## Idea ## The commutative version of the [[A3-space]] up to homotopy, without any higher commutative coherences. ## Definition ## A __commutative $A_3$-space__ or __commutative $A_3$-algebra in homotopy types__ or __commutative 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 $$\alpha:\prod_{(a_1:A)} \prod_{(a_2:A)} \prod_{(a_3:A)} \mu(\mu(a_1, a_2),a_3)=\mu(a_1,\mu(a_2,a_3))$$ * A commutator $$\kappa:\prod_{(a_1:A)} \prod_{(a_2:A)} \mu(a_1, a_2)=\mu(a_2, a_1)$$ One could also speak of commutative $A_3$-spaces where commutativity is mere property rather than structure, which is a commutative $A_3$-space as defined above with additional structure specifying that the type $\prod_{(a_1:A)} \prod_{(a_2:A)} \mu(a_1, a_2)=\mu(a_2, a_1)$ is [[contractible type|contractible]]: $$c_\kappa: \prod_{b:\prod_{(a_1:A)} \prod_{(a_2:A)} \mu(a_1, a_2)=\mu(a_2, a_1)} \kappa = b$$ ### Homomorphisms of commutative $A_3$-spaces ### A __homomorphism of commutative $A_3$-spaces__ between two commutative $A_3$-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: $$\phi_\alpha(\alpha_A) = \alpha_B$$ * A function $$\phi_\kappa:\left(\prod_{(a_1:A)} \prod_{(a_2:A)} \mu(a_1, a_2)=\mu(a_2, a_1)\right) \to \left(\prod_{(b_1:B)} \prod_{(b_2:B)} \mu(b_1, b_2)=\mu(b_2, b_1)\right)$$ such that the commutator is preserved: $$\phi_\kappa(\kappa_A) = \kappa_B$$ ### Tensor product of commutative $A_3$-spaces ### (...) ## Examples ## * The [[integers]] are a commutative $A_3$-space. * A [[commutative monoid]] is a 0-truncated commutative $A_3$-space. * A [[H-rig]] is a $A_3$-algebra in commutative $A_3$-spaces. ## See also ## * [[Synthetic homotopy theory]] * [[A3-space]] * [[commutative monoid]] * [[commutative grouoplike A3-space]]