restriction and extension of sheaves



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

(f t) *:PSh(X,A)PSh(Y,A)(f^t)_* : PSh(X,A) \to PSh(Y,A)direct image
(f t) :PSh(Y,A)PSh(X,A)(f^t)^\dagger : PSh(Y,A) \to PSh(X,A)left adjoint to direct image
(f t) :PSh(Y,A)PSh(X,A)(f^t)^\ddagger : PSh(Y,A) \to PSh(X,A)right adjoint to direct image

Recall moreover that for f:XYf : 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)Sh(X,A). f^{-1} : Sh(Y,A) \to Sh(X,A) \,.

Now, if the morphism of sites ff happens to be restriction to a sub-site f:XUf : X \to U with UPSh(X,A)U \in PSh(X,A) with UU 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 XX with underlying category S XS_X and given a presheaf UPSh(X)U \in PSh(X) with the induced sub-site j UX:XUj_{U \to X} : X \to U corresponding to the forgetful functor j UX t:(Y S X/U)S Xj^t_{U \to X} : (Y_{S_X}/U) \to S_X from the comma category S U=(Y S X/U)S XS_U = (Y_{S_X}/U) \to S_X underlying the site UU (as discussed at site) the right adjoint functor

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

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

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

whose action may suggestively be denoted

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

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

one calls

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

the extension of sheaves on UU to sheaves on XX.

To summarize notation and terminology:

morphism of sitesj UX:XUj_{U \to X} : X \to U
underlying functorj UX t:(Y S X/U)S Xj^t_{U \to X} : (Y_{S_X}/U) \to S_X
sheaf restriction(j UX) *:Sh(X)Sh(U)(j_{U \to X})_* : Sh(X) \to Sh(U)direct image
sheaf extensionj UX :Sh(U)Sh(X)j^{\ddagger}_{U \to X} : Sh(U) \to Sh(X)right adjoint to direct image
sheaf inverse image(j t) UX ¯:Sh(U)Sh(X)\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

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


For instance section 17.6 of

Revised on December 16, 2017 05:10:48 by Mike Shulman (