nLab suspensions are H-cogroup objects



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Basic facts




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


Let 𝒞\mathcal{C} be an (∞,1)-category with finite (∞,1)-colimits and with a zero object. Write Σ:X0X0\Sigma \colon X \mapsto 0 \underset{X}{\coprod} 0 for the reduced suspension functor.

Then the pinch map

ΣX0X00XXX00X0X0ΣXΣX \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 ΣX\Sigma X in the homotopy category Ho(𝒞)Ho(\mathcal{C}).

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

Ho(𝒞)(ΣX,):YHo(𝒞)(ΣX,Y)Ho(𝒞)(X,ΩY)π 1Ho(𝒞)(X,Y). 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) \,.


Positive-dimensional spheres in Grpd\infty Grpd

All n-spheres, regarded as their homotopy types in 𝒞=\mathcal{C} = ∞Grpd suspensions for n1n \geq 1: S n+1ΣS nS^{n+1}\simeq \Sigma S^n, hence they carry cogroup structure in the classical homotopy category. Under the ΣΩ\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 17:58:12. See the history of this page for a list of all contributions to it.