Existence theorem for limits
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits.
Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize.
Limits and colimits in a category C are defined by means of diagrams in C. Formally, a diagram of shape J in C is a functor from J to C:
The category J is thought of as an index category, and the diagram F is thought of as indexing a collection of objects and morphisms in C patterned on J.
One is most often interested in the case where the category J is a small or even finite category. A diagram is said to be small or finite whenever J is.
Let F : J → C be a diagram of shape J in a category C. A cone to F is an object N of C together with a family ψX : N → F(X) of morphisms indexed by the objects X of J, such that for every morphism f : X → Y in J, we have F(f) o ψX = ψY.
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