#
nLab

geometric transformation

### Context

#### Topos Theory

**topos theory**

## Background

## Toposes

## Internal Logic

## Topos morphisms

## Cohomology and homotopy

## In higher category theory

## Theorems

# 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

Last revised on February 24, 2014 at 11:41:04.
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