Could not include topos theory - contents
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.
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 $D$.
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
If $D$ is a full subcategory then the second condition is automatic.
The following theorem is known as the 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
with $f^*$ given by precomposition with $f$ and $f_*$ given by right Kan extension induces an equivalence of categories between the categories of sheaves
This appears as (Johnstone, theorm C2.2.3).
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 $C = TopManifold$ 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.
Section C2.2