nLab
restriction and extension of sheaves

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), $\mathrm{PSh}(X,A)=[S_X^{\mathrm{op}},A]$, every functor $f^t:S_Y\to S_X$ induces three functors of presheaf catgeories:

Notation		Definition
$(f^t{)}_{*}:\mathrm{PSh}(X,A)\to \mathrm{PSh}(Y,A)$		direct image
$(f^t{)}^{†}:\mathrm{PSh}(Y,A)\to \mathrm{PSh}(X,A)$		left adjoint to direct image
$(f^t{)}^{‡}:\mathrm{PSh}(Y,A)\to \mathrm{PSh}(X,A)$		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 $\overline{(-)}$ is the inverse image map of sheaves:

$$f^{-1}:\mathrm{Sh}(Y,A)\to \mathrm{Sh}(X,A).$$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 $U\in \mathrm{PSh}(X,A)$ 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 $U\in \mathrm{PSh}(X)$ with the induced sub-site $j_{U\to X}:X\to U$ corresponding to the forgetful functor $j_{U\to X}^t:(Y_{S_X}/U)\to S_X$ from the comma category $S_U=(Y_{S_X}/U)\to S_X$ underlying the site $U$ (as discussed at site) the right adjoint functor

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

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

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

whose action may suggestively be denoted

$$(j_{U\to X}{)}_{*}:F\mapsto F{\mid }_U$$(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

$$j_{U\to X}^{‡}:\mathrm{Sh}(U)\to \mathrm{Sh}(X)$$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:

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

Remarks

Notice the difference to the inverse image operation

$$j_{U\to X}^{-1}:\mathrm{Sh}(U)\to \mathrm{Sh}(X).$$j^{-1}_{U \to X} : Sh(U) \to Sh(X) \,.

References

For instance section 17.6 of

• Kashiwara, Schapira, Categories and Sheaves