The augmented two-sided bar complex

Let $k$ be a field and $A$ a unital associative $k$-algebra. Set $A^{{\otimes }_{k}0}:=k$ and, for every $n\ge 0$, set

${\mathrm{Bar}}_n(A):=A{\otimes }_k{A}^{{\otimes }_{k}n}{\otimes }_{k}A\mathrm{.}$

Write a pure tensor as $a_0\otimes \cdots \otimes a_{n+1}$. For $n\ge 1$ define

$d_n:={\sum }_{r=0}^n(-1{)}^r{\mu }_{r,r+1}:{\mathrm{Bar}}_n(A)\to {\mathrm{Bar}}_{n-1}(A),$

where ${\mu }_{r,r+1}$ multiplies slots $r$ and $r+1$ and leaves the other slots in order. The augmentation is

$\epsilon :=\mu :{\mathrm{Bar}}_0(A)=A{\otimes }_{k}A\to A,a_0\otimes a_1\mapsto a_0{a}_1\mathrm{.}$

In degree one, $d_1(a_0\otimes a_1\otimes a_2)=a_0{a}_1\otimes a_2-a_0\otimes a_1{a}_2\mathrm{.}$

The empty middle tensor convention makes ${\mathrm{Bar}}_0(A)=A{\otimes }_{k}A$; there is no unaugmented differential out of degree zero.

Using Enveloping algebra and the bimodule–module dictionary, each term has the following left and right $A^e$-module structures, considered separately:

$(c\otimes d^{\mathrm{o}\mathrm{p}})\cdot (a_0\otimes \cdots \otimes a_{n+1})=c{a}_0\otimes a_1\otimes \cdots \otimes a_{n+1}d,$

$(a_0\otimes \cdots \otimes a_{n+1})\cdot (c\otimes d^{\mathrm{o}\mathrm{p}})=d{a}_0\otimes a_1\otimes \cdots \otimes a_{n+1}c\mathrm{.}$

Every adjacent-multiplication face is linear for each of these module structures: at the first and last faces this is associativity, and at internal faces the outer factors are unchanged. The augmentation is linear on both sides, since $\epsilon (c{a}_0\otimes a_{1}d)=c(a_0{a}_1)d$ and $\epsilon (d{a}_0\otimes a_{1}c)=d(a_0{a}_1)c$. This item specifies the bar terms, maps, and outer actions; the asserted zero-composite identities are addressed by the following bar-boundary lemma.