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 unitalmagma 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.