In one of the answers to this question, $E_1$ spaces are mentioned. In that context, they are described as a homotopically invariant version of topological group. To me, this would imply that we have a topological group, where the multiplication only satisfies associativity, identity and invertibility up to homotopy.
The wiki page for $E_{\infty}$ operads also contains a definition for $E_1$ operad. My first question: Are these the same as what are described as $E_1$ spaces in the aforementioned link?
Secondly, the above wiki page goes on to say that in an $E_1$ space we have a multiplication which is homotopy coherently associative. I would like to check that my understanding of this is correct.
Let $X$ be an $E_1$ space, and let $x,y,z \in X$. Given that we have homotopy associativity, we know that there exists a homotopy $F:X \times I \rightarrow X, ( x \circ y ) \circ z \simeq x \circ (y \circ z)$.
Now suppose instead that we have $w,x,y,z \in X$. Homotopy associativity tells us that
$w \circ ( x \circ (y \circ z)) \simeq w \circ (( x \circ y ) \circ z) \simeq (w \circ ( x \circ y )) \circ z \simeq ((w \circ x) \circ y ) \circ z \simeq (w \circ x) \circ (y \circ z) $.
If I could, I would now draw a pentagon giving a graphical depiction of this situation.
Is the homotopy coherence telling us that the homotopies available between these different things are homotopic? That is, two different ways around the pentagon, between two particular composition orders, are homotopic?
