Limit (category theory)

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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.

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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.

Definition

Limits and colimits in a category are defined by means of diagrams in . Formally, a diagram of shape in is a functor from to :

The category is thought of as an index category, and the diagram is thought of as indexing a collection of objects and morphisms in patterned on .

One is most often interested in the case where the category is a small or even finite category. A diagram is said to be small or finite whenever is.

Limits

Let be a diagram of shape in a category . A cone to is an object of together with a family of morphisms indexed by the objects of , such that for every morphism in , we have .

A limit of the diagram is a cone to such that for every other cone to there exists a unique morphism such that for all in .

A universal cone Functor cone (extended).svg
A universal cone

One says that the cone factors through the cone with the unique factorization . The morphism 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 has to be general enough to allow any other cone to factor through it; on the other hand, 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.

Colimits

The dual notions of limits and cones are colimits and co-cones. Although it is straightforward to obtain the definitions of these by inverting all morphisms in the above definitions, we will explicitly state them here:

A co-cone of a diagram is an object of together with a family of morphisms

for every object of , such that for every morphism in , we have .

A colimit of a diagram is a co-cone of such that for any other co-cone of there exists a unique morphism such that for all in .

A universal co-cone Functor co-cone (extended).svg
A universal co-cone

Colimits are also referred to as universal co-cones . They can be characterized as initial objects in the category of co-cones from .

As with limits, if a diagram has a colimit then this colimit is unique up to a unique isomorphism.

Variations

Limits and colimits can also be defined for collections of objects and morphisms without the use of diagrams. The definitions are the same (note that in definitions above we never needed to use composition of morphisms in ). This variation, however, adds no new information. Any collection of objects and morphisms defines a (possibly large) directed graph . If we let be the free category generated by , there is a universal diagram whose image contains . The limit (or colimit) of this diagram is the same as the limit (or colimit) of the original collection of objects and morphisms.

Weak limit and weak colimits are defined like limits and colimits, except that the uniqueness property of the mediating morphism is dropped.

Examples

Limits

The definition of limits is general enough to subsume several constructions useful in practical settings. In the following we will consider the limit (L, φ) of a diagram F : JC.

Pullback categories.svg
x is a topological limit of A if and only if A is a categorical limit of

Colimits

Examples of colimits are given by the dual versions of the examples above:

Properties

Existence of limits

A given diagram F : JC may or may not have a limit (or colimit) in C. Indeed, there may not even be a cone to F, let alone a universal cone.

A category C is said to have limits of shape J if every diagram of shape J has a limit in C. Specifically, a category C is said to

A complete category is a category that has all small limits (i.e. all limits of shape J for every small category J).

One can also make the dual definitions. A category has colimits of shape J if every diagram of shape J has a colimit in C. A cocomplete category is one that has all small colimits.

The existence theorem for limits states that if a category C has equalizers and all products indexed by the classes Ob(J) and Hom(J), then C has all limits of shape J. In this case, the limit of a diagram F : JC can be constructed as the equalizer of the two morphisms

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.

Universal property

Limits and colimits are important special cases of universal constructions.

Let C be a category and let J be a small index category. The functor category CJ may be thought of as the category of all diagrams of shape J in C. The diagonal functor

is the functor that maps each object N in C to the constant functor Δ(N) : JC to N. That is, Δ(N)(X) = N for each object X in J and Δ(N)(f) = idN for each morphism f in J.

Given a diagram F: JC (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 : NF(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:

Adjunctions

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 : FG the unique morphism lim η : lim F → lim G commuting with the corresponding universal cones. This functor is right adjoint to the diagonal functor Δ : CCJ. 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. This fails if J is not connected. For example, if J is a discrete category, the components of the unit are the diagonal morphisms δ : NNJ.

Dually, if every diagram of shape J has a colimit in C (for J small) there exists a colimit functor

which assigns each diagram its colimit. This functor is left adjoint to the diagonal functor Δ : CCJ, 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.

Note that both the limit and the colimit functors are covariant functors.

