Reduced graded symmetric coalgebra with its shuffle coproduct

Let $k$ be a characteristic-zero field (Field), meaning $\mathrm{char}(k)=0$ under The characteristic of a ring: the least $n\ge 1$ with $n\cdot 1_R=0$ when one exists, and $0$ otherwise. By Every field is a commutative ring with $1\mathrm{\ne }0$; it is an integral domain, and it is a commutative division ring, $k$ is a commutative ring. Let $W={⨁}_{j\in \mathbb{Z}}{W}^j$ be a graded $k$-vector space. For $n\ge 1$, define the graded symmetric power

$S^n(W)=W^{{\otimes }_{k}n}\mathrm{/}{J}_n,$

where $J_n$ is spanned by the differences obtained by replacing any adjacent homogeneous factors $x\otimes y$ by $(-1{)}^{\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }}y\otimes x$ inside a pure tensor. Write $w_1\odot \cdots \odot w_n$ for the image of a tensor and set

$S^c(W)={⨁}_{n\ge 1}{S}^n(W)\mathrm{.}$

The reduced shuffle coproduct is the degree-zero linear map $\bar{\mathrm{\Delta }}:S^c(W)\to S^c(W){\otimes }_k{S}^c(W)$ given on a homogeneous pure symmetric tensor by

$\bar{\mathrm{\Delta }}(w_1\odot \cdots \odot w_n)={\sum }_{p=1}^{n-1}{\sum }_{\sigma \in \mathrm{Sh}(p,n-p)}ϵ(\sigma ;w_1,\dots ,w_n)(w_{\sigma (1)}\odot \cdots \odot w_{\sigma (p)})\otimes (w_{\sigma (p+1)}\odot \cdots \odot w_{\sigma (n)}),$

where $\mathrm{Sh}(p,n-p)=\{\sigma \in S_n:\sigma (1)<\cdots <\sigma (p),\sigma (p+1)<\cdots <\sigma (n)\}$, and $ϵ$ is the Koszul sign of Koszul sign rule for graded tensor products and permutations. For $n=1$ the sum is empty and $\bar{\mathrm{\Delta }}(w)=0$. This is a reduced, hence noncounital coalgebra convention; no arity-zero summand is included. The next lemmas verify its coproduct laws, conilpotence, and cofree universal property.