The dense topology The definition of the dense topology confuses me. If $C$ is a category and $X \in C$, a sieve $S$ on $X$ is a covering for the dense topology iff for every $f : Y \to X$ there is some morphism $g : Z \to Y$ such that $f \circ g \in S$ (Mac Lane, Moerdijk, Sheaves in Geometry and Logic, III.2).


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*What is the intuition behind this definition? What does it have to do with density in the usual sense? When $C$ is a partially ordered set, it says something like that the elements in $S$ don't approximate $X$, but rather that they are arbitrary small?!

*What is an explicit basis for this topology? I have to admit that I have no intuiton for sieves, but I have some intuition for bases of topologies (often just called (pre)topologies).

*How can I understand sheaves for this topology?

*What is the connection to the following topology on the category of spaces: A family $\{f_i : X_i \to X\}$ is a covering if $\cup_i f_i(X_i)$ is dense in $X$.

 A: I think I can answer the terminilogical part.
Take $C$ to be the poset $\mathcal O (T) \setminus \{\emptyset\}$ of open sets of a topological space $T$ (the arrows being the inclusion). Then a dense sieve on a open set $U \subseteq T$ is a sieve $\mathsf S$ on $U$ such that : for every open set $V \subseteq U$ there is a open set $W \in \mathsf S$ inside $V$.
So a open set $U' \subseteq U$ is (topologically) dense inside $U$ if and only if the generated sieve $\langle U' \rangle = \{ W \text{ open } \subseteq U' \} $ is dense (as a sieve).
This should answer also the last question.
As for the sheaves on this topology, you can keep reading Sheaves in Geometry and Logic : the dense topology, also called $\neg\neg$-topology, is the topology one put on the poset of Cohen's forcing conditions. The category of sheaves on this site is (almost[1]) a ZFC model in which $\neg$CH holds.

[1] One should quotient the resulting topos to make its subobject classifier 2-valued.
A: I would say that this topology comes more from logic: as Pece mentioned, this corresponds precisely to the Lawvere–Tierney topology $\lnot \lnot : \Omega \to \Omega$ on the presheaf topos. However, the name comes from topology. Let me try to explain the connection.
Let $X$ be a space (by which I really mean locale) and let $\mathcal{O}$ be the category of open subspaces. Then $\mathcal{O}$ is a complete Heyting algebra; in particular, it has an operation $\lnot$ that sends an open subspace $U \subseteq X$ to the interior of its complement, i.e. the largest open subspace $\lnot U$ such that $U \cap \lnot U = \emptyset$. Clearly, $\lnot U = \emptyset$ if and only if $U$ is a dense open subspace in $X$; and more generally, for open subspaces $U \subseteq V \subseteq X$, $U$ is dense in $V$ if and only if $V \subseteq \lnot \lnot U$, i.e. if and only if $V$ is contained in the interior of the closure of $U$.
Now, let $\mathcal{O}_{\lnot \lnot}$ be the full subcategory of those $U$ such that $\lnot \lnot U = U$. (In topology, these are called "regular open sets".) It turns out that $\mathcal{O}_{\lnot \lnot}$ is also a complete Heyting algebra, so is the category of open subspaces of a space $X_{\lnot \lnot}$ (which is really just a locale in general, i.e. not necessarily spatial). In fact, $X_{\lnot \lnot}$ is a subspace of $X$ (but not necessarily open), and it can be shown that $X_{\lnot \lnot}$ is the smallest dense subspace of $X$ (where we have now extended the meaning of dense to not-necessarily-open subspaces).
Thus, we should think of sheaves on $X_{\lnot \lnot}$ as being sheaves that are defined "generically" or "almost everywhere" on $X$. Of course, the direct image functor allows us to embed sheaves on $X_{\lnot \lnot}$ as a full subcategory of sheaves on $X$, and these will be seen to be precisely those sheaves that are right orthogonal to all dense open inclusions $U \hookrightarrow V$ (where $V \subseteq X$), i.e. sheaves $F$ on $X$ such that the restriction map $F (V) \to F (U)$ is a bijection whenever $U \subseteq V$ is dense. For example, if $X$ is an irreducible sober space (e.g. the underlying space of an integral scheme), then the sheaves on $X_{\lnot \lnot}$ can be identified with constant sheaves on $X$. In particular, $X_{\lnot \lnot}$ is isomorphic to a point – and indeed, the inclusion $X_{\lnot \lnot} \hookrightarrow X$ is none other than the inclusion of the generic point of $X$.

