Example of non-"propositional" local operators on a topos?

Question

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!

Answer 1

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" 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 $eX$.

Answer 2

Here are examples that are really not propositional in the sense that they are not obtained by combining propositional modalities.

Take a "non-commutative torus" I.e Takes the circle $S^1$ and makes $\mathbb{Z} $ acts on it by rotation by an angle not commensurable with $\pi$. And consider the topos of $\mathbb{Z}$-equivariant sheaves over the circle.

In this topos the subobject classifier has no global section other than true and false : indeed a subterminal object is an invariant open subsets of $S^1$, and those does not exists.

So the only non trivial modality that you will get with your construction is the double negation topology.

But any invariant sublocale of $S^1$ would give you a local operator on the topos.

Answer 3

In a comment exchange, another class of easily describable examples emerged. Let $C$ be a small category, and consider the presheaf topos $\mathbf{Set}^{C^{\mathrm{op}}}$. In there, the subobject classifier $\Omega$ is the presheaf which to an object $c$ of $C$ assigns the poset of all sieves for $c$, i.~e. subsets $R\subseteq|C/c|$ which satisfy $(\gamma:c'\to c)\in R\ \Rightarrow\ \forall\ \gamma':c''\to c'\ (\gamma\gamma':c''\to c)\in R$.

Now any subset $S\subseteq|C|$ of the set of objects of $C$ defines a local operator $j_S:\Omega\to\Omega$ given by $$ j_S(R)=\{\gamma:c'\to c\mid\forall\ s\in S\ \forall\ \sigma:s\to c'\ (\gamma\sigma:s\to c)\in R\}. $$ The corresponding subtopos is the topos of presheaves on the full subcategory of $C$ with $S$ as the set of objects.

Clearly in general there are many such $j_S$ which do not correspond to any subterminals: for example, if $\hom(c,c')$ is nonempty for all objects $c$, $c'$, then there are no notrivial subterminals at all, but there might clearly be lots of full subcategories giving rise to various different local operators.