There are seemingly 4 different levels of linearity that a category can satisfy: preadditivity: hom-sets are abelian groups and composition is bilinear additivity: all finite biproducts exist pre-Abelian: all morphisms have kernels and cokernels Abelian: every monic is a kernel and every epic is a cokernel. and these are all nice properties, but I wonder how useful these classifications are, since every preadditive category I've ever met is actually Abelian. OK, well I saw an additive category which wasn't Abelian in a homework exercise, but it was a rather contrived example. So does anyone have a bucket of fun examples of categories that are preadditive but not additive, additive but not pre-Abelian, or pre-Abelian but not Abelian? Do such categories occur in mathematics, or are they only for use in proving abstract nonsense theorems in category theory?