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 $\left(C,J\right)$ 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 $\left(C,J\right)$ 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 $\left\{{U}_{i}\to U\right\}$ 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 $\left\{{f}_{i}:{U}_{i}\to U\right\}$ such that the composites $f\circ {f}_{i}$ are in $D$.

###### Remark

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

###### Theorem (comparison lemma)

Let $\left(C,J\right)$ 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

$\left({f}^{*}⊣{f}_{*}\right):\mathrm{PSh}\left(D\right)\stackrel{\stackrel{{f}^{*}}{←}}{\underset{{f}_{*}}{\to }}\mathrm{PSh}\left(C\right)$(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

$\left({f}_{*}⊣{f}^{*}\right):{\mathrm{Sh}}_{{J}_{D}}\left(D\right)\underset{\underset{{f}_{*}}{\to }}{\overset{\stackrel{{f}^{*}}{←}}{\simeq }}{\mathrm{Sh}}_{J}C\phantom{\rule{thinmathspace}{0ex}}.$(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 $\mathrm{Op}\left(X\right)$ regarded as a site with the canonical coverage ($\left\{{U}_{i}\to U\right\}$ covers if the join of the ${U}_{i}$ us $U$). Let $\mathrm{bOp}\left(X\right)$ be a basis for the topology of $X$: a complete join-semilattice such that every object of $\mathrm{Op}\left(X\right)$ is the join of objects of $\mathrm{bOp}\left(X\right)$. Then $\mathrm{bOp}\left(X\right)$ is a dense sub-site.

• For $X$ a locally contractible space, $\mathrm{Op}\left(X\right)$ its category of open subsets and $\mathrm{cOp}\left(X\right)$ the full subcategory of contractible open subsets, we have that $\mathrm{cOp}\left(X\right)$ is a dense sub-site.
• For $C=\mathrm{TopManifold}$ the category of all paracompact topological manifolds equipped with the open cover coverage, the category CartSp${}_{\mathrm{top}}$ 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 January 15, 2012 11:56:39 by Urs Schreiber (82.113.98.116)