Cocycle condition for transition functions

Let $\{(U_i,\Phi_i)\}$ be a bundle atlas for a smooth fiber bundle with typical fiber $F$, and let $t_{ij}:U_{ij}\to \mathrm{Diff}(F)$ be the associated transition functions . They satisfy the following identities:

1. Identity on the diagonal: $t_{ii}(x)=\mathrm{id}_F$ for all $x\in U_i$.
2. Inverse on overlaps: on $U_{ij}$, one has $t_{ji}(x)=t_{ij}(x)^{-1}$.
3. Cocycle condition on triple overlaps: on $U_{ijk}=U_i\cap U_j\cap U_k$, $t_{ij}(x)\circ t_{jk}(x)=t_{ik}(x)\qquad \text{for all }x\in U_{ijk}.$

These conditions are exactly the statement that the changes of trivialization compose consistently, so that the local products $U_i\times F$ glue to a well-defined smooth fiber bundle .

Examples

1. Trivial bundle: all $t_{ij}$ are the identity, so the cocycle condition holds tautologically.
2. Möbius line bundle: with $t_{12}\equiv-1$, the cocycle identity on a triple overlap reduces to $(-1)\cdot(-1)=1$.
3. Tangent bundle: on triple overlaps of coordinate charts, the cocycle condition is the chain rule for Jacobians of coordinate changes.