ScalingStacks

A.9.2 The bar construction and relative tensor product of bimodules[00J6]

We describe the relative tensor product of bimodules in the presentably monoidal case, though the theory works much more generally.

Given bimodules \({}_{A}M_B\) and \({}_{B}N_C\) between algebras \(A,B,C\) in a monoidal \(\infty\)-category \(\mathcal M\) the bar construction defines a simplicial object \({\textup{Bar}}(M,B,N)\in {}_{A} \mathrm{BMod}_C(\mathcal C)\) with \({\textup{Bar}}(M, B, N)_n \coloneqq M \otimes B^{\otimes n} N\) with face and degeneracy maps given by multiplication, actions and the unit. The relative tensor product ([Lur17, Prop. 4.4.2.14]) \(M\otimes_B N\) is defined as the geometric realization of \({\textup{Bar}}(M, B, N)\) in \({}_{A} \mathrm{BMod}_C(\mathcal C)\). If \(\mathcal M\) is a presentably monoidal \(\infty\)-category, this defines a functor in \(\mathrm{Pr}^\mathrm{L}\): \[-\otimes_B - \colon {}_A\mathrm{BMod}_B(\mathcal C) \otimes {}_B\mathrm{Mod}_C(\mathcal C) \rightarrow{}_A\mathrm{BMod}_C(\mathcal C).\] For \(A=B=C\), this induces a presentably monoidal structure on \({}_{A}\mathrm{BMod}_A(\mathcal C)\).