bounded geometric morphism


Topos Theory

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Given a geometric morphism f:𝒮f : \mathcal{E} \to \mathcal{S}, we can 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:𝒮f : \mathcal{E} \to \mathcal{S} between toposes is called bounded if there exists an object BB \in \mathcal{E} – called a bound of ff – such that for every AA \in \mathcal{E} the following equivalent conditions hold:

  • AA is a subquotient of an object of the form (f *I)×B(f^* I) \times B for some ISI \in S: this means that there exists a diagram

    S epi A mono (f *I)×B. \array{ S &\stackrel{epi}{\to}& A \\ {}^{\mathllap{mono}}\downarrow \\ (f^* I) \times B } \,.
  • (one more)

  • (yet one more)

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.


If f:=Γ:Setf := \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 SetSet-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.


definition B3.1.7 in

Revised on May 8, 2013 10:13:00 by David Roberts (