Limit (category theory)

In category theory, the limit of a functor generalizes the notions of inverse limit and product used in various parts of mathematics. The dual notion, colimit, generalizes direct limits and direct sums. Limits and colimits are defined via universal properties and as such provide many examples of adjoint functors.

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)

for every object N of C.

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.