Group structures from the wedge axiom

Remark 3.4.2. Suppose that \(F\) satisfies the wedge axiom. The cogroup structure on \(\Sigma X\) from Remark 2.4.21 makes \(F(\Sigma X)\) a group: the pinch map induces its multiplication, while the collapse and reflection maps induce its unit and inverses. The two cogroup structures on \(\Sigma ^2X\) agree by the Eckmann–Hilton argument, so \(F(\Sigma ^2X)\) is an abelian group. A natural transformation between two functors satisfying the wedge axiom preserves these group structures.