Homotopy Type Theory A3-space > history (Rev #2, changes)

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Definition

An $A_3$-space or H-monoid consists of

• A type $A$,
• A basepoint $e:A$
• A binary operation $\mu : A \to A \to A$
• for A every left unitor$a:A$
  $$\lambda:\prod_{(a:A)} \mu(e,a)=a$$
  , equalities $\mu(e,a)=a$ and $\mu(a,e)=a$
• for A every right unitor$a:A$
  $$\rho:\prod_{(a:A)} \mu(a,e)=a$$
  , $b:A$, $c:A$, an equality $\mu(\mu(a, b),c)=\mu(a,\mu(b,c))$
• An asssociator
  $$\alpha:\prod_{(a:A)} \prod_{(b:A)} \prod_{(c:A)} \mu(\mu(a, b),c)=\mu(a,\mu(b,c))$$

Examples

• The integers are an $A_3$-space.

• Every loop space is naturally an $A_3$-space with path concatenation as the operation. In fact every loop space is a group.

• The type of endofunctions $A \to A$ has the structure of an $A_3$-space, with basepoint $id_A$, operation function composition.

• A monoid is a 0-truncated $A_3$-space.

See also

• Synthetic homotopy theory
• H-space
• commutative A3-space
• monoid

On the nlab

Classically, an A3-space is a homotopy type equipped with the structure of a monoid in the homotopy category (only).