nLab fibred natural transformation

Idea

Fibred natural transformations are natural transformations between fibred functors. They are 2-cells in the 2-category $\mathbf{Fib}$.

Definition

Given fibrations $p$ and $p'$ and fibred functors $F,G:p \to p'$, a fibred natural transformation $\alpha=(\alpha_1,\alpha_0):F \Rightarrow G$ is a pair of 2-cells as follows: This means $\alpha_0:F_0 \Rightarrow G_0$, $\alpha_1:F_1 \Rightarrow G_1$ and $p'(\alpha_1) = \alpha_0 p$.

When looking at fibrations over a fixed base $\mathcal{B}$, then $F_0=G_0=1_{\mathcal{B}}$, and also $\alpha_0$ is the identity natural transformation. In that case, $\alpha$ reduces to a natural transformation between the functor $F_1$ and $G_1$ whose components are vertical, i.e. stay in the fibers.

See also

• fibred category, fibred functor

• natural transformation