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 $\{U_i \to U\}$ in $J$ with all $U_i$ in $D$;

2. for every morphism $f : U \to d$ in $C$ with $d \in D$ there is a covering family $\{f_i : U_i \to U\}$ 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 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, theorm C2.2.3).

###### Examples

• Let $X$ be a locale with frame $Op(X)$ regarded as a site with the canonical coverage ($\{U_i \to U\}$ covers if the join of the $U_i$ us $U$). Let $bOp(X)$ be a basis for the topology of $X$: a complete join-semilattice such that every object of $Op(X)$ is the join of objects of $bOp(X)$. Then $bOp(X)$ is a dense sub-site.

• For $X$ a locally contractible space, $Op(X)$ its category of open subsets and $cOp(X)$ the full subcategory of contractible open subsets, we have that $cOp(X)$ is a dense sub-site.
• For $C = TopManifold$ the category of all paracompact topological manifolds equipped with the open cover coverage, the category CartSp${}_{top}$ is a dense sub-site: every paracompact topological manifold has a good open cover by open balls homeomorphic to a Cartesian space.

## References

Section C2.2

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