Bar construction

Construction 19.2.8 (Bar construction). Let \(B\) be an associative algebra in a monoidal \(\infty \)-category \(C\), let \(M\) be a right \(B\)-module, and let \(N\) be a left \(B\)-module. The two-sided bar construction for \(M\) and \(N\) over \(B\), denoted \(\Bar _B(M,N)_{\bullet }\), is a simplicial object in \(C\) described as follows:

• In degree \([n] \in \simp \catop \), it is given by \(\Bar _B(M,N)_n := M \otimes B^{\otimes n} \otimes N\);

• The face maps \(d_i \colon \Bar _B(M,N)_n \to \Bar _B(M,N)_{n-1}\) are given by:

  • \(d_0 = \act _M \otimes \id _{B^{\otimes (n-1)}} \otimes \id _N \colon M \otimes B \otimes B^{\otimes (n-1)} \otimes N \to M \otimes B^{\otimes (n-1)} \otimes N\).
  • For \(0 < i < n\), \(d_i\) is induced by the multiplication map \(m\colon B \otimes B \to B\) on the \(i\)-th and \((i+1)\)-th factors of \(B^{\otimes n}\).
  • \(d_n = \id _M \otimes \id _{B^{\otimes (n-1)}} \otimes \act _N \colon M \otimes B^{\otimes (n-1)} \otimes B \otimes N \to M \otimes B^{\otimes (n-1)} \otimes N\).
• The degeneracy maps \(s_i \colon \Bar _B(M,N)_n \to \Bar _B(M,N)_{n+1}\) are induced by inserting the unit \(e\colon \unit \to B\) at the \((i+1)\)-th position.