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Informally, an $A_\infty$-space is a monoid whose laws are only satisfied up to homotopy.

Let’s define now what I will call a "homotopic monoid" to be a space $M$ together with a point $e\in{}M$ and a multiplication $m:M\times {}M\to{}M$ with the monoid laws satisfied "up to a path". More precisely, if $F$ is the fibration over $M^3$ such that the fiber over $(x,y,z)$ is the space of paths (in $M$) going from $m(m(x,y),z)$ to $m(x,m(y,z))$, then I want a section of this fibration (the section is part of the structure of homotopic monoid). And of course I want the same thing for the laws with $e$.
At least from the point of view of homotopy type theory, this is a very natural homotopy-theoretic generalization of the notion of monoid (we just replaced equality by existence of a path).

My questions are:

  • Do those "homotopic monoids" have already been studied somewhere?
  • What is the relationship between an $A_\infty$-space structure and a homotopic monoid structure? I think it is easy to prove that every $A_\infty$ space has a homotopic monoid structure, but I’m not sure of the converse.
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For the relation between homotopy monoids and A∞-spaces one place to look at is also chapter 3 (in particular 3.4-3.5) of Tom Leinster's "Homotopy algebras for operads". In particular it is brought up that conceptually an A∞-space is a homotopy semigroup in Top∗ (spaces with basepoint) rather than a homotopy monoid in Top. – Gjergji Zaimi Aug 17 '11 at 16:33
up vote 6 down vote accepted

Ignoring the unit conditions for a moment, what you are defining is called an $A_3$-structure on the space $M$. It is the third stage of a whole iteratively defined chain of structures on a space (the $n$-th being called an $A_n$-structure). These were studied by Stasheff in the 1960s.

An equivalent way to write your condition is to specify a homotopy $I \times M^3 \to M$ from $(({-},{-}),{-})$ to $({-},({-},{-}))$ (here I am using $({-},{-})$ to designate your multiplication $m$).

In this context, the unit interval functions as the second space in the Stasheff associahedron.

Regarding units, Stasheff did not deal with unit conditions up to homotopy, since, under mild hypotheses, one can assume the basepoint is a strict unit for the multiplication. The full unit conditions appear later in the book by Boardman and Vogt.

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It seems to me that rewriting this in terms of ordinary homotopies, you get something very like what is described in Segal's old paper (with categories and cohomology theories in the title I think). It occurs also in work by Vogt about 1973 and would then seem to be a truncated version of a homotopy coherent associativity law. My one doubt is that the existence of the path should not be enough to get a good analogue of a monoid as how would you get the quadruple associativity law... or would you not have one (you would not in general get one path but two ... hence you need to have a homotopy between them and so on.) This may not answer your question but I suggest looking at the work on homotopy coherence (e.g. [[homotopy coherent diagram]] as the combinatorial considerations in that area would seem relevant to your query.)

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