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suspensions are H-cogroup objects
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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.

Last revised on March 7, 2018 at 12:58:12.
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