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clarified pointed spaces
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Jeff Strom
Jeff Strom
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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.

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 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.

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.

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.

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Jeff Strom
Jeff Strom
  • 12.5k
  • 3
  • 48
  • 76

Terminology for an kind-of principal fibration

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 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.

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.