My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets.
Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, on the right. The action divides $X$ into maximal orbits, which I think of as the equivalence classes with respect to the equivalence relation $\sim$ generated by the basic equivalences of the form $x \sim y \cdot m$ for $x, y\in x$ and $m\in M$. There is then a quotient space and quotient map $q:X\to X // M$, which the quotient of $X$ by the equivalence relation.
Roughly speaking, a principal fibration is a fibration sequence of the form $M \to X\to X//M$; this "definition" is good enough for the analogy I want to make. Here $M\to X$ sends $m$ to $*\cdot m$, where $*\in X$ is the basepoint.
Our favorite family of examples of fibrations (I say!) is the Moore path-loop fibrations, $\Omega_M (X) \to \mathcal{P}_M(X) \to X$. Here the fiber $\Omega_M(X)$ is a topological monoid acting on the total space $\mathcal{P}_M(X)$, but the quotient $\mathcal{P}_M(X)//\Omega_M (X)$ is not equal to $X$. However, there is a factorization $$ \mathcal{P}_M(X) \to \mathcal{P}_M(X)//\Omega_M (X) \to X . $$
QUESTION: Here is a concept: fibration sequences $M \to X\to B$ in which
- $M$ is a topological monoid acting on $X$
- there is a factorization $X\to X//M \to B$.
Is there an existing term for this? Or perhaps it is very close to some other concept that has a name? I don't want to go renaming the wheel, so I'd appreciate knowing what these are called, and especially references to places where the terminology is introduced, or at least used.
