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Idea
The invertible version of the A3-space up to homotopy, without any higher coherences for inverses.
Definition
A grouplike -space or grouplike -algebra in homotopy types or H-group consists of
A type ,
A basepoint
A binary operation
A unary operation
A left unitor
A right unitor
An asssociator
A left invertor
A right invertor
One could also speak of grouplike -spaces where the existence of left and right inverses are mere property rather than structure, which is a grouplike -space as defined above with additional structure specifying that the types and are contractible: