Limit (category theory)
Limits

Let F: J → C {\displaystyle F:J\to C} be a diagram of shape J {\displaystyle J} in a category C {\displaystyle C}. A cone to F {\displaystyle F} is an object N {\displaystyle N} of C {\displaystyle C} together with a family ψ X: N → F (X) {\displaystyle \psi _{X}:N\to F(X)} of morphisms indexed by the objects X {\displaystyle X} of J {\displaystyle J}, such that for every morphism f: X → Y {\displaystyle f:X\to Y} in J {\displaystyle J}, we have F (f) ∘ ψ X = ψ Y {\displaystyle F(f)\circ \psi _{X}=\psi _{Y}}.

A limit of the diagram F: J → C {\displaystyle F:J\to C} is a cone (L, ϕ) {\displaystyle (L,\phi)} to F {\displaystyle F} such that for every cone (N, ψ) {\displaystyle (N,\psi)} to F {\displaystyle F} there exists a unique morphism u: N → L {\displaystyle u:N\to L} such that ϕ X ∘ u = ψ X {\displaystyle \phi _{X}\circ u=\psi _{X}} for all X {\displaystyle X} in J {\displaystyle J}.

One says that the cone (N, ψ) {\displaystyle (N,\psi)} factors through the cone (L, ϕ) {\displaystyle (L,\phi)} withthe unique factorization u {\displaystyle u}. The morphism u {\displaystyle 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 {\displaystyle L} has to be general enough to allow any cone to factor through it; on the other hand, L {\displaystyle L} 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 category of cones 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. For this reason one often speaks of the limit of F.
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