Sometimes we can equip a type with a certain structure, called a H-space, allowing us to derive some nice properties about the type or even construct fibrations

Definition

A H-Space consists of

A type $A$,

A basepoint $e:A$

A binary operation $\mu : A \to A \to A$

A left unitor

$\lambda:\prod_{(a:A)} \mu(e,a)=a$

A right unitor

$\rho:\prod_{(a:A)} \mu(a,e)=a$

Properties

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

Examples

There is a H-space structure on the circle. See Lemma 8.5.8 of the HoTT book.

Proof. We define $\mu : S^1 \to S^1 \to S^1$ by circle induction:

where $h : \prod_{x : S^1} x = x$ is defined in Lemma 6.4.2 of the HoTT book. We now need to show that $\mu(x,e)=\mu(e,x)=x$ for every $x : S^1$. Showing $\mu(e,x)=x$ is quite simple, the other way requires some more manipulation. Both of which are done in the book.

The type of maps?$A \to A$ has the structure of a H-space, with basepoint $id_A$, operation function composition.

An A3-space$A$ is an H-space with an equality $\mu(\mu(a, b),c)=\mu(a,\mu(b,c))$ for every $a:A$, $b:A$, $c:A$ representing associativity up to homotopy.