Given a geometric morphism $f : \mathcal{E} \to \mathcal{S}$, we can regard $\mathcal{E}$ as a toposover$\mathcal{S}$ via $f$. The geometric morphism $f$ being bounded is the “over $\mathcal{S}$” version of $\mathcal{E}$ being a Grothendieck topos.

Definition

A geometric morphism$f : \mathcal{E} \to \mathcal{S}$ between toposes is called bounded if there exists an object $B \in \mathcal{E}$ – called a bound of $f$ – such that for every $A \in \mathcal{E}$ the following equivalent conditions hold:

$A$ is a subquotient of an object of the form $(f^* I) \times B$ for some $I \in S$: this means that there exists a diagram

$\array{
S &\stackrel{epi}{\to}& A
\\
{}^{\mathllap{mono}}\downarrow
\\
(f^* I) \times B
}
\,.$

(one more)

(yet one more)

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.

Properties

If $f := \Gamma\colon \mathcal{E}\to Set$ 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”.

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.