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.



Homotopic monoids and $A_\infty$ spaces
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.


