G-fibration

{{Short description|Concept in algebraic topology}}

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,{{cite book|last=James|first=I.M.|title=Handbook of Algebraic Topology|url=https://books.google.com/books?id=xoM5DxQZihQC&pg=PA833|year=1995|publisher=Elsevier|isbn=978-0-08-053298-1|page=833}} 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) $p(x\; g)\; =\; p(x)$ for all x in P and g in G.
• (2) For each x in P, the map $G\; \backslash to\; p^\{-1\}(p(x)),\; g\; \backslash mapsto\; xg$ 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 $P\text{'}X$ be the space of paths of various length in a based space X. Then the fibration $p:\; P\text{'}X\; \backslash to\; X$ that sends each path to its end-point is a G-fibration with G the space of loops of various lengths in X.