Relative tensor product

Definition 19.2.9 (Relative tensor product). Let \(C\) be a monoidal \(\infty \)-category. Assume that \(C\) admits geometric realizations and that the tensor product \(- \otimes -\colon C \times C \to C\) preserves geometric realizations in both variables. For an associative algebra \(B\), a right \(B\)-module \(M\) and a left \(B\)-module \(N\), their relative tensor product is the object of \(C\) given by the geometric realization of the bar construction: \[ M \otimes _B N := \abs {\Bar _B(M,N)_{\bullet }} = \colim _{[n] \in \simp \catop } \Bar _B(M,N)_n. \] More generally, let \(A,B,D\in \Alg (C)\). If \(M\in {}_A\BMod _B(C)\) and \(N\in {}_B\BMod _D(C)\), then the bar construction lifts to a simplicial object of \({}_A\BMod _D(C)\). Geometric realizations of bimodules are created on underlying objects by [Lurie (2017), Proposition 4.3.3.9], so its realization defines an \((A,D)\)-bimodule \[ M\otimes _BN\in {}_A\BMod _D(C). \]