## 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 $$\alpha:\prod_{(a:A)} \prod_{(b:A)} \prod_{(c:A)} \mu(\mu(a, b),c)=\mu(a,\mu(b,c))$$ ### Homomorphisms of $A_3$-spaces ### A __homomorphism of $A_3$-spaces__ between two $A_3$-spaces $A$ and $B$ consists of * A function $\phi:A \to B$ such that * The basepoint is preserved $$\phi(e_A) = e_B$$ * The binary operation is preserved $$\prod_{(a:A)} \prod_{(b:A)} \phi(\mu_A(a, b)) = \mu_B(\phi(a),\phi(b))$$ * A function $$\phi_\lambda:\left(\prod_{(a:A)} \mu(e_A,a)=a\right) \to \left(\prod_{(b:B)} \mu(e_B,b)=b\right)$$ such that the left unitor is preserved: $$\phi_\lambda(\lambda_A) = \lambda_B$$ * A function $$\phi_\rho:\left(\prod_{(a:A)} \mu(a, e_A)=a\right) \to \left(\prod_{(b:B)} \mu(b, e_B)=b\right)$$ such that the right unitor is preserved: $$\phi_\rho(\rho_A) = \rho_B$$ * A function $$\phi_\alpha:\left(\prod_{(a_1:A)} \prod_{(a_2:A)} \prod_{(a_3:A)} \mu(\mu(a_1, a_2),a_3)=\mu(a_1,\mu(a_2,a_3))\right) \to \left(\prod_{(b_1:B)} \prod_{(b_2:B)} \prod_{(b_3:B)} \mu(\mu(b_1, b_2),b_3)=\mu(b_1,\mu(b_2,b_3))\right)$$ such that the associator is preserved: $$\phi_\alpha(\alpha_A) = \alpha_B$$ ## 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 ## * [[Higher algebra]] * [[H-space]] * [[commutative A3-space]] * [[grouplike A3-space]] * [[monoid]] * [[ring]] * [[homotopy precategory]] ### 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).