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 × G → P such that
(1) \( {\displaystyle p(xg)=p(x)} for all x in P and g in G.
(2) For each x in P, the map \( {\displaystyle G\to p^{-1}(p(x)),g\mapsto xg} \) 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 \( {\displaystyle P'X} \) be the space of paths of various length in a based space X. Then the fibration \( {\displaystyle p:P'X\to X} \) that sends each path to its end-point is a G-fibration with G the space of loops of various lengths in X.
References
James, I.M. (1995). Handbook of Algebraic Topology. Elsevier. p. 833. ISBN 978-0-08-053298-1.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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