is a subquotient of an object of the form for some : this means that there exists a diagram
(yet one more)
If we regard as a topos over via , then when is bounded we call a bounded -topos.
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 -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