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:
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:
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
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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