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:
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.
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
to the direct image or, in this case, restriction functor
whose action may suggestively be denoted
happens to take sheaves to sheaves (when $U$ is equipped with the canonical induced topology as described at site):
one calls
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 |
Notice the difference to the inverse image operation
For instance section 17.6 of
Kashiwara, Schapira, Categories and Sheaves
Stacks Project, Tag 00VC and Tag 00XF.