Let $\mathcal{E}$ be a topos, and let $\top\colon1\to\Omega$ be its subobject classifier. We refer to global elements $P\colon 1\to\Omega$ as propositions; they form a poset, denoted $(|\Omega|,\leq)$. There are also connectives $\Omega^2\to\Omega$, such as $\Rightarrow,\wedge,\vee$.


A *Lawvere-Tierney* topology on $\mathcal{E}$, also known as a *local operator* or a *modality*, is a morphism $j\colon\Omega\to\Omega$ such that $P\leq j(P)$, $jj(P)= j(P)$, and $j(P\wedge Q)=j(P)\wedge j(Q)$ for all $P,Q\in|\Omega|$.


For any proposition $\phi\in|\Omega|$, there are three well-known modalities that one can associate to $\phi$:


1. The open modality for $\phi$, denoted $o\_\phi\colon\Omega\to\Omega$, given by the formula $o\_\phi(P):=(\phi\Rightarrow P)$.
2. The closed modality for $\phi$, denoted $c\_\phi\colon\Omega\to\Omega$, given by the formula $c\_\phi(P):=(\phi\vee P)$.
3. The quasi-closed modality for $\phi$, denoted $q\_\phi\colon\Omega\to\Omega$, given by the formula $q\_\phi(P):=((P\Rightarrow\phi)\Rightarrow\phi)$.


Let's call the above "propositional" modalities, for want of a better term. Let's also include the "union" of two propositional modalities as propositional (the union of $j\_1$ and $j\_2$ is given by $(j\_1\wedge j\_2)(-)$), as well as various intersections that can be defined internally. For example, it is easy to check that $(j\_1\circ j\_2)$ is again a modality (called the intersection of $j\_1$ and $j\_2$) if either $j\_1$ is open or $j\_2$ is closed. [Thanks to Simon Henry for reminding me of intersections.]


I certainly wouldn't expect that all modalities are propositional in the above sense.


**Question:** Can you supply an example of a non-propositional modality on a topos $\mathcal{E}$?


Thanks!



Example of non-"propositional" local operators on a topos?
An answer similar to that of Simon Henry. Take (left) $M$-sets for a monoid $M$: the terminal is a singleton set, so has only two subobjects, and the only nontrivial propositional in your sense modality you can get is double negation. However there can be other modalities: $\Omega$ can be taken to be the Heyting algebra of left ideals of $M$, i. e. $M$-subsets $\mathfrak a\subseteq M$ of $M$ with the action on itself by left multiplication. This $\Omega$ is an $M$-set via
$$
m\mathfrak a=\{m'\in M\mid m'm\in\mathfrak a\}
$$
and it is clearly an $M$-equivariant Heyting algebra, i. e. $M$ acts on it via Heyting algebra endomorphisms. Thus for example analogs of all kinds of propositional modalities that you list can be realized in it, e. g. for an ideal $\mathfrak a\in\Omega$ we have modalities $\mathfrak a\cup-$, $\mathfrak a\Rightarrow -$, $(-\Rightarrow\mathfrak a)\Rightarrow\mathfrak a$ which in general cannot be obtained from any subobjects of the terminal, i. e. they are essentially "$M$-propositional" rather than "$1$-propositional".


**Later** - in a comment below Simon revealed a misconception of mine: the actual examples I proposed were actually wrong. While still not understanding well what goes on, but inspired by his further comment, I decided to add some (hopefully) valid examples, with the aid of "[Remarks on quintessential and persistent localizations](http://www.tac.mta.ca/tac/volumes/1996/n8/n8.pdf)" by Johnstone (TAC **2** (1996) pp. 90–99). It is shown there that for $M$-sets, those local operators $j$ on $\Omega$ for which the associated sheaf functor is not only left but also right adjoint to the inclusion of $j$-sheaves, are in one-to-one correspondence with central idempotents of $M$.


Explicitly, if $e$ is a central idempotent of $M$ then $\mathfrak a\mapsto e\mathfrak a$ is a local operator, with sheaves those $M$-sets on which $e$ acts by identity, the associated sheaf of an $M$-set $X$ being $eX$.


