I}} ob({\cal C}_{i})$ and $arr({{\cal C}'})=\mathbin{\mathop{\textrm{\huge $\cap$}}\limits_{i\in I}} arr({\cal C}_{i})$ i.e. given an arrow $f:a\to b$ in $\cal C$, $f$ belongs to $arr({{\cal C}'})$ if and only if it belongs to $arr({{\cal C}_{i}})$ for every $i\in I$. It is immediate to see that ${\cal C}'$ is a subcategory of $\cal C$.

We note that, for any Grothendieck topology $J$ on $\cal C$, any finite intersection of subcategories of $\cal C$ which are $J$-dense is again $J$-dense; indeed, this easily follows from the fact that a finite intersection of $J$-covering sieves is again $J$-covering.

The following result provides a couple of useful facts about dense subcategories.

\[Onetoposmany\_denseness\] Let $\cal C$ be a category. Then

\(i) $\mathfrak{Groth}_{{\cal C}'}({\cal C})$ is closed in $\mathfrak{Groth}({\cal C})$ under arbitrary (non-empty) intersections (i.e. meets in $\mathfrak{Groth}({\cal C})$) and under taking larger topologies, and hence it is an Heyting algebra inheriting the Heyting operations from $\mathfrak{Groth}({\cal C})$ whose maximal element is the maximal Grothendieck topology on $\cal C$ and minimal element is the intersection of all the topologies in $\mathfrak{Groth}_{{\cal C}'}({\cal C})$;

\(ii) Let ${\cal C}'$ and ${\cal C}''$ subcategories of $\cal C$ such that ${\cal C}'$ is a subcategory of ${\cal C}''$. Then ${\cal C}'$ is 