Limit (category theory)

In , a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as , and inverse limits. The of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, s 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 are defined by means of diagrams in . Formally, a of shape in is a functor from to : The category is thought of as an , and the diagram is thought of as indexing a collection of objects and morphisms in patterned on . One is most often interested in the case where the category is a or even finite category. A diagram is said to be small or finite whenever is. Inverse limit Let be a diagram of shape in a category . A to is an object of together with a family of morphisms indexed by the objects of , such that for every morphism in , we have . A limit of the diagram is a cone to such that for every other cone to there exists a unique morphism such that for all in . One says that the cone factors through the cone with the unique factorization . The morphism 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 has to be general enough to allow any other cone to factor through it; on the other hand, has to be sufficiently specific, so that only one such factorization is possible for every cone. Limits may also be characterized as terminal objects in the to F. It is possible that a diagram does not have a limit at all. However, if a diagram does have a limit then this limit is essentially unique: it is unique up to a unique isomorphism.

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