nLab
direct image

Contents

Idea

The right adjoint part f * of any geometric morphism

(f *f *):E 1E 2(f^* \dashv f_*) \;\; : E_1 \leftrightarrows E_2

of toposes is called a direct image.

Moe generally, pairs of adjoint functors between the categories of sheaves appear in various other setups apart from geometric morphisms of topoi, for instance on abelian categories of quasicoherent sheaves, bounded derived categories of coherent sheaves and the term direct image is used for the right adjoint part of these, too.

Specifically for Grothendieck toposes: a morphism of sites f:XY induces a geometric morphism of Grothendieck toposes

(p *p *):Sh(X)p *p *Sh(Y)(p^* \dashv p_*) \;\;\; : \;\;\; Sh(X) \stackrel{}{\stackrel{\overset{p_*}{\to}}{\overset{p^*}{\leftarrow}}} Sh(Y)

between the categories of sheaves on the sites, with

Definition

Given a morphism of sites f:XY coming from a functor f t:S YS X, the direct image operation on presheaves is the functor

f *:PSh(X)PSh(Y)f_* : PSh(X) \to PSh(Y)
f *F:S Y opf tS X opFSet.f_* F : S_Y^{op} \stackrel{f^t}{\to} S_X^{op} \stackrel{F}{\to} Set \,.

The restriction of this operation to sheaves, which respects sheaves, is the direct image of sheaves

f *:Sh(X)Sh(Y).f_* : Sh(X) \to Sh(Y) \,.

Examples

Global sections

For X a site with a terminal object, let the morphism of sites be the canonical morphism p:X*.

Restriction and extension of sheaves

See

for the moment.