Editing Category (mathematics) (section)

==Types of categories==
* In many categories, e.g. '''[[category of abelian groups|Ab]]''' or [[K-Vect|'''Vect'''<sub>''K''</sub>]], the hom-sets hom(''a'', ''b'') are not just sets but actually [[abelian group]]s, and the composition of morphisms is compatible with these group structures; i.e. is [[Bilinear form|bilinear]]. Such a category is called [[preadditive category|preadditive]]. If, furthermore, the category has all finite [[product (category theory)|products]] and [[coproduct]]s, it is called an [[additive category]]. If all morphisms have a [[kernel (category theory)|kernel]] and a [[cokernel]], and all epimorphisms are cokernels and all monomorphisms are kernels, then we speak of an [[abelian category]]. A typical example of an abelian category is the category of abelian groups.
* A category is called [[complete category|complete]] if all small [[limit (category theory)|limits]] exist in it. The categories of sets, abelian groups and topological spaces are complete.
* A category is called [[cartesian closed category|cartesian closed]] if it has finite direct products and a morphism defined on a finite product can always be represented by a morphism defined on just one of the factors. Examples include '''[[Category of sets|Set]]''' and '''CPO''', the category of [[complete partial order]]s with [[Scott continuity|Scott-continuous functions]].
* A [[topos]] is a certain type of cartesian closed category in which all of mathematics can be formulated (just like classically all of mathematics is formulated in the category of sets). A topos can also be used to represent a logical theory.