Weakly extensive adequate triple

Definition 13.3.3. An adequate triple \((C,C_L,C_R)\) is called weakly extensive if the following conditions are satisfied:

(1)

The \(\infty \)-category \(C\) admits finite coproducts;

(2)

The coproduct functor \(C \times C \to C\) is a morphism of adequate triples (i.e., morphisms in \(C_L\) and \(C_R\) are closed under coproducts, and coproducts of pullback squares of morphisms in \(C_L\) along morphisms in \(C_R\) are again pullback squares);

(3)

For every object \(X \in C\), the morphisms \(\emptyset \to X\) and \(\nabla \colon X \sqcup X \to X\) are in \(C_L\);

(4)

For every morphism \(r\colon X \to Y\) in \(C_R\), the two squares in Equation 13.2 are pullback squares.

We say that the triple is weakly coextensive if \((C,C_R,C_L)\) is weakly extensive, and we say it is extensive if it is both weakly extensive and weakly coextensive.