[[!redirects invertible A3-space]] ## Idea ## The invertible version of the [[A3-space]] up to homotopy, without any higher coherences for inverses. ## Definition ## A __grouplike $A_3$-space__ or __grouplike $A_3$-algebra in homotopy types__ or __H-group__ consists of * A type $A$, * A basepoint $e:A$ * A binary operation $\mu : A \to A \to A$ * A unary operation $\iota: A \to A$ * A left unitor $$\lambda_u:\prod_{(a:A)} \mu(e,a)=a$$ * A right unitor $$\rho_u:\prod_{(a:A)} \mu(a,e)=a$$ * An asssociator $$\alpha:\prod_{(a:A)} \prod_{(b:A)} \prod_{(c:A)} \mu(\mu(a, b),c)=\mu(a,\mu(b,c))$$ * A left invertor $$l:\prod_{(a:A)} \mu(\iota(a), a)=e$$ * A right invertor $$r:\prod_{(a:A)} \mu(a,\iota(a))=e$$ One could also speak of grouplike $A_3$-spaces where the existence of left and right inverses are mere property rather than structure, which is a grouplike $A_3$-space as defined above with additional structure specifying that the types $\prod_{(a:A)} \mu(\iota(a), a)=e$ and $\prod_{(a:A)} \mu(a,\iota(a))=e$ are [[contractible]]: $$c_l: \sum_{l:\prod_{(a:A)} \mu(\iota(a), a)=e} \prod_{b:\prod_{(a:A)} \mu(\iota(a), a)=e} l = b$$ $$c_r: \sum_{r:\prod_{(a:A)} \mu(a,\iota(a))=e} \prod_{b:\prod_{(a:A)} \mu(a,\iota(a))=e} r = b$$ or equivalently, $$c_l:\prod_{(a:A)} \sum_{(l:\mu(\iota(a), a)=e)} \prod_{(b:\mu(\iota(a), a)=e)} l = b$$ $$c_r:\prod_{(a:A)} \sum_{(r:\mu(a,\iota(a))=e)} \prod_{(r:\mu(a,\iota(a))=e)} r = b$$ ## Examples ## * The [[integers]] are an grouplike $A_3$-space. * Every [[loop space]] is naturally an grouplike $A_3$-space with [[path]] concatenation as the operation. In fact every [[loop space]] is a $\infty$-group. * A [[group]] is a 0-truncated grouplike $A_3$-space. ## See also ## * [[Higher algebra]] * [[A3-space]] * [[commutative grouplike A3-space]] * [[group]]