Definition 5.90.
A functor \(F\colon C \to T\) is called excisive if it sends pushout squares in \(C\) to pullback squares in \(T\).
For \(n \geq 0\), we say \(F\) is \(n\)-excisive if it sends strongly cocartesian \((n+1)\)-cubes in \(C\) to cartesian \((n+1)\)-cubes in \(T\). Recall that an \((n+1)\)-cube \(X\colon [1]^{n+1} \to C\) is called strongly cocartesian if it is left Kan extended from its restriction to the \(n+1\) edges \(\{0\}^{k} \times [1] \times \{0\}^{n-k}\), and it is cartesian if it is a limit cone.
We denote by \(\Exc^n(C,T) \subseteq \Fun(C,T)\) the full subcategory of \(n\)-excisive functors.