G-fibration

In algebraic topology, a G-fibration or principal fibration is a generalization of a principal G-bundle, just as a fibration is a generalization of a fiber bundle. By definition, given a topological monoid G, a G-fibration is a fibration p: P→B together with a continuous right monoid action P × G → P such that

• (1) [math]\displaystyle{ p(x g) = p(x) }[/math] for all x in P and g in G.
• (2) For each x in P, the map [math]\displaystyle{ G \to p^{-1}(p(x)), g \mapsto xg }[/math] is a weak equivalence.

A principal G-bundle is a prototypical example of a G-fibration. Another example is Moore's path space fibration: namely, let [math]\displaystyle{ P'X }[/math] be the space of paths of various length in a based space X. Then the fibration [math]\displaystyle{ p: P'X \to X }[/math] that sends each path to its end-point is a G-fibration with G the space of loops of various lengths in X.