Editing Associated bundle (section)

===Fiber bundle associated with a principal bundle===

Let <math>\pi \colon P \to X</math> be a [[principal bundle|principal ''G''-bundle]] and let <math>\rho \colon G \to \text{Homeo}(F)</math> be a continuous [[Group action (mathematics)|left action]] of <math>G</math> on a space <math>F</math> (in the smooth category, we should have a smooth action on a smooth manifold). Without loss of generality, we can take this action to be effective.

Define a right action of <math>G</math> on <math>P \times F</math> via<ref>Husemoller, Dale (1994), p. 45.</ref><ref>Sharpe, R. W. (1997), p. 37.</ref>
:<math>(p,f)\cdot g = (p\cdot g, \rho(g^{-1})f)\, .</math>
We then [[Quotient space (topology)|identify]] by this action to obtain the space <math>E = P \times_\rho F = (P \times F) / G</math>. Denote the equivalence class of <math>(p, f)</math> by <math>[p, f]</math>. Note that
:<math>[p\cdot g,f] = [p,\rho(g)f] \mbox{ for all } g\in G.</math>
Define a projection map <math>\pi_\rho \colon E \to X</math> by <math>\pi_\rho([p, f]) = \pi(p)</math>. Note that this is [[well-defined]].

Then <math>\pi_\rho \colon E \to X</math> is a fiber bundle with fiber <math>F</math> and structure group <math>G</math>. The transition functions are given by <math>\rho(t_{ij})</math> where <math>t_{ij}</math> are the transition functions of the principal bundle <math>P</math>.

This construction can also be seen [[Category theory|categorically]].
More precisely, there are two continuous maps <math>P \times G \times F \to P \times F</math>, given by acting with <math>G</math> on the right on <math>P</math> and on the left on <math>F</math>.
The associated vector bundle <math>P \times_\rho F</math> is then the [[coequalizer]] of these maps.