to stop us taking M=A, that's a perfectly good A-module; and on doing so, lo and behold, we get the same simplicial object F(A).
Thus, Hochschild homology, regardless of the choice of coefficients, can be thought of as "coming from" a comonad - namely, that induced on A-mod by the forgetful functor from A-mod to k-mod. In my opinion, that is probably the (co)monad they are talking about.
It so happens that, since F(A) is contractible in A-mod and hence a fotiori in k-mod, the "chain-complex-ification" of beta(A) is, as a chain complex in R-bimod, a resolution of R by k-relatively projective R-bimodules -- and hence applying RrmHomR(cdot,X) to it and taking homology coincides with taking k-relative Tor of R and X as R-bimodules. Hence the point of view that Hochschild homology is a special case of relative Tor.
Finally, I actually agree with Reid that this is not the best example to motivate (co)monad (co)homology. Group cohomology with coefficients in the ground field; or indeed André-Quillen cohomology, which is given by a "free algebra" adjunction but only for commutative algebras, or sheaf cohomology, would be better. (No originality in my choices; I've cribbed them out of Weibel Section 8.6).
(Apologies for the length and the tediousness, by the way.)
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