Smash product of pointed animae

Definition 2.4.10 (Smash product). Let \(X\) and \(Y\) be pointed animae. Their smash product is the pointed anima \[ X \wedge Y := \cofib (X \vee Y \longrightarrow X \times Y), \] where the map is induced by \((\id _X,0)\colon X\to X\times Y\) and \((0,\id _Y)\colon Y\to X\times Y\). Interchanging the two factors induces a natural swap map \[ \sigma _{X,Y}\colon X\wedge Y \xrightarrow {\cong } Y\wedge X. \]