Limit (category theory) 

For the formal definition, consider a covariant functor F : J -> C. A limit of F is an object L of C, together with morphisms φ_X : L -> F(X) for every object X of J, such that for every morphism f : X -> Y in J, we have F(f) φ_X = φ_Y, and such that the following universal property is satisfied:

for any object N of C and any set of morphisms ψ_X : N -> F(X) such that for every morphism f : X -> Y in J, we have F(f) ψ_X = ψ_Y, there exists precisely one morphism u : N -> L such that φ_X u = ψ_X for all X.

If F has a limit (which it need not), then the limit is defined up to a unique isomorphism, and is denoted by lim F.

If J is a small category and every functor from J to C has a limit, then the limit operation forms a functor from the functor category (see category theory) C^J to C. For example, if J is a discrete category and C is the category Ab of abelian groups, then lim : Ab^J -> Ab is the functor which assigns to every family of abelian groups its direct product. More generally, if J is a small category arising from a partially ordered set, then lim: Ab^J -> Ab assigns to every system of abelian groups its inverse limit.

The limit functor lim : C^J -> C (if it exists) has as left adjoint the diagonal functor C -> C^J which assigns to every object N of C the constant functor whose value is always N on objects and id_N on morphisms.

If every functor from every small category J to C has a limit, then the category C is called complete. Many important categories are complete: groups, abelian groups, sets, modules over some ring, topological spaces and compact Hausdorff spaces. In general, a category C is complete iff it contains arbitrary products, and equalizers[?] for any pair of parallel morphisms.

A functor G : C -> D is continuous if it maps limits to limits, in the following sense: whenever I is a small category and a functor F : I -> C has limit L together with morphisms φ_X : L -> F(X), then the functor GF : I -> D has limit G(L) with maps G(φ_X). Important examples of continuous functors are given by the representable ones: if U is some object of C, then the functor G_U : C -> Set with G_U(V) = Mor_D(U, V) for all objects V in D is continuous.

The importance of adjoint functors lies in the fact that every functor which has a left adjoint (and therefore is a right adjoint) is continuous. In the category Ab of abelian groups, this for example shows that the kernel of a product of homomorphisms is naturally identified with the product of the kernels. Also, limit functors themselves are continuous.

Colimits are defined analogous to limits: A colimit of the functor F : J -> C is an object L of C, together with morphisms φ_X : F(X) -> L for every object X of J, such that for every morphism f : X -> Y in J, we have φ_Y F(f) = φ_X, and such that the following universal property is satisfied:

for any object N of C and any set of morphisms ψ_X : F(X) -> N such that for every morphism f : X -> Y in J, we have ψ_Y F(f) = ψ_X, there exists precisely one morphism u : L -> N such that u φ_X = ψ_X for all X.

The colimit of F, unique up to unique isomorphism if it exists, is denoted by colim F.

Limits and colimits are related as follows: A functor F : J -> C has a colimit if and only if for every object N of C, the functor X |-> Mor_C(F(X),N) (which is a covariant functor on the dual category J^op) has a limit. If that is the case, then

Mor_C(colim F, N) = lim Mor_C(F(-), N)

The category C is called cocomplete if every functor F : J -> C with small J has a colimit. The following categories are cocomplete: sets, groups, abelian groups, modules over some ring and topological spaces.

A covariant functor G : C -> D is cocontinuous if it transforms colimits into colimits. Every functor which has a right adjoint (and is a left adjoint) is cocontinuous. As an example in the category Grp of groups: the functor F : Set -> Grp which assigns to every set S the free group over S has a right adjoint (the forgetfull functor Grp -> Set) and is therefore cocontinuous. The free product[?] of groups is an example of a colimit construction, and it follows that the free product of a family of free groups is free.

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