nLab
dense sub-site

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.

Theorem (comparison lemma)

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

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

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

  • For $X$ a locally contractible space, $\mathrm{Op}(X)$ its category of open subsets and $\mathrm{cOp}(X)$ the full subcategory of contractible open subsets, we have that $\mathrm{cOp}(X)$ 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.

  • Similarly for $C=$ Diff the category of paracompact smooth manifolds equipped with the good open cover coverage, the full subcategory CartSp${}_{\mathrm{smooth}}$ is a dense sub-site: every such smooth manifold has a differentially good open cover (see there): a good cover by open balls each of which are diffeomorphic to a Cartesian space.

References

Section C2.2

• Peter Johnstone, Sketches of an Elephant .