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,^[1] given a topological monoid G, a G-fibration is a fibration p: P→B together with a continuous right monoid action P ×ばつ G → P such that

• (1) ${\displaystyle p(xg)=p(x)}${\displaystyle p(xg)=p(x)} for all x in P and g in G.
• (2) For each x in P, the map ${\displaystyle G\to p^{-1}(p(x)),g\mapsto xg}${\displaystyle G\to p^{-1}(p(x)),g\mapsto xg} 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 ${\displaystyle P'X}${\displaystyle P'X} be the space of paths of various length in a based space X. Then the fibration ${\displaystyle p:P'X\to X}${\displaystyle p:P'X\to X} that sends each path to its end-point is a G-fibration with G the space of loops of various lengths in X.