nLab
dense sub-site

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Notions of subcategory

Contents

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 sieve generated by maps U iUU_i \to U with U iDU_i \in D.

  2. for every morphism f:UVf : U \to V in CC with U,VDU, V \in D there is a covering sieve {f i:U iU}\{f_i : U_i \to U\} of UU in DD 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 the 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, theorem C2.2.3).

Problems with another definition

The nLab following Johnstone (2002,p.546) had initially the following form of condition 2 in definition 2:

2’. For every morphism f:UVf : U \to V in CC with VDV \in D there is a cover SJ(U)S\in J(U) in CC generated by a family of morphisms {f i:U iU}\{f_i : U_i \to U\} in CC such that the composites ff if \circ f_i are in DD.

But this is too weak to prove the comparison lemma as the following example shows:

Let CC be any groupoid, with the trivial topology (only maximal sieves cover), and let DD be the discrete category on the same objects. Then for any morphism f:UVf:U\to V, its inverse f 1:VUf^{-1}:V\to U generates the maximal sieve on UU, and the composite ff 1=1 Vf f^{-1} = 1_V is in DD, so the conditions 1 and 2’ of the definition are satisfied. But the restriction Set C opSet D opSet^{C^{op}} \to Set^{D^{op}} is not generally an equivalence.

See the dicussion here.

Examples

Remark

There is also the notion of dense subcategory, which is however only remotly related to the concept of a dense sub-site by both vaguely invoking the topological concept of a dense subspace.

References

The comparison lemma originates with the exposé III by Verdier in

A more general form is used to give a site characterization for étendue toposes in

  • A. Kock, I. Moerdijk, Presentations of Etendues , Cah. Top. Géom. Diff. Cat. XXXII 2 (1991) pp.145-164. (numdam, pp.151f)

A proof of the comparison lemma together with a nice list of examples is in

See also

Revised on August 21, 2016 08:14:43 by Thomas Holder (176.5.139.45)