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.
Last revised on August 6, 2020 at 02:53:31. See the history of this page for a list of all contributions to it.