So much for spaces. There are several complications in generalising this idea to sites. First of all, the "dense" topology as you have defined it is for bare categories – so even for preorders we only get the special case where $X$ is an Alexandrov space. The second complication is that not every subpresheaf of a representable presheaf is representable (whereas every subsheaf of a representable sheaf on a space is representable), so we really do need to think about sieves (which I identify with subpresheaves of representable presheaves).
To begin, we need to define $\lnot$ on sieves. Let $C$ be an object in $\mathcal{C}$ and let $\mathfrak{U}$ be a sieve on $C$. Then $\lnot \mathfrak{U}$ is defined to be the largest sieve on $C$ such that $\mathfrak{U} \cap \lnot \mathfrak{U} = \emptyset$. Explicitly, $\lnot \mathfrak{U}$ consists of all those morphisms $C' \to C$ such that, for all $C'' \to C'$, the composite $C'' \to C' \to C$ is not in $\mathfrak{U}$. Thus, $\lnot \lnot \mathfrak{U}$ consists of all those morphisms $C' \to C$ such that, for some $C'' \to C'$, the composite $C'' \to C' \to C$ is in $\mathfrak{U}$. The covering sieves on $C$ in the "dense" topology are therefore all those $\mathfrak{U}$ such that $\lnot \lnot \mathfrak{U} = \mathcal{C}(-, C)$.
We can mechanically convert this to a Grothendieck pretopology by defining the covering families to be those that generate a covering sieve; more explicitly, $\{ C'_i \to C : i \in I \}$ is a covering family of $C$ if, for any $C' \to C$ in $\mathcal{C}$, for some $C'_i \to C$, there is a commutative square in $\mathcal{C}$ of the form below:
$$\begin{array}{ccc}
C'' & \rightarrow & C' \\
\downarrow & & \downarrow \\
C'_i & \rightarrow & C
\end{array}$$
In particular, if $\mathcal{C}$ is a category with pullbacks, or even just one that satisfies the appropriate Ore condition, then every non-empty family covers. This is also called the atomic topology on $\mathcal{C}$ and has some applications – for instance, the topos of $G$-sets for any topological group $G$ admits a site of this form; and in particular, the petit étale topos for a field admits a site of this form.

The common generalisation of the two ideas is, as I mentioned, the $\lnot \lnot$-topology on (elementary) toposes. Let $\mathcal{E}$ be a topos and let $U \rightarrowtail V$ be a monomorphism in $\mathcal{E}$. Then $\lnot (U \rightarrowtail V)$ is defined to be the largest subobject of $V$ whose intersection with $U \rightarrowtail V$ is $0 \rightarrowtail V$, where $0$ is the initial object in $\mathcal{E}$. This always exists in a topos. A monomorphism $U \rightarrowtail V$ is said to be $\lnot \lnot$-dense if $\lnot \lnot (U \rightarrowtail V)$ is $V$ itself (as a subobject of $V$). A $\lnot \lnot$-sheaf is an object that is right orthogonal to all $\lnot \lnot$-dense monomorphisms, and a theorem of Lawvere and Tierney is that the full subcategory of $\lnot \lnot$-sheaves is a subtopos $\mathcal{E}_{\lnot \lnot} \subseteq \mathcal{E}$.
In the case where $\mathcal{E} = \mathbf{Sh} (\mathcal{C}, J)$ for some site, $\mathcal{E}_{\lnot \lnot}$ can be identified with the topos of sheaves for a "bigger" topology. And of course, if $\mathcal{E} = \mathbf{Sh} (X)$ for some space $X$, then $\mathcal{E}_{\lnot \lnot} \simeq \mathbf{Sh} (X_{\lnot \lnot})$ for $X_{\lnot \lnot}$ as described above. In particular, $\mathbf{Sh} (X_{\lnot \lnot})$ can be described as the topos of sheaves on $\mathcal{O}$ for the pretopology you ask about in (4).
