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Idea

A dense sub-site is a subcategory of a site such that a natural functor between the corresponding categories of sheaves is an equivalence of categories.

Definition

Definition

For (C,J)(C,J) a site with coverage JJ and DCD \to C any subcategory, the induced coverage J DJ_D on DD has as covering sieves the intersections of the covering sieves of CC with the morphisms in DD.

Definition

Let (C,J)(C,J) be a site (possibly large). A subcategory DCD \to C (not necessarily full) is called a dense sub-site with the induced coverage J DJ_D if

  1. every object UCU \in C has a covering {U iU}\{U_i \to U\} in JJ with all U iU_i in DD;

  2. for every morphism f:Udf : U \to d in CC with dDd \in D there is a covering family {f i:U iU}\{f_i : U_i \to U\} such that the composites ff if \circ f_i are in DD.

Remark

If DD is a full subcategory then the second condition is automatic.

The following theorem is known as the comparison lemma.

Theorem

Let (C,J)(C,J) be a (possibly large) site with CC a locally small category and let f:DCf : D \to C be a small dense sub-site. Then pair of adjoint functors

(f *f *):PSh(D)f *f *PSh(C) (f^* \dashv f_*) : PSh(D) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} PSh(C)

with f *f^* given by precomposition with ff and f *f_* given by right Kan extension induces an equivalence of categories between the categories of sheaves

(f *f *):Sh J D(D)f *f *Sh JC. (f_* \dashv f^*) : Sh_{J_D}(D) \underoverset {\underset{f_*}{\to}}{\overset{f^*} {\leftarrow}} {\simeq} Sh_J{C} \,.

This appears as (Johnstone, theorm C2.2.3).

Examples

 

References

Section C2.2

Revised on May 29, 2013 18:04:39 by Urs Schreiber (89.204.155.181)