Cohomology and homotopy
In higher category theory
Given a geometric morphism , we may regard as a topos over via . The geometric morphism being bounded is the “over ” version of being a Grothendieck topos.
A geometric morphism between toposes is called bounded, if any of the following three equivalent condition holds.
There exists an object – called a bound of – such that every is a subquotient of an object of the form for some : this means that there exists a diagram
The gluing fibration? has a separating family.
The exists a such that for every the composite
is epic, where is the partial map classifier of .
Proof of the equivalence.
Lemma B3.1.6 in the Elephant
If we regard as a topos over via , then when is bounded we call a bounded -topos.
As relative Grothendieck toposes
If 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 is a Grothendieck topos. As such we can also call Grothendieck toposes “bounded Set-toposes”.
More generally, bounded toposes over are precisely the toposes of -valued internal sheaves on internal sites in (Johnstone, Section B3.3).
If is bounded and is a Grothendieck topos, then is a Grothendieck topos as well. This is a consequence of prop. 1.
Stability under composition
Bounded geometric morphisms are stable under composition.
Assume that is bounded by , and is bounded by . Let . Then there exist and , and subquotient spans and . By applying to the second subquotient and forming a pullback, we get the diagram
where the vertical arrows are monos and the horizontal ones are epis (using the fact that epis are stable under , products, and pullbacks), from which we can see that is bounded by .
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