Let $R$ be a $k$-algebra and $M$ be an $(R,R)$-bimodule. Let $[n] \mapsto M \otimes R^{\otimes n}$ be the simplicial $k$-module which defines the Hochschild homology $H\_\*(R,M)$. Is it possible to write down explicitly the $k$-linear map $f^\* : M \otimes R^{\otimes n} \to M \otimes R^{\otimes m}$ associated to a monotonic map $f : [m] \to [n]$? Does this give a more slick way to construct the simplicial module? I imagine that once we have a formula, one can prove $(f \circ g)^\* = g^\* \circ f^\*$ directly and does not have to verify all the simplicial identities.



Slick construction of Hochschild complex
The category Δ\_a of finite linearly ordered sets with disjoint union is a monoidal category.


A monoid M in a monoidal category F is a strong monoidal functor Δ\_a → F.


Discarding the monoidal structure on such a functor yields an augmented simplicial object,
which is (by definition) the bar construction of M.


The reversal functor Δ\_a → Δ\_a turns a monoid into its opposite.


The category Δ\_e of pointed objects in Δ\_a, where the basepoint is the smallest element,
is a right module over Δ\_a.


A right module X over a monoid M in a monoidal category F can now be defined
as a strong right module over M in the sense of strong monoidal functors,
with structure maps X(x)⊗M(m) → X(x⊗m).
There is a canonical inclusion Δ\_e → Δ\_a (add a basepoint as the smallest element),
and restricting along it gives back M.


A left module is defined in the same way, but using Δ^e, where the basepoint is now the largest element.
We also have an inclusion Δ^e→Δ\_a.
The reversal functor Δ^e → Δ\_e turns a right module into a left module,
and vice versa.


Finally, M-N-bimodules can be defined using Δ\_\*, the category of pointed objects in Δ (or Δ\_a).
We have two inclusions Δ\_e → Δ\_\* and Δ^e → Δ\_\*, which recover M and N.
The reversal functor Δ\_\* → Δ\_\* sends an M-N-bimodule into an N^op-M^op-bimodule.


For M-M bimodules we can “glue” together the left and right action
and define an M-cyclic module as a functor C\_\* → F
whose restriction along the inclusion Δ\_\* → C\_\* yields an M-M-bimodule.
Here C\_\* denotes the pointed cyclic category (definition left as an exercise)
and Δ\_\* → C\_\* closes a finite linearly ordered set into a cycle.


(If the bimodule is M with its left and right M-action,
then cycles need not be pointed, and we recover the cyclic bar construction C → F of M this way.)


The underlying simplicial object of an M-cyclic module
can be now recovered by precomposing with the functors Δ → Δ\_\* → C\_\*.


All of this is easy to remember if one keeps in mind
that all these combinatorial categories
are small models for the monoidal Weiss sites of various topological manifolds with boundaries and defects:
the interval (0,1) for Δ\_a,
[0,1) for Δ\_e,
(0,1] for Δ^e,
(-1,1) with {0} pointed for Δ\_\*,
S^1\_\* for C\_\*,
S^1 for C.


