nLab
geometric transformation

Context

Topos Theory

Could not include topos theory - contents

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

and

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

two geometric morphisms, a geometric transformation

η:fg \eta : f \Rightarrow g

is a natural transformation between the inverse image functors

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

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

g *f *. g_* \Rightarrow f_* \,.

References

Section A4.1 of

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