bounded geometric morphism in nLab

Context

Topos Theory

• Idea
• Definition
• Properties
• As relative Grothendieck toposes
• Stability under composition
• Related concepts
• References

Definition

A geometric morphism f=(f *⊣f *):ℰ⟶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 B∈ℰB \in \mathcal{E} – called a bound of ff – such that every A∈ℰA \in \mathcal{E} is a subquotient of an object of the form (f *I)×B(f^* I) \times B for some I∈𝒮I \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? (ℰ/f *)→𝒮(\mathcal{E}/f^*)\to \mathcal{S} has a fibered separating family.

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

  f *f *(A˜ B)×B→A˜ B×B→A˜ 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

Proposition

Bounded geometric morphisms are stable under composition.

Proof

Assume that f:ℱ→𝒮f : \mathcal{F} \to \mathcal{S} is bounded by B∈ℱB\in\mathcal{F}, and g:𝒢→ℱg:\mathcal{G}\to\mathcal{F} is bounded by C∈𝒢C\in\mathcal{G}. Let A∈𝒢A\in\mathcal{G}. Then there exist J∈ℱJ\in \mathcal{F} and I∈𝒮I\in\mathcal{S}, and subquotient spans g *J×C←•→Ag^*J\times C\leftarrow\bullet\rightarrow A and f *I×B←•→Jf^*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.