Limit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits.
The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts 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.
As with every universal property, the above definition describes a balanced state of generality: The limit object
Limits and colimits can also be defined for collections of objects and morphisms without the use of diagrams.
Any collection of objects and morphisms defines a (possibly large) directed graph
Weak limit and weak colimits are defined like limits and colimits, except that the uniqueness property of the mediating morphism is dropped.
The definition of limits is general enough to subsume several constructions useful in practical settings.
[1]: §V.2 Thm.1 In this case, the limit of a diagram F : J → C can be constructed as the equalizer of the two morphisms[1]: §V.2 Thm.2 given (in component form) by There is a dual existence theorem for colimits in terms of coequalizers and coproducts.
Both of these theorems give sufficient and necessary conditions for the existence of all (co)limits of shape J.
Limits and colimits are important special cases of universal constructions.
Given a diagram F: J → C (thought of as an object in CJ), a natural transformation ψ : Δ(N) → F (which is just a morphism in the category CJ) is the same thing as a cone from N to F. To see this, first note that Δ(N)(X) = N for all X implies that the components of ψ are morphisms ψX : N → F(X), which all share the domain N. Moreover, the requirement that the cone's diagrams commute is true simply because this ψ is a natural transformation.
(Dually, a natural transformation ψ : F → Δ(N) is the same thing as a co-cone from F to N.) Therefore, the definitions of limits and colimits can then be restated in the form: Like all universal constructions, the formation of limits and colimits is functorial in nature.
In other words, if every diagram of shape J has a limit in C (for J small) there exists a limit functor which assigns each diagram its limit and each natural transformation η : F → G the unique morphism lim η : lim F → lim G commuting with the corresponding universal cones.
This adjunction gives a bijection between the set of all morphisms from N to lim F and the set of all cones from N to F which is natural in the variables N and F. The counit of this adjunction is simply the universal cone from lim F to F. If the index category J is connected (and nonempty) then the unit of the adjunction is an isomorphism so that lim is a left inverse of Δ.
For example, if J is a discrete category, the components of the unit are the diagonal morphisms δ : N → NJ.
This functor is left adjoint to the diagonal functor Δ : C → CJ, and one has a natural isomorphism The unit of this adjunction is the universal cocone from F to colim F. If J is connected (and nonempty) then the counit is an isomorphism, so that colim is a left inverse of Δ.
One can use the above relationship to define the limit of F in C. The first step is to observe that the limit of the functor Hom(N, F–) can be identified with the set of all cones from N to F: The limiting cone is given by the family of maps πX : Cone(N, F) → Hom(N, FX) where πX(ψ) = ψX.
If one is given an object L of C together with a natural isomorphism Φ : Hom(L, –) → Cone(–, F), the object L will be a limit of F with the limiting cone given by ΦL(idL).
In fancy language, this amounts to saying that a limit of F is a representation of the functor Cone(–, F) : C → Set.
Dually, if a diagram F : J → C has a colimit in C, denoted colim F, there is a unique canonical isomorphism which is natural in the variable N and respects the colimiting cones.
For any bifunctor there is a natural isomorphism In words, filtered colimits in Set commute with finite limits.
If C is a complete category, then, by the above existence theorem for limits, a functor G : C → D is continuous if and only if it preserves (small) products and equalizers.
Dually, G is cocontinuous if and only if it preserves (small) coproducts and coequalizers.
If the categories C and D have all limits of shape J then lim is a functor and the morphisms τF form the components of a natural transformation The functor G preserves all limits of shape J if and only if τ is a natural isomorphism.
In this sense, the functor G can be said to commute with limits (up to a canonical natural isomorphism).
Preservation of limits and colimits is a concept that only applies to covariant functors.
A functor G : C → D is said to Dually, one can define creation and reflection of colimits.
The following statements are easily seen to be equivalent: There are examples of functors which lift limits uniquely but neither create nor reflect them.
