Extensivity is a fundamental categorical notion which captures the behaviour of coproducts in the category of sets. Although extensivity is a property of a category, there are many interesting cases of non-extensive categories which possess classes of morphisms exhibiting extensive behaviour. The class of morphisms of pointed sets with trivial kernels is one such example. The class of product projections of centerless groups is another such example. The aim of this talk is to introduce a concept of extensive morphism, as well as its dual concept of coextensive morphism. We will investigate these categorical concepts in various concrete situations, and discuss several general results concerning them. The main result of this talk will show how coextensive categories can be characterised in terms of monomorphism being coextensive, given suitable categorical conditions.