Let $F$ be a topological space and $G$ be a topological group which acts on $F$ on the left. A fiber bundle with fiber $F$ and structure group $G$ consists of the following data:

• a topological space $B$ called the base space, a space $E$ called the total space and a continuous surjective map $\pi:E\to B$ called the projection of the bundle,

• an open cover $\{U_{i}\}$ of $B$ along with a collection of continuous maps $\{\phi_{i}:\pi^{{-1}}U_{i}\to F\}$ called local trivializations and

• a collection of continuous maps $\{g_{{ij}}:U_{i}\cap U_{j}\to G\}$ called transition functions

which satisfy the following properties

1. 1.

   the map $\pi^{{-1}}U_{i}\to U_{i}\times F$ given by $e\mapsto(\pi(e),\phi_{i}(e))$ is a homeomorphism for each $i$,

2. 2.

   for all indices $i,j$ and $e\in\pi^{{-1}}(U_{i}\cap U_{j})$, $g_{{ji}}(\pi(e))\cdot\phi_{i}(e)=\phi_{j}(e)$ and

3. 3.

   for all indices $i,j,k$ and $b\in U_{i}\cap U_{j}\cap U_{k}$, $g_{{ij}}(b)g_{{jk}}(b)=g_{{ik}}(b)$.

Readers familiar with Čech cohomology may recognize condition 3), it is often called the cocycle condition. Note, this imples that $g_{{ii}}(b)$ is the identity in $G$ for each $b$, and $g_{{ij}}(b)=g_{{ji}}(b)^{{-1}}$.

If the total space $E$ is homeomorphic to the product $B\times F$ so that the bundle projection is essentially projection onto the first factor, then $\pi:E\to B$ is called a trivial bundle. Some examples of fiber bundles are vector bundles and covering spaces.

There is a notion of morphism of fiber bundles $E,E^{{\prime}}$ over the same base $B$ with the same structure group $G$. Such a morphism is a $G$-equivariant map $\xi:E\to E^{{\prime}}$, making the following diagram commute

Thus we have a category of fiber bundles over a fixed base with fixed structure group.