H-space ($m:Y\times Y\to Y$)

A based topological space with a continuous map $m$ whose left and right restrictions at the basepoint are homotopic to the identity. Associativity and inverses up to homotopy are additional conditions, not part of the bare definition.

An H-space whose multiplication is homotopy associative and admits a homotopy inverse. These operations give $[X,Y{]}_{\ast }$ a group structure for every based CW complex $X$. A loop space is the basic example: concatenate paths, use reversal for inversion, and use reparametrization for homotopy associativity. A bare H-space should not automatically be treated as a group object.