Left tensoring

Definition 19.1.3. Let \(C\) be a monoidal \(\infty \)-category. A left tensoring of an \(\infty \)-category \(D\) over \(C\) consists of an \(\oLMod \)-monoidal \(\infty \)-category, given by a cocartesian fibration of \(\infty \)-operads \(p_D\colon D^{\otimes } \to \oLMod ^{\otimes }\), together with the following identifications:

• The underlying monoidal \(\infty \)-category \(\fa ^*(p_D)\colon \fa ^*(D^{\otimes }) \to \Assoc ^{\otimes }\) is equivalent to \(C\);
• The underlying \(\Triv \)-monoidal \(\infty \)-category \(\fm ^*(p_D)\colon \fm ^*(D^{\otimes }) \to \Triv ^{\otimes }\) corresponds to \(D\) under the equivalence \(\Alg _{\Triv }(\Cat _{\infty }) \simeq \Cat _{\infty }\).

We say that \(D\) is left-tensored over \(C\) if it is equipped with such a left tensoring.