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

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

and

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

two geometric morphisms, a geometric transformation

$$\eta :f\Rightarrow g$$\eta : f \Rightarrow g

is a natural transformation between the inverse image functors

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

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

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

References

Section A4.1 of

• Peter Johnstone, Sketches of an Elephant