Lemma 13.3.4. Let \(C\) be an extensive \(\infty \)-category and let \((C,C_L,C_R)\) be an adequate triple such that \(C_L\) and \(C_R\) are closed under finite coproducts.
- (1)
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If the morphisms \(\emptyset \to X\) and \(\nabla \colon X \sqcup X \to X\) lie in \(C_L\) for every \(X \in C\), then the triple is weakly extensive.
- (2)
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If the morphisms \(\emptyset \to X\) and \(\nabla \colon X \sqcup X \to X\) lie in \(C_R\) for every \(X \in C\), then the triple is weakly coextensive.
Consequently, if these morphisms lie in both classes, then the triple is extensive.
Proof. Part (1) of Lemma 13.3.2 shows that the coproduct functor preserves the pullback squares required by adequacy, while part (2) verifies condition (4) of Definition 13.3.3. Together with the closure hypotheses and the assumption on the fold and initial maps, this verifies conditions (1)–(4) of that definition and proves part (1). Interchanging \(C_L\) and \(C_R\) proves part (2), and the final assertion follows by combining the two parts. □
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