Shuffle comparison and symmetric powers of simplicial modules

On unnormalized chains use the front/back formula $\mathrm{AW}(x\otimes y)={\sum }_{p+q=n}{d}_{p+1}\cdots d_{n}x\otimes d_0^{p}y$ and the shuffle formula obtained by inserting the complementary degeneracies into $x\in X_p$ and $y\in Y_q$, with the shuffle permutation sign. Every summand is separately linear in $x,y$. The simplicial identities show these are chain maps: internal faces cancel in pairs differing by transposition of adjacent shuffle labels, and the uncancelled endpoint faces are exactly the two tensor differentials. Degenerate inputs go to sums with at least one degenerate factor. The maps therefore induce the asserted normalized maps.

Here is an acyclic-model construction of the homotopies, including its linear descent. In degree $n$, unnormalized diagonal chains are represented bilinearly by $k\mathrm{\Delta }[n],k\mathrm{\Delta }[n]$; tensor chains of bidegree $(p,q)$ are represented by $k\mathrm{\Delta }[p],k\mathrm{\Delta }[q]$. The target on a diagonal model is the chain complex of $\mathrm{\Delta }[p]\times \mathrm{\Delta }[q]$, contracted to $(0,0)$ by prepending $(0,0)$ to its two vertex lists. On a tensor model use $c=h_p\otimes 1+{\iota }_p{ϵ}_p\otimes h_q$, where $h_p$ prepends vertex $0$, ${ϵ}_p$ is augmentation and ${\iota }_p$ includes vertex $0$; tensor signs are the ordinary chain-complex signs. In either target $\mathrm{\partial }c+c\mathrm{\partial }=1-\iota ϵ$. This identity follows by separating the first face of the prepended list from the remaining faces. It applies to zero-augmented cycles also in degree zero.

For either pair $u,v$ of composites and identity, start $H_{-1}=0$. On a universal generator $t$ in degree $n$, having defined lower degrees, set $z=(u-v)t-H_{n-1}(\mathrm{\partial }t)$ and ${\tilde{H}}_n(t)=c(z)$. Its boundary is $z$: indeed $\mathrm{\partial }z=0$ by the lower-degree homotopy identity and ${\mathrm{\partial }}^2=0$, while $ϵz=0$ since $u,v$ induce the same augmentation. Extend this formula by evaluating the universal simplices at the actual $x,y$. All evaluated expressions are finite sums of tensors of faces and degeneracies, separately linear in $x,y$, so this defines a natural transformation on arbitrary modules, including their linear relations. This construction does not require the chosen contractions of the models to commute with simplicial maps. Naturality follows from the evaluation prescription on the representable models.

The same recursion works on normalized chains using their natural projector $P$ supplied by [F1]. More explicitly, at each degree extend the model formula to unnormalized generators and precompose it with $P$; on the tensor source use $P\otimes P$. The residual $z$ already factors through that projector, since lower homotopies do and the normalized differential commutes with it. Projection in the target then gives $\mathrm{\partial }{H}_n+H_{n-1}\mathrm{\partial }=u-v$ on the normalized direct summand. This proves both normalized chain-homotopy identities without assuming the universal contraction descends to the degenerate quotient.

Composing shuffles of two blocks indexes both associative composites by the same ordered three-block shuffles with the same signs. Interchanging two blocks multiplies by the graded interchange sign; the zero-degree unit is fixed. Thus the $r$-fold shuffle is ${\mathrm{\Sigma }}_r$-equivariant and a quasi-isomorphism by iteration. For $r=0$ it is the identity on $k$. Tensor products preserve quasi-isomorphisms over a field. Averaging by $1\mathrm{/}r!$ makes coinvariants exact, so taking ${\mathrm{\Sigma }}_r$-coinvariants preserves this quasi-isomorphism. Exact normalization commutes with coinvariants and direct sums. This gives the displayed symmetric-power comparison for every $r$, with no boundedness or finite-dimensionality assumption on $X$.