nLab bounded geometric morphism

Idea

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.

Definition

Definition

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

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

• The gluing fibration? $(\mathcal{E}/f^*)\to \mathcal{S}$ has a fibered separating family.

• There exists a $B\in\mathcal{E}$ such that for every $A\in\mathcal{E}$ the composite

  $$f^*f_*(\tilde A^B) \times B \to \tilde A^B\times B\to \tilde A$$

  is an epimorphism, where $\tilde A$ is the partial map classifier of $A$.

Proof of the equivalence.

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.

Properties

As relative Grothendieck toposes

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. .

Stability under composition

Proposition

Bounded geometric morphisms are stable under composition.

Proof

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

$$\begin{matrix} \bullet & \to & \bullet & \to A \\ \downarrow && \downarrow \\ \bullet & \rightarrow & g*J\times C \\ \downarrow \\ g^*f^* I\times g^* B \times C \end{matrix}$$

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.

Related concepts

• base topos
• internal site
• internal sheaf
• unbounded topos

References

definition B3.1.7 in

• Peter Johnstone, Sketches of an Elephant