Lemma 2.4.16 (Basic properties of smash products). For a pointed anima \(X\) and animae \(Y,Z\), write \(Y_+:=Y\sqcup *\) and \(Z_+:=Z\sqcup *\) for the result of adjoining a disjoint basepoint. There are natural isomorphisms \[ S^0 \wedge X \cong X, \qquad Y_+ \wedge Z_+ \cong (Y \times Z)_+, \qquad S^1 \wedge X \cong \Sigma X. \] Moreover, the functor \((-) \wedge X\colon \An _*\to \An _*\) preserves pushouts, and there are natural isomorphisms \(S^m\wedge S^n\cong S^{m+n}\) for all \(m,n\geq 0\).
Proof. The required calculations from the pushout descriptions of wedges and cofibers are left to Chapterexercise 2.9. □
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