nLab
closed monoidal structure on sheaves

Definition

Let XX be a site with underlying category S XS_X. Write PSh(X)=[S op,Set]PSh(X) = [S^{op}, Set] for the presheaf category of XX and Sh(X)Sh(X) for the corresponding subcategory of sheaves.

The closed monoidal structure on presheaves restricts under the inclusion Sh(X)PSh(X)Sh(X) \hookrightarrow PSh(X) to a closed monoidal structure on sheaves.

Properties

f *hom(f 1G,F)hom(G,f *F). f_* hom(f^{-1}G, F) \simeq hom(G, f_* F) \,.

Here f *f_* is the direct image and f 1f^{-1} the inverse image operation.

Revised on April 2, 2009 18:16:00 by Urs Schreiber (134.100.222.156)