What is "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) p(xg) = p(x) for all x in P and g in G.
• (2) For each x in P, the map G → p(p(x)), g ↦ 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 PʹX be the space of paths of various length in a based space X. Then the fibration p : PʹX → X that sends each path to its end-point is a G-fibration with G the space of loops of various lengths in X.