show/hide this revision's text 3 fixed LaTeX and caught a couple of typos

-- here is the Hochschild chain complex for an algebra A $A$ and bimodule M, $M$, as defined in Hochschild's original papers;

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). $A=M$). 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 $k$-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 once one can also obtain Hn(A,M) $H_n(A,M)$ 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) $k$-modules) that yields the Hochschild chain complex, arises by applying a certain Hom-functor (namely AHomA(__,X) ${}_A{\rm Hom}_A(\ \cdot\ ,X)$ ) to another simplicial object, say \beta(A), $\beta(A)$, in the category of A-bimodules.$A$-bimodules.

Now \beta(A) $\beta(A)$ is not contractible in the category of A-bimodules, $A$-bimodules, in general, and doesn't come from a (co)monad on that category. However, \beta(A) $\beta(A)$ can be identified with another simplicial object $F(A)$, which lives in the category of A-modules.$A$-modules.

What is F(A)?$F(A)$?

Well, take a step back and consider the adjunction between k-modules $k$-modules and A-modules $A$-modules (maybe you need k $k$ to be a field at this point, maybe not). That gives rise to a bar construction in A-mod, $A$-mod, namely for any given M $M$ in A-mod $A$-mod one obtains a simplicial object F(M) $F(M)$ which is given in each degree byF{-1}(M)=M, Fn(M)

$$ F_{-1}(M)=M\quad,\quad F_n(M) = M \otimes AA^{\otimes n+1} \ otimes (n+1){\rm for }\ n \geq 0. $$

Note that this is contractible in A-mod $A$-mod by the general machinery of the bar resolution associated to a monad. There was nothing to stop us taking M=A, $M=A$, that's a perfectly good A-module; $A$-module; and on doing so, lo and behold, we get the same simplicial object F(A).$F(A)$.

Thus, Hochschild homology, regardless of the choice of coefficients, can be thought of as "coming from" a comonad - namely, that induced on A-mod $A$-mod by the forgetful functor from A-mod $A$-mod to k-mod. $k$-mod. In my opinion, that is probably the (co)monad they are talking about.

It so happens that, since F(A) $F(A)$ is contractible in A-mod $A$-mod and hence a fotiori in k-mod, $k$-mod, the "chain-complex-ification" of \beta(A) $\beta(A)$ is, as a chain complex in R-bimod, $R$-bimod, a resolution of R $R$ by k-relatively $k$-relatively projective R-bimodules $R$-bimodules -- and hence applying RHomR(__,X) ${}_R{\rm Hom}_R(\ \cdot \ ,X)$ to it and taking homology coincides with taking k-relative $k$-relative Tor of R $R$ and X $X$ 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 Andre-Quillen 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).
show/hide this revision's text 2 slight tweak

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 is 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.) You can then use On reflecting a bit more, because the Connes-Tsygan exact sequence to get Hochschild homology out of cyclic homologya free (=tensor) algebra is confined to degrees 0 and 1, so I agree that perhaps once can also obtain Hn(A,M) as Tyler suggests, by taking the free algebra resolution of A (in some sense Hochschild homology with certain coefficients comes out the category of k-algebras) and then hitting the aforementioned adjunction. But resulting simplicial object with a suitable functor - but this seems indirect and leaves open trickier than in the problem commutative case (Andre-Quillen) and I can't get hold of that coefficient module..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 AHomA(__,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)=M, Fn(M) = M \otimes A\otimes (n+1) for n \geq 0. 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 RHomR(__,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 Andre-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.)
show/hide this revision's text 1

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 recollection is 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.) You can then use the Connes-Tsygan exact sequence to get Hochschild homology out of cyclic homology, so I agree that in some sense Hochschild homology with certain coefficients comes out of the aforementioned adjunction. But this seems indirect and leaves open the problem of that coefficient module...

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 AHomA(__,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)=M, Fn(M) = M \otimes A\otimes (n+1) for n \geq 0. 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 RHomR(__,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 Andre-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.)