In algebraic topology, a ''G''-fibration or principal fibration is a generalization of a principal ''G''-bundle, just as a

fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all map ...

is a generalization of a

fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...

. By definition, given a topological monoid ''G'', a ''G''-fibration is a fibration ''p'': ''P''→''B'' together with a continuous right

monoid action In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup (usi ...

''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 \to p^(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

P'X

be the space of paths of various length in a based space ''X''. Then the fibration

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

Algebraic topology Differential geometry Fiber bundles {{topology-stub