An explicit construction of the colimit of a filtered diagram in the category of topoi and logical morphisms is given and then used to construct a family of topoi with a fixed Boolean algebra of truth values but with varying amounts of cocompleteness. This same construction, when applied to the diagram of complete Boolean algebras in a quantum logic Q gives a partial topos, a noncategory which is a close to being a model of set theory with algebra of truth values Q as a noncategory can be.
Stout, Lawrence, "Laminations, or How to Build a Quantum-Logic-Valued Model of Set Theory" (1979). Scholarship. 85.