Homotopy Type Theory
H-space (Rev #5, changes)

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Contents

Idea

Classically, H-spaces an are simply types equipped with the structure of a magma (from classical Algebra). They are useful classically in constructing fibrations.H-space is a homotopy type equipped with the structure of a unital magma in the homotopy category (only).

Definition

A H-Space consists of

  • A type AA,
  • A basepoint e:Ae:A
  • A binary operation μ:AAA\mu : A \to A \to A
  • for every a:Aa:A, equalities μ(e,a)=a\mu(e,a)=a and μ(a,e)=a\mu(a,e)=a

Properties

Let AA be a connected H-space. Then for every a:Aa:A, the maps μ(a,),μ(,a):AA\mu(a,-),\mu(-,a):A \to A are equivalences.

See also

synthetic homotopy theory? hopf fibration

On the nlab

Classically, an H-space is a homotopy type equipped with the structure of a unital magma in the homotopy category (only).

References

HoTT book

category: homotopy theory

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