I've read some articles and books for the definition and use of Lawvere-Tierney topologies, but I still don't understand their role.
Some people introduce these topologies as modal operators for logic. Others speak about them as a way to define "locally true" propositions, although I don't see what it means.
When others introduce them as topologies on a category, I don't understand if it allows to define open sets on the category or on the image sets by any presheaf.
Could someone give me some clarifications about this ? In particular I'd like to see some examples of them in simple cases (I've been working with the Sierpinsky topos for example) and their use and advantage.
Best Answer
A Lawvere–Tierney topology, or local operator, can be regarded as definition of what it means for something to be "locally true" in an elementary topos; that is to say, it is a generalisation of the notion of "locality" from ordinary topology.
So let's first recall what it means for something to be locally true in a topological space. Usually, when we say a topological space $X$ has property $P$ locally at a point $x$, what we mean is that every neighbourhood $V$ of $x$ contains a neighbourhood $U$ such that $x \in U$ and $U$ itself has property $P$. In more sophisticated language, we say $X$ is locally $P$ at $x$ if the set of $P$-neighbourhoods of $x$ is cofinal in the neighbourhood filter of $x$. We say simply that a topological space has property $P$ locally if it has property $P$ locally at every point. For example, a (second-countable Hausdorff) space $X$ is locally homeomorphic to euclidean space in this sense if and only if $X$ is a topological manifold.
Now, let us restrict our attention to "good" properties $P$ that satisfy the following condition:
For example, the property of being a topological manifold is a "good" property, whereas the property of being simply connected is not "good". (One is tempted to call these properties "hereditary", but the established meaning of "hereditarily $P$" requires all subsets to have $P$, not just open subsets.)
For "good" properties $P$, a topological space $X$ has property $P$ locally if and only if $X$ can be covered by open subsets that have property $P$. We now consider the effects of replacing $P$ by locally-$P$:
But lo and behold: these are precisely the axioms for a local operator! After all, a local operator is a morphism $j : \Omega \to \Omega$ in an elementary topos with subobject classifier $\Omega$ satisfying the following conditions: \begin{gather} x \le y \text{ implies } j(x) \le j(y) \\ x \le j(x) \\ j(j(x)) = j(x) \\ j(x \land y) = j(x) \land j(y) \end{gather}
However, let me emphasise that neither Lawvere–Tierney topologies nor Grothendieck topologies generalise the notion of "topology on a set". For one thing, a topos is essentially a "generalised space", and so already comes equipped with a "topology", in some sense.
More precisely, let us fix a set $X$. A topology on $X$ is a subframe of the powerset $\mathscr{P}(X)$, and so induces an injective frame homomorphism $f^* : \textrm{Ouv}(X) \hookrightarrow \mathscr{P}(X)$. By general nonsense this has a right adjoint $f_* : \mathscr{P}(X) \to \textrm{Ouv}(X)$ and so induces a left exact monad $f_* \circ f^*$ on ... $\textrm{Ouv}(X)$. Not $\mathscr{P}(X)$. (The induced monad on $\textrm{Ouv}(X)$ is not even interesting: it is the identity monad.) So although one might naïvely expect there to be a natural bijection between topologies on $X$ and local operators on $\textbf{Set}^X = \textbf{Sh}(X^\textrm{disc})$, in fact there is no relation at all.
Instead, what is true is that, for example, for each subset $Y$ of $X$ (now with a fixed topology) there is a corresponding local operator $j_Y : \textrm{Ouv}(X) \to \textrm{Ouv}(X)$ given by $V \mapsto ((X \setminus Y) \cup V)^\circ$. In general these are not guaranteed to be different for different subsets $Y$, but they can distinguish between any two open subsets or any two closed subsets. In particular, if $x$ is a closed point of $X$, then there is a local operator $j_{\{x\}}$ given by $$j_{\{x\}}(V) = \begin{cases} X \setminus \{ x \} & \text{if } x \notin V \\ X & \text{if } x \in V \end{cases}$$ and this local operator corresponds to the modality "true for some open neighbourhood of $x$", i.e. "locally true at $x$" for the above-discussed "good" properties. More generally, for any topos $\mathcal{E}$, there is an order-reversing bijection between local operators and replete subtoposes of $\mathcal{E}$ (using the Lawvere–Tierney sheafification construction).
So what is the notion of a local operator good for? The simple answer is that it gives us a way to construct new toposes from old ones; a better answer is that it gives a somewhat more tractable complete classification of subtoposes in a given topos. (After all, there could be a proper class of subtoposes in principle, but there is only a set of local operators in any locally small topos.) For example, by examining the local operators in the Sierpiński topos, we may conclude that there are only three subtoposes (up to equivalence): the degenerate topos $\mathbf{1} \cong \textbf{Sh}(\emptyset)$, the base topos $\textbf{Set} \cong \textbf{Sh}(1)$, and the Sierpiński topos itself.