According to nLab

Proposition 2.3. A complete Boolean algebra is completely distributive iff it is atomic (a CABA), i.e., is a power set as a Boolean algebra.

This is basically an infinitary version of the statement that full disjunctive normal forms exist and are unique.

For example, let $B$ denote the boolean algebra freely generated by $X = \{x,y\}$. Since $X$ is finite, hence $B$ is a completely distributive boolean algebra. Hence by (more useful way of stating the above theorem), $B$ is isomorphic to the powerset of its atoms.

The atoms of $B$ are $$\{x \wedge y, \neg x \wedge y, x \wedge \neg y, \neg x \wedge \neg y\}.$$ By the above theorem, every element of $B$ can be expressed as a join of these four elements in a unique way. In other words, we've obtained the existence and uniqueness of full disjunctive normal form as a special case of Proposition 2.3 above.

I'd like to know if there's anything like this in which power sets are replaced by slice categories and, and complete Boolean algebras are replaced by topoi (or some variant thereof). The theorem should read something like:

A bicomplete locally cartesian closed category satisfying $P$ satisfies $Q$ iff it is a slice category.

where $P$ is a technical condition and $Q$ is the condition "completely distributive" or some variant on that, like the statement that every epimorphism splits (which is a version of the axiom of choice).

Question. Do there exist conditions $P$ and $Q$ that make this true, and if so, what are they?

According to nLab

Proposition 2.3. A complete Boolean algebra is completely distributive iff it is atomic (a CABA), i.e., is a power set as a Boolean algebra.

This is basically an infinitary version of the statement that full disjunctive normal forms exist and are unique.

For example, let $B$ denote the boolean algebra freely generated by $X = \{x,y\}$. Since $X$ is finite, hence $B$ is a completely distributive boolean algebra. Hence by (a slight refinement of) the above theorem, $B$ is isomorphic to the powerset of its atoms.

The atoms of $B$ are $$\{x \wedge y, \neg x \wedge y, x \wedge \neg y, \neg x \wedge \neg y\}.$$ By the above theorem, every element of $B$ can be expressed as a join of these four elements in a unique way. In other words, we've obtained the existence and uniqueness of full disjunctive normal form as a special case of Proposition 2.3 above.

I'd like to know if there's anything like this in which power sets are replaced by slice categories and, and complete Boolean algebras are replaced by topoi (or some variant thereof). The theorem should read something like:

A bicomplete locally cartesian closed category satisfying $P$ satisfies $Q$ iff it is a slice category.

where $P$ is a technical condition and $Q$ is the condition "completely distributive" or some variant on that, like the statement that every epimorphism splits (which is a version of the axiom of choice).

Question. Do there exist conditions $P$ and $Q$ that make this true, and if so, what are they?