## Idea ## Sometimes we can equip a [[type]] with a certain structure, called an $A_3$-algebra structure, allowing us to derive some nice properties about the type and 0-truncate it to form [[monoid|monoids]]. ## Definition ## An __$A_3$-space__ or __$A_3$-algebra in homotopy types__ or __H-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 $$\alpha:\prod_{(a:A)} \prod_{(b:A)} \prod_{(c:A)} \mu(\mu(a, b),c)=\mu(a,\mu(b,c))$$ ## 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 [[nLab:A-n space|A3-space]] is a [[nLab:homotopy type]] equipped with the structure of a [[nLab:monoid]] in the [[nLab:homotopy category]] (only).