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A∞-operad
In the theory of operads in algebra and algebraic topology, an A∞-operad is a parameter space for a multiplication map that is homotopy coherently associative. (An operad that describes a multiplication that is both homotopy coherently associative and homotopy coherently commutative is called an E∞-operad.)
== Definition ==
In the (usual) setting of operads with an action of the symmetric group on topological spaces, an operad ''A'' is said to be an ''A''∞-operad if all of its spaces ''A''(''n'') are Σ''n''-equivariantly homotopy equivalent to the discrete spaces Σ''n'' (the symmetric group) with its multiplication action (where ''n'' ∈ N). In the setting of non-Σ operads (also termed nonsymmetric operads, operads without permutation), an operad ''A'' is ''A''∞if all of its spaces ''A''(''n'') are contractible. In other categories than topological spaces, the notions of ''homotopy'' and ''contractibility'' have to be replaced by suitable analogs, such as homology equivalences in the category of chain complexes.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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