nLab fibred natural transformation




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


Given fibrations pp and pp' and fibred functors F,G:ppF,G:p \to p', a fibred natural transformation α=(α 1,α 0):FG\alpha=(\alpha_1,\alpha_0):F \Rightarrow G is a pair of 2-cells as follows: This means α 0:F 0G 0\alpha_0:F_0 \Rightarrow G_0, α 1:F 1G 1\alpha_1:F_1 \Rightarrow G_1 and p(α 1)=α 0pp'(\alpha_1) = \alpha_0 p.

When looking at fibrations over a fixed base \mathcal{B}, then F 0=G 0=1 F_0=G_0=1_{\mathcal{B}}, and also α 0\alpha_0 is the identity natural transformation. In that case, α\alpha reduces to a natural transformation between the functor F 1F_1 and G 1G_1 whose components are vertical, i.e. stay in the fibers.

See also

Last revised on December 12, 2023 at 00:09:43. See the history of this page for a list of all contributions to it.