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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 $A$,
• A basepoint $e:A$
• A binary operation $\mu : A \to A \to A$
• for every $a:A$, equalities $\mu(e,a)=a$ and $\mu(a,e)=a$

Properties

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

References

HoTT book