nLab geometric transformation

Idea

A geometric transformation is a morphism between geometric morphisms between toposes: a 2-morphism in the 2-category Topos.

Definition

For

$$f = (f^* \dashv f_*) : \mathcal{E} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathcal{F}$$

and

$$g = (g^* \dashv g_*) : \mathcal{E} \stackrel{\overset{g^*}{\leftarrow}}{\underset{g_*}{\to}} \mathcal{F}$$

two geometric morphisms, a geometric transformation

$$\eta : f \Rightarrow g$$

is a natural transformation between the inverse image functors

$$f^* \Rightarrow g^* \,.$$

By mate-calculus, these are in bijection to natural transformations of the direct image functors

$$g_* \Rightarrow f_* \,.$$

References

Section A4.1 of

• Peter Johnstone, Sketches of an Elephant