topos theory

# Contents

## 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

Revised on February 24, 2014 11:41:04 by Auke Booij? (131.111.184.30)