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

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. 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. 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 Cech 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 $E,E'$ over the same base $B$ with the same structure group $G$ Such a morphism is a $G$ equivariant map $\xi:E\to E'$ making the following diagram commute

$$\xymatrix{E\ar[rr]^\xi\ar[dr]_\pi& &E'\ar[dl]^{\pi'}\\ &B&}.$$

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