nLab suspensions are H-cogroup objects

Idea

Suspension objects are canonically cogroup objects up to homotopy, via their “pinch map”. In particular this is the case for positive dimensional n-spheres.

Statement

Let $\mathcal{C}$ be an (∞,1)-category with finite (∞,1)-colimits and with a zero object. Write $\Sigma \colon X \mapsto 0 \underset{X}{\coprod} 0$ for the reduced suspension functor.

Then the pinch map

$$\Sigma X \simeq 0 \underset{X}{\sqcup} 0 \simeq 0 \underset{X}{\sqcup} X \underset{X}{\sqcup} 0 \longrightarrow 0 \underset{X}{\sqcup} 0 \underset{X}{\sqcup} 0 \simeq \Sigma X \coprod \Sigma X$$

exhibits a cogroup structure on the image of $\Sigma X$ in the homotopy category $Ho(\mathcal{C})$.

This is equivalently the group-structure of the first (fundamental) homotopy group of the values of the functor co-represented by $\Sigma X$:

$$Ho(\mathcal{C})(\Sigma X, -) \;\colon\; Y \mapsto Ho(\mathcal{C})(\Sigma X, Y) \simeq Ho(\mathcal{C})(X, \Omega Y) \simeq \pi_1 Ho(\mathcal{C})(X, Y) \,.$$

Examples

Positive-dimensional spheres in $\infty Grpd$

All n-spheres, regarded as their homotopy types in $\mathcal{C} =$∞Grpd suspensions for $n \geq 1$: $S^{n+1}\simeq \Sigma S^n$, hence they carry cogroup structure in the classical homotopy category. Under the $\Sigma \dashv \Omega$-adjunction, this cogroup structure turns into the group structure on all homotopy groups in positive degree.