nLab
bounded geometric morphism

Idea

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

Definition

A geometric morphism $f:ℰ\to 𝒮$ between toposes is called bounded if there exists an object $B\in ℰ$ – called a bound of $f$ – such that for every $A\in ℰ$ 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

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

• (one more)

• (yet one more)

If we regard $ℰ$ as a topos over $𝒮$ via $f$, then when $f$ is bounded we call $ℰ$ a bounded $𝒮$-topos.

Properties

If $f:=\Gamma :ℰ\to \mathrm{Set}$ 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 $ℰ$ is a Grothendieck topos. As such we can also call Grothendieck toposes “bounded $\mathrm{Set}$-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.

Related concepts

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

References

definition B3.1.7 in

• Peter Johnstone, Sketches of an Elephant