Limit (category theory)

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. 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) ∘ ψX = ψY. A limit of the diagram F : J → C is a cone (L, φ) to F such that for any other cone (N, ψ) to F there exists a unique morphism u : N → L such that φX o u = ψX for all X in J. One says that the cone (N, ψ) factors through the cone (L, φ) withthe unique factorization u. The morphism u is sometimes called the mediating morphism. Limits are also referred to as universal cones, since they are characterized by a universal property (see below for more information). As with every universal property, the above definition describes a balanced state of generality: The limit object L has to be general enough to allow any other cone to factor through it; on the other hand, L has to be sufficiently specific, so that only one such factorization is possible for every cone.