Extensive category

Summary

An extensive category is a category with finite coproducts that behave like disjoint unions of sets. Coproducts are disjoint and stable under pullback. So an object over $A + B$ splits uniquely into a part over $A$ and a part over $B$. Sets, topological spaces, small categories and every topos are extensive, while categories of modules are not. Extensivity is the categorical form of the idea that a space can be cut into disjoint pieces.