nLab bounded geometric morphism



Topos Theory

topos theory



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In higher category theory




Given a geometric morphism f:𝒮f \colon \mathcal{E} \longrightarrow \mathcal{S}, we may regard \mathcal{E} as a topos over 𝒮\mathcal{S} via ff. The geometric morphism ff being bounded is the “over 𝒮\mathcal{S}” version of \mathcal{E} being a Grothendieck topos.



A geometric morphism f=(f *f *):𝒮f = (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 BB \in \mathcal{E} – called a bound of ff – such that every AA \in \mathcal{E} is a subquotient of an object of the form (f *I)×B(f^* I) \times B for some I𝒮I \in \mathcal{S}: this means that there exists a diagram

    S epi A mono (f *I)×B. \array{ S &\xrightarrow{\; epi \;}& A \\ {}^{\mathllap{mono}} \big\downarrow \\ (f^* I) \times B } \,.
  • The gluing fibration? (/f *)𝒮(\mathcal{E}/f^*)\to \mathcal{S} has a fibered separating family.

  • There exists a BB\in\mathcal{E} such that for every AA\in\mathcal{E} the composite

    f *f *(A˜ B)×BA˜ B×BA˜ f^*f_*(\tilde A^B) \times B \to \tilde A^B\times B\to \tilde A

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

Proof of the equivalence.

Lemma B3.1.6 in the Elephant

If we regard \mathcal{E} as a topos over 𝒮\mathcal{S} via ff, then when ff is bounded we call \mathcal{E} a bounded 𝒮\mathcal{S}-topos.


As relative Grothendieck toposes

If fΓ:Setf \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 𝒮\mathcal{S} are precisely the toposes of 𝒮\mathcal{S}-valued internal sheaves on internal sites in 𝒮\mathcal{S} (Johnstone, Section B3.3).

If f:𝒮f \colon \mathcal{E}\to\mathcal{S} is bounded and 𝒮\mathcal{S} is a Grothendieck topos, then \mathcal{E} is a Grothendieck topos as well. This is a consequence of prop. .

Stability under composition


Bounded geometric morphisms are stable under composition.


Assume that f:𝒮f : \mathcal{F} \to \mathcal{S} is bounded by BB\in\mathcal{F}, and g:𝒢g:\mathcal{G}\to\mathcal{F} is bounded by C𝒢C\in\mathcal{G}. Let A𝒢A\in\mathcal{G}. Then there exist JJ\in \mathcal{F} and I𝒮I\in\mathcal{S}, and subquotient spans g *J×CAg^*J\times C\leftarrow\bullet\rightarrow A and f *I×BJf^*I\times B\leftarrow\bullet\rightarrow J. By applying g *()×Cg^*(-)\times C to the second subquotient and forming a pullback, we get the diagram

A g*J×C g *f *I×g *B×C \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 *g^*, products, and pullbacks), from which we can see that fgf g is bounded by g *B×Cg^*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 February 12, 2024 at 14:13:30. See the history of this page for a list of all contributions to it.