Associahedra, Multiplihedra and units in $A_{\infty}$ form

Abstract

A higher associativity was introduced by Jim Stasheff in [Sta63] with higher coherence conditions and now becomes one of the most important structures on spaces and algebras. He also claims that the condition on unit can be weakened, using James retractile arguments [Jam60], while the proof given in [Sta63] for the equivalence of two definitions is not very clear for us. We had been puzzled for years, and decided to prove it in a different way by constructing an $A_{m}$-structure. To justify that our construction is natural, we bring our ideas into the theory of an internal precategory which is a weak version of Aguiar's internal category [Agu97]. Using that construction, we show the equivalence of two definitions under the `loop-like' condition. That condition is not necessary to manipulate higher forms using retractile arguments as is performed in [Sta63], but is necessary to construct an $A_{m}$-structure from the given $A_{m}$-form with {\em strict-unit} as is mentioned in Stasheff [Sta70].