The suspension as a cogroup

Remark 2.4.21 (The suspension as a cogroup). For every pointed anima \(X\), the suspension \(\Sigma X\) is a cogroup object in the homotopy category \(\Ho (\An _*)\): there are pointed maps \[ \Sigma X \to *, \qquad \mathrm {pinch}\colon \Sigma X \to \Sigma X \vee \Sigma X, \qquad i\colon \Sigma X \to \Sigma X, \] satisfying counitality, coassociativity, and the coinverse axiom in \(\Ho (\An _*)\). To see this, use the homotopy hypothesis to choose a topological space \(Y\) whose underlying anima is isomorphic to the underlying anima of \(X\), and choose a point of \(Y\) in the component corresponding to the basepoint of \(X\). Applying the factorization from Theorem 2.3.8 to this basepoint map, we may further assume that \(Y\) is a cell complex whose basepoint is a \(0\)-cell. Then \(\Sigma X \simeq \Pi _{\infty }(\Sigma Y)\) by Proposition 2.4.12, Remark 2.4.13, and the maps above are the images under \(\Pi _{\infty }\) of the classical pinch comultiplication on the topological suspension \(\Sigma Y\). The classical pointed homotopies expressing the cogroup axioms are carried by \(\Pi _{\infty }\) to homotopies in \(\An _*\). The double suspension \(\Sigma ^2 X\) carries comultiplications in two a priori distinct ways, but by the Eckmann–Hilton argument they agree and the resulting comultiplication is cocommutative.

Consequently, for every pointed anima \(Y\) the representable functor \([-,Y]_*\) sends wedges to products, so the cogroup structure on \(\Sigma X\) makes \([\Sigma X, Y]_*\) into a group; this group is abelian when the source is a double suspension, as for \([\Sigma ^2 X,Y]_*\).