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.