Toward Homotopy-Algebra Model using A-infinity Algebra

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There are two different definitions of A∞structures on a space, both are introduced by J. D. Stasheff. One requires a strict unit and the other requires only a homotopy unit. We give a proof for that an A∞space with homotopy unit is homotopy equivalent to an A∞space with strict unit. We also give a definition of a two-sided Borel construction for A∞actions, which must imply that no cohomological difference shall appear after deforming a true action as A∞action.