Given a geometric morphism $f \colon \mathcal{E} \longrightarrow \mathcal{S}$, we may regard $\mathcal{E}$ as a topos over $\mathcal{S}$ via $f$. The geometric morphism $f$ being bounded is the “over $\mathcal{S}$” version of $\mathcal{E}$ being a Grothendieck topos.
A geometric morphism $f = (f^*\dashv f_*) \colon \mathcal{E} \longrightarrow \mathcal{S}$ between toposes is called bounded, if any of the following three equivalent condition holds.
There exists an object $B \in \mathcal{E}$ – called a bound of $f$ – such that every $A \in \mathcal{E}$ is a subquotient of an object of the form $(f^* I) \times B$ for some $I \in \mathcal{S}$: this means that there exists a diagram
The gluing fibration? $\partial_0 : (\mathcal{E}/f^*)\to\mathcal{S}$ has a separating family#fibered.
The exists a $B\in\mathcal{E}$ such that for every $A\in\mathcal{E}$ the composite
is epic, where $\tilde A$ is the partial map classifier of $A$.
Lemma B3.1.6 in the Elephant
If we regard $\mathcal{E}$ as a topos over $\mathcal{S}$ via $f$, then when $f$ is bounded we call $\mathcal{E}$ a bounded $\mathcal{S}$-topos.
If $f \coloneqq \Gamma\colon \mathcal{E}\to Set$ is the global section geometric morphism of a topos (such a geometric morphism being unique if it exists), then it is bounded if and only if $\mathcal{E}$ is a Grothendieck topos. As such we can also call Grothendieck toposes “bounded Set-toposes”.
More generally, bounded toposes over $\mathcal{S}$ are precisely the toposes of $\mathcal{S}$-valued internal sheaves on internal sites in $\mathcal{S}$ (Johnstone, Section B3.3).
If $f \colon \mathcal{E}\to\mathcal{S}$ is bounded and $\mathcal{S}$ is a Grothendieck topos, then $\mathcal{E}$ is a Grothendieck topos as well. This is a consequence of prop. .
Bounded geometric morphisms are stable under composition.
Assume that $f : \mathcal{F} \to \mathcal{S}$ is bounded by $B\in\mathcal{F}$, and $g:\mathcal{G}\to\mathcal{F}$ is bounded by $C\in\mathcal{G}$. Let $A\in\mathcal{G}$. Then there exist $J\in \mathcal{F}$ and $I\in\mathcal{S}$, and subquotient spans $g^*J\times C\leftarrow\bullet\rightarrow A$ and $f^*I\times B\leftarrow\bullet\rightarrow J$. By applying $g^*(-)\times C$ to the second subquotient and forming a pullback, we get the diagram
where the vertical arrows are monos and the horizontal ones are epis (using the fact that epis are stable under $g^*$, products, and pullbacks), from which we can see that $f g$ is bounded by $g^*B\times C$.
Almost all geometric morphisms in practice are bounded, so that often when people work in the 2-category Topos of toposes and geometric morphisms, they mean that the geometric morphisms are bounded. See unbounded topos for the few examples of unbounded geometric morphisms.
definition B3.1.7 in
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