Contents

topos theory

# 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 $(C,J)$ a site with coverage $J$ and $D \to C$ any subcategory, the induced coverage $J_D$ on $D$ has as covering sieves the intersections of the covering sieves of $C$ with the morphisms in $D$.

###### Definition

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

1. every object $U \in C$ has a covering sieve generated by maps $U_i \to U$ with $U_i \in D$.

2. for every morphism $f : U \to V$ in $C$ with $U, V \in D$ there is a covering sieve $\{f_i : U_i \to U\}$ of $U$ in $D$ such that the composites $f \circ f_i$ are in $D$.

###### Remark

If $D$ is a full subcategory then the second condition is automatic.

The following theorem is known as the comparison lemma.

###### Theorem

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

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

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

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

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’. For every morphism $f : U \to V$ in $C$ with $V \in D$ there is a cover $S\in J(U)$ in $C$ generated by a family of morphisms $\{f_i : U_i \to U\}$ in $C$ such that the composites $f \circ f_i$ are in $D$.

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

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

See the dicussion here.

## Warning

Replacing sheaves by (∞,1)-sheaves of spaces produces a strictly stronger notion. See (∞,1)-comparison lemma for a sufficient criterion for a dense inclusion of (∞,1)-sites.

## 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.

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