# Contents

## Idea

Recall that for presheaves on a site $X$ with values in a category $A$ that admits small limits and small colimits (so in particular for $A=$ Set), $\mathrm{PSh}\left(X,A\right)=\left[{S}_{X}^{\mathrm{op}},A\right]$, every functor ${f}^{t}:{S}_{Y}\to {S}_{X}$ induces three functors of presheaf catgeories:

NotationDefinition
$\left({f}^{t}{\right)}_{*}:\mathrm{PSh}\left(X,A\right)\to \mathrm{PSh}\left(Y,A\right)$direct image
$\left({f}^{t}{\right)}^{†}:\mathrm{PSh}\left(Y,A\right)\to \mathrm{PSh}\left(X,A\right)$left adjoint to direct image
$\left({f}^{t}{\right)}^{‡}:\mathrm{PSh}\left(Y,A\right)\to \mathrm{PSh}\left(X,A\right)$right adjoint to direct image

Recall moreover that for $f:X\to Y$ any morphism of sites, the left adjoint to direct image followed by sheafification $\overline{\left(-\right)}$ is the inverse image map of sheaves:

${f}^{-1}:\mathrm{Sh}\left(Y,A\right)\to \mathrm{Sh}\left(X,A\right)\phantom{\rule{thinmathspace}{0ex}}.$f^{-1} : Sh(Y,A) \to Sh(X,A) \,.

Now, if the morphism of sites $f$ happens to be restriction to a sub-site $f:X\to U$ with $U\in \mathrm{PSh}\left(X,A\right)$ with $U$ carrying the induced topology, then

• the direct image is called restriction of sheaves;

• the right adjoint takes sheaves to sheaves and is called extension of sheaves.

## Definition

Given a site $X$ with underlying category ${S}_{X}$ and given a presheaf $U\in \mathrm{PSh}\left(X\right)$ with the induced sub-site ${j}_{U\to X}:X\to U$ corresponding to the forgetful functor ${j}_{U\to X}^{t}:\left({Y}_{{S}_{X}}/U\right)\to {S}_{X}$ from the comma category ${S}_{U}=\left({Y}_{{S}_{X}}/U\right)\to {S}_{X}$ underlying the site $U$ (as discussed at site) the right adjoint functor

${j}_{U\to X}^{‡}:\mathrm{PSh}\left(U\right)\to \mathrm{PSh}\left(X\right)$j^{\ddagger}_{U \to X} : PSh(U) \to PSh(X)

to the direct image or, in this case, restriction functor

$\left({j}_{U\to X}{\right)}_{*}:\mathrm{Sh}\left(X\right)\to \mathrm{Sh}\left(U\right)$(j_{U \to X})_* : Sh(X) \to Sh(U)

whose action may suggestively be denoted

$\left({j}_{U\to X}{\right)}_{*}:F↦F{\mid }_{U}$(j_{U \to X})_* : F \mapsto F|_U

happens to take sheaves to sheaves (when $U$ is equipped with the canonical induced topology as described at site):

one calls

${j}_{U\to X}^{‡}:\mathrm{Sh}\left(U\right)\to \mathrm{Sh}\left(X\right)$j^{\ddagger}_{U \to X} : Sh(U) \to Sh(X)

the extension of sheaves on $U$ to sheaves on $X$.

To summarize notation and terminology:

TerminologyNotationDefinition
morphism of sites${j}_{U\to X}:X\to U$
underlying functor${j}_{U\to X}^{t}:\left({Y}_{{S}_{X}}/U\right)\to {S}_{X}$
sheaf restriction$\left({j}_{U\to X}{\right)}_{*}:\mathrm{Sh}\left(X\right)\to \mathrm{Sh}\left(U\right)$direct image
sheaf extension${j}_{U\to X}^{‡}:\mathrm{Sh}\left(U\right)\to \mathrm{Sh}\left(X\right)$right adjoint to direct image
sheaf inverse image$\overline{\left({j}^{t}{\right)}_{U\to X}^{†}}:\mathrm{Sh}\left(U\right)\to \mathrm{Sh}\left(X\right)$left adjoint to direct image followed by sheafification

## Remarks

Notice the difference to the inverse image operation

${j}_{U\to X}^{-1}:\mathrm{Sh}\left(U\right)\to \mathrm{Sh}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$j^{-1}_{U \to X} : Sh(U) \to Sh(X) \,.

## References

For instance section 17.6 of

Revised on April 13, 2010 15:21:15 by Garlef Wegart? (79.216.240.51)