Slick construction of Hochschild complex

Question

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.

Answer 1

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.