nLab restriction and extension of sheaves

Idea

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

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

$$f^{-1} : Sh(Y,A) \to Sh(X,A) \,.$$

Now, if the morphism of sites $f$ happens to be restriction to a sub-site $f : X \to U$ with $U \in PSh(X,A)$ 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 $U \in PSh(X)$ with the induced sub-site $j_{U \to X} : X \to U$ corresponding to the forgetful functor $j^t_{U \to X} : (Y_{S_X}/U) \to S_X$ 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

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

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

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

whose action may suggestively be denoted

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

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

one calls

$$j^{\ddagger}_{U \to X} : Sh(U) \to Sh(X)$$

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		$j^t_{U \to X} : (Y_{S_X}/U) \to S_X$
sheaf restriction		$(j_{U \to X})_* : Sh(X) \to Sh(U)$		direct image
sheaf extension		$j^{\ddagger}_{U \to X} : Sh(U) \to Sh(X)$		right adjoint to direct image
sheaf inverse image		$\overline{(j^t)^{\dagger}_{U \to X}} : Sh(U) \to Sh(X)$		left adjoint to direct image followed by sheafification

Remarks

Notice the difference to the inverse image operation

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

References

For instance section 17.6 of

• Kashiwara, Schapira, Categories and Sheaves

• Stacks Project, Tag 00VC and Tag 00XF.