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An -space or H-monoid consists of
The integers are an -space.
Every loop space is naturally an -space with path concatenation as the operation. In fact every loop space is a group.
The type of endofunctions has the structure of an -space, with basepoint , operation function composition.
A monoid is a 0-truncated -space.
Classically, an A3-space is a homotopy type equipped with the structure of a monoid in the homotopy category (only).