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Recently I was learning about $A_\infty$-spaces. The slogan what they are is: "Homotopy associative H-spaces with higher coherences for the associativity".

When I first heard the slogan, I thought the definition would be something like this: You have a homotopy associative H-space. 'Higher coherences' then means that two different associators are homotopic to each other, choices of these homotopies are then again homotopic to each other, etc. (Or phrased a bit more concretely, a suitable space of associators is contractible.)

But the actual definition is completely different: For the actual definition we fix(!) an associator and then the higher coherences mean something completely different than what I thought (they are related to applying the associator to expressions in several variables).

Is there any relation between the actual definition of an $A_\infty$-space and what I tought the definition should be after hearing the slogan?

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    $\begingroup$ Good question but it would help if you included a link to the definition of $A_\infty$-space that you are referring to. Different sources may have different definitions (i.e. the details matter and may be different in different sources.). $\endgroup$
    – Tim Porter
    Commented Aug 14, 2023 at 9:04
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    $\begingroup$ The original one by Stasheff. The nLab has the reference: ncatlab.org/nlab/show/A-infinity-space $\endgroup$
    – AlexE
    Commented Aug 14, 2023 at 9:12
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    $\begingroup$ A choice of an associator is the same as an A_3 structure. Your definition is asking for an essentially unique A_3 structure, so it is both stronger and weaker. $\endgroup$
    – Tilman
    Commented Aug 14, 2023 at 9:25
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    $\begingroup$ If $G$ is actually a topological group then there is a trivial associator but the full space of all possible associators is $\Omega\text{Map}(G^3,G)$, which contains $\Omega G$ as a retract. Thus, the space of associators cannot be contractible unless $G$ is weakly equivalent to a discrete space. $\endgroup$
    – Neil Strickland
    Commented Aug 14, 2023 at 11:19
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    $\begingroup$ You’re kind of thinking of a case where there’s an essentially unique A-infinity space structure with a given binary product. That’s not as common. $\endgroup$
    – Fernando Muro
    Commented Aug 14, 2023 at 11:34

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