I am looking for ways to internalize the modal operator of necessity $\Box$, ending up with a morphism $\Box: \Omega \to \Omega$ satisfying the necessitation rule (if $\phi$, then $\Box \phi$) and the distributive under implication $(\Box(\phi \to \psi)\to (\Box \phi \to \Box \psi))$. The reason why is I would like to study categories in which their internal logic, instead of classical ($\Omega$ is a Boolean algebra) or intuitionistic ($\Omega$ is a Heyting algebra), are modal logics, satisfying specifics modal axioms. However, seems like it is a little bit trick to define categorical versions of modal operators.

In the paper Topos Theoretic Approaches to Modality by Reyes and Zolfaghary, they say

''There is a fundamental difference between modal operators in a topos such as $\Box$ (necessity) and other logical operators such as $\neg$ (negation): while $\neg$, is functorial in the sense that it commutes with pull-backs (and hence it defines a map $\neg: \Omega \to \Omega$), this is not so for $\Box$. Indeed, the only operator $\Box: \Omega \to \Omega$ such that $\Box p \subseteq p$ and $\Box T = T$ is the identity''

Then, they define modal operators in the image of a subobject classifier by a geometric morphism.

There are also other constructs, like in the paper Elementary Axioms for Local Maps of Toposes by Awodey and Birkedal, where they notice that the internal logic for local maps could be viewed as the modal logic $S_{4}$.

I would be glad if someone could explain in more detail why it is not possible to define modalities $(\Box,\Diamond)$ internally in a topos, using its subobject classifier.

Many thanks.

whole pointof $\Box$ is that the truth value of $\Box p$ doesn't just depend on the truth value of $p$. As such, it is unreasonable to try to make $\Box$ into a function from the object $\Omega$ of truth value to itself. $\endgroup$cum grano salis, however, because a Lawvere-Tierney topology (=local operator) $j\colon\Omega\to\Omega$ certainly behaves as a sort of modal operator (whose meaning is something like “locally”), and by definition it makes sense at the truth-value level.) $\endgroup$1more comment