Homotopy Type Theory monoid > history (Rev #1)

Definition

A monoid 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
  $$\rho:\prod_{(a:A)} \mu(a,e)=a$$
• An asssociator
  $$\alpha:\prod_{(a:A)} \prod_{(b:A)} \prod_{(c:A)} \mu(\mu(a, b),c)=\mu(a,\mu(b,c))$$
• A 0-truncator
  $$\tau_0: \prod_{(a:A)} \prod_{(b:A)} \prod_{(c:a=b)} \prod_{(d:a=b)} \sum_{(x:c=d)} \prod_{(y:c=d)} x=y$$

Examples

• The integers are a monoid.

• Every loop space is naturally a monoid with path concatenation as the operation. In fact every loop space is a group.

• Given a set $A$, the type of endofunctions $A \to A$ has the structure of an monoid, with basepoint $id_A$, operation function composition.

See also

• Synthetic homotopy theory
• A3-space
• commutative monoid