As representations of functors

One can use Hom functors to relate limits and colimits in a category C to limits in Set, the category of sets. This follows, in part, from the fact the covariant Hom functor Hom(N, ) : CSet preserves all limits in C. By duality, the contravariant Hom functor must take colimits to limits.

If a diagram F : JC has a limit in C, denoted by lim F, there is a canonical isomorphism

which is natural in the variable N. Here the functor Hom(N, F) is the composition of the Hom functor Hom(N, ) with F. This isomorphism is the unique one which respects the limiting cones.

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) : CSet.

Dually, if a diagram F : JC 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. Identifying the limit of Hom(F, N) with the set Cocone(F, N), this relationship can be used to define the colimit of the diagram F as a representation of the functor Cocone(F, ).

Interchange of limits and colimits of sets

Let I be a finite category and J be a small filtered category. For any bifunctor

there is a natural isomorphism

In words, filtered colimits in Set commute with finite limits. It also holds that small limits commute with small limits. [1]

Functors and limits

If F : JC is a diagram in C and G : CD is a functor then by composition (recall that a diagram is just a functor) one obtains a diagram GF : JD. A natural question is then:

“How are the limits of GF related to those of F?”

Preservation of limits

A functor G : CD induces a map from Cone(F) to Cone(GF): if Ψ is a cone from N to F then is a cone from GN to GF. The functor G is said to preserve the limits of F if (GL, ) is a limit of GF whenever (L, φ) is a limit of F. (Note that if the limit of F does not exist, then G vacuously preserves the limits of F.)

A functor G is said to preserve all limits of shape J if it preserves the limits of all diagrams F : JC. For example, one can say that G preserves products, equalizers, pullbacks, etc. A continuous functor is one that preserves all small limits.

One can make analogous definitions for colimits. For instance, a functor G preserves the colimits of F if G(L, φ) is a colimit of GF whenever (L, φ) is a colimit of F. A cocontinuous functor is one that preserves all small colimits.

If C is a complete category, then, by the above existence theorem for limits, a functor G : CD 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.

An important property of adjoint functors is that every right adjoint functor is continuous and every left adjoint functor is cocontinuous. Since adjoint functors exist in abundance, this gives numerous examples of continuous and cocontinuous functors.

For a given diagram F : JC and functor G : CD, if both F and GF have specified limits there is a unique canonical morphism

which respects the corresponding limit cones. The functor G preserves the limits of F if and only this map is an isomorphism. 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. For contravariant functors the corresponding notions would be a functor that takes colimits to limits, or one that takes limits to colimits.

Lifting of limits

A functor G : CD is said to lift limits for a diagram F : JC if whenever (L, φ) is a limit of GF there exists a limit (L, φ) of F such that G(L, φ) = (L, φ). A functor Glifts limits of shape J if it lifts limits for all diagrams of shape J. One can therefore talk about lifting products, equalizers, pullbacks, etc. Finally, one says that Glifts limits if it lifts all limits. There are dual definitions for the lifting of colimits.

A functor Glifts limits uniquely for a diagram F if there is a unique preimage cone (L, φ) such that (L, φ) is a limit of F and G(L, φ) = (L, φ). One can show that G lifts limits uniquely if and only if it lifts limits and is amnestic.

Lifting of limits is clearly related to preservation of limits. If G lifts limits for a diagram F and GF has a limit, then F also has a limit and G preserves the limits of F. It follows that:

The dual statements for colimits are equally valid.

Creation and reflection of limits

Let F : JC be a diagram. A functor G : CD 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.

Examples

A note on terminology

Older terminology referred to limits as "inverse limits" or "projective limits," and to colimits as "direct limits" or "inductive limits." This has been the source of a lot of confusion.

There are several ways to remember the modern terminology. First of all,

are types of colimits, whereas

are types of limits. Second, the prefix "co" implies "first variable of the ". Terms like "cohomology" and "cofibration" all have a slightly stronger association with the first variable, i.e., the contravariant variable, of the bifunctor.

See also

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References

  1. "limit". nLab.