that converge to x. The filters F are given a small and thin category structure by adding an arrow A → B if and only if A ⊆ B. The injection becomes a functor and the following equivalence holds : x is a topological limit of A if and only if A is a categorical limit of Colimits Examples of colimits are given by the dual versions of the examples above: Initial objects are colimits of empty diagrams. Coproducts are colimits of diagrams indexed by discrete categories. Copowers are colimits of constant diagrams from discrete categories. Coequalizers are colimits of a parallel pair of morphisms. Cokernels are coequalizers of a morphism and a parallel zero morphism. Pushouts are colimits of a pair of morphisms with common domain. Direct limits are colimits of diagrams indexed by directed sets. Properties Existence of limits A given diagram F : J → C may or may not have a limit (or colimit) in C. Indeed, there may not even be a cone to F, let alone a universal cone. A category C is said to have limits of shape J if every diagram of shape J has a limit in C. Specifically, a category C is said to have products if it has limits of shape J for every small discrete category J (it need not have large products), have equalizers if it has limits of shape (i.e. every parallel pair of morphisms has an equalizer), have pullbacks if it has limits of shape (i.e. every pair of morphisms with common codomain has a pullback). A complete category is a category that has all small limits (i.e. all limits of shape J for every small category J). One can also make the dual definitions. A category has colimits of shape J if every diagram of shape J has a colimit in C. A cocomplete category is one that has all small colimits. The existence theorem for limits states that if a category C has equalizers and all products indexed by the classes Ob(J) and Hom(J), then C has all limits of shape J. In this case, the limit of a diagram F : J → C can be constructed as the equalizer of the two morphisms given (in component form) by There is a dual existence theorem for colimits in terms of coequalizers and coproducts. Both of these theorems give sufficient and necessary conditions