Connectivity-preserving symmetric monoidal structures

Definition 8.2.1 (Connectivity-preserving symmetric monoidal structure). Let \(C\) be a stably symmetric monoidal \(\infty \)-category, meaning that \(C\) is stable and its tensor product is exact separately in both variables, and equip \(C\) with a t-structure. We say that the symmetric monoidal structure on \(C\) is connectivity-preserving if the following two conditions are satisfied:

(1)

The monoidal unit \(\unit \) of \(C\) is connective;

(2)

For two connective objects \(X,Y \in C_{\geq 0}\), the tensor product \(X \otimes Y\) is again connective.

This condition is often expressed by saying that the t-structure is compatible with the symmetric monoidal structure. If the symmetric monoidal structure is connectivity-preserving, then \(C_{\geq 0}\) inherits a symmetric monoidal structure making the inclusion \(C_{\geq 0} \hookrightarrow C\) into a symmetric monoidal functor; this is proved in Part II in Lemma 14.5.2. As a consequence, the truncation functor \(\tau _{\geq 0}\colon C \to C_{\geq 0}\) is canonically lax symmetric monoidal (see Proposition 14.3.6).