t-flat objects

Definition 6.5.1. Let \(C\), \(D\), and \(E\) be stable \(\infty \)-categories equipped with t-structures. Let \(- \otimes - \colon C \times D \to E\) be a functor that is exact in each variable separately, and assume it restricts to \(C_{\geq 0} \times D_{\geq 0} \to E_{\geq 0}\), i.e. the tensor product of connective objects is connective.

A connective object \(N \in D_{\geq 0}\) is called t-flat (with respect to \(\otimes \)) if the functor \((-) \otimes N \colon C \to E\) is t-exact. Similarly, \(M \in C_{\geq 0}\) is t-flat if the functor \(M \otimes (-) \colon D \to E\) is t-exact.