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suspensions are H-cogroup objects

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Group Theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

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Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

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 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) \,.

Examples

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 12:58:12. See the history of this page for a list of all contributions to it.