is an essentially small category.

Some geometric invariants
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Let us start with some general facts about dense subcategories. We recall from [@El2] the following definition.

Let $({\cal C}, J)$ be a site. We say a subcategory $\cal D$ of $\cal C$ is $J$-dense if

\(i) every object $c$ of $\cal C$ has a covering sieve $R \in J(c)$ generated by morphisms whose domains are in $\cal D$; and

\(ii) for any morphism $f:c \to d$ in $\cal C$ with $d\in {\cal D}$, there is a covering sieve $R \in J(c)$ generated by morphisms $g: b \to c$ for which the composite $f\circ g$ is in $\cal D$.

Let us denote by $\mathfrak{Groth}({\cal C})$ the Heyting algebra of Grothendieck topologies on a category $\cal C$ (cfr. [@OC6]). Given a subcategory ${\cal C}'$ of $\cal C$, we denote by $\mathfrak{Groth}_{{\cal C}'}({\cal C})$ the subset of $\mathfrak{Groth}({\cal C})$ formed by the Grothendieck topologies $J$ on $\cal C$ such that ${\cal C}'$ is $J$-dense.

There is an obvious notion of intersection of subcategories; specifically, given a collection $\{{\cal C}_{i} \textrm{ | } i\in I\}$ of subcategories of a category $\cal C$, we can define their intersection ${\cal C}'$ by putting $ob({{\cal C}'})=\mathbin{\mathop{\textrm{\huge $\cap$}}\limits_{i\in 