I am new to simplicial methods, and have some naive questions. As such, the questions may be malformed and the terminology might not even be right. I assume these are answered in some reference and would appreciate one.
1. Question.
Suppose you are given a sequence $S_\bullet$ of topological spaces, equipped with maps $d^i_j$ that do not necessarily satisfy the requisite commutativity axioms to form a semisimplicial space on the nose, but nevertheless are homotopy commutative in such a way that the homotopies satisfy higher coherences.
I want to form the fat geometric realization of this object. I believe I can still take the disjoint union of the $S_n \times \Delta^n$ and quotient by $(d^i_j s,t) \sim (s,\partial^i_j t)$, irrespective of the fact the normal identities don't hold, but suspect the resulting object is—in some sense that would also need to be made precise—no good.
But there are these homotopies. Is there a canonical way to use them construct a well-defined object up to homotopy (or better, homeomorphism) out of this object in an analogous way to geometric realization?
In case I'm taking away too much structure in asking this question, I'll post a motivating example here. I will also post a second, related question.
2. Example.
Suppose $G$ is an $H$-space, associative up to higher homotopies (an $A_\infty$-something?) "acting" on a space $X$ in such a way that $g(g'x) = (gg')x$ can't be assumed to hold on the nose, but holds up to higher homotopies. One has maps between the $M^n \times X$ obtained by performing multiplication of some two elements in some order, and these multiplications seem to resemble face maps and satisfy the appropriate axioms up to homotopy.
An answer to the second question is indeed known, and is discussed for example in a paper by Eduardo Hoefel, Muriel Livernet, and Jim Stasheff as Definition 2.9:
http://arxiv.org/abs/1312.7155.
Briefly, suppose $M$ is an $A_\infty$-space and has an $A_\infty$-action on $X$. Then the coherences induce maps
$$K_{n+1} \times M^n \to M$$ and $$K_{n+1} \times M^{n-1} \times X \to X,$$
where $K_n$ is an associahedron, and these maps restrict amongst one another in a fairly natural way. Then one can define a homotopy orbit space or one-sided bar construction $B(*,M,X)$ as the quotient of the union over $n$ of the domains
$$K_{n+1} \times M^{n-1} \times X$$
under the expected identifications. In the event $M$ is a topological monoid and $M \times X \to X$ is an action, so everything is actually associative, this returns, up to homotopy, the classical homotopy quotient $EM \times_M X$.