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.
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. 1.
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
Last revised on June 13, 2016 at 11:45:19. See the history of this page for a list of all contributions to it.