Modules reverse algebra

Observation 19.2.5. There are equivalences of \(\infty \)-operads \[ \rev \colon \Assoc \iso \Assoc , \qquad \rev \colon \oLMod \iso \oRMod \qquadtext { and } \rev \colon \oBMod \iso \oBMod \] given on objects by \(\fa \mapsto \fa \), \(\fa _- \mapsto \fa _+\), \(\fa _+ \mapsto \fa _-\) and \(\fm \mapsto \fm \), and on multimorphisms by reversing the order on the finite set. The map \(\Assoc \to \Comm \) coequalizes \(\rev \) and the identity, so the symmetric monoidal structure on \(C\) canonically identifies the two induced monoidal structures. Consequently, restriction along \(\rev \) defines an endofunctor \(\rev ^*\colon \Alg (C) \to \Alg (C)\). It sends an associative algebra \(A\) to its opposite algebra, denoted \(A^{\rev }\), which has the same underlying object but whose multiplication is given by \[ A \otimes A \iso A \otimes A \xrightarrow {m_A} A, \] where the first map is the swap map. For associative algebras \(A\) and \(B\) we thus get equivalences \[ \LMod _A(C) \iso \RMod _{A^{\rev }}(C) \qquadtext { and } {}_A\BMod _B(C) \iso {}_{B^{\rev }}\BMod _{A^{\rev }}(C). \]