## 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 $$\alpha:\prod_{(a:A)} \prod_{(b:A)} \prod_{(c:A)} \mu(\mu(a, b),c)=\mu(a,\mu(b,c))$$ * A commutator $$\kappa:\prod_{(a:A)} \prod_{(b:A)} \mu(a, b)=\mu(b, a)$$ ### Homomorphisms of commutative $A_3$-spaces ### A __homomorphism of commutative $A_3$-spaces__ between two commutative $A_3$-spaces $A$ and $B$ is a function $\phi:A \to B$ such that * The basepoint is preserved $$\phi(e_A) =_B e_B$$ * The binary operation is preserved $$\prod_{(a:A)} \prod_{(b:A)} \phi(\mu_A(a, b)) =_B \mu_B(\phi(a),\phi(b))$$ (...) ### 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]]