limit

Definition

Limits and colimits in a category

C

{\displaystyle C}

are defined by means of diagrams in

C

{\displaystyle C}

. Formally, a diagram of shape

J

{\displaystyle J}

in

C

{\displaystyle C}

is a functor from

J

{\displaystyle J}

to

C

{\displaystyle C}

:

F

:

J

→

C

.

{\displaystyle F:J\to C.}

The category

J

{\displaystyle J}

is thought of as an index category, and the diagram

F

Limits

Let

F

:

J

→

C

{\displaystyle F:J\to C}

be a diagram of shape

J

{\displaystyle J}

in a category

C

{\displaystyle C}

. A cone to

F

{\displaystyle F}

is an object

N

{\displaystyle N}

of

C

{\displaystyle C}

together with a family

ψ

{\displaystyle \psi }

of morphisms

ψ

X

:

N

→

F

(

X

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

F

:

J

→

C

{\displaystyle F:J\to C}

is an object

N

{\displaystyle N}

of

C

{\displaystyle C}

together with a family of morphisms

ψ

X

:

F

(

X

)

→

N

{\displaystyle \psi _{X}:F(X)\to N}

for every object

X

{\displaystyle X}

of

J

{\displaystyle J}

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

J

{\displaystyle J}

). This variation, however, adds no new information. Any collection of objects and morphisms defines a (possibly large) directed graph

G

{\displaystyle G}

. If we let

J

{\displaystyle J}

be the free category generated by

G

{\displaystyle G}

, there is a universal diagram

F

:

J

→

C

{\displaystyle F:J\to C}

whose image contains

G

{\displaystyle G}

. 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 : J → C.

Terminal objects. If J is the empty category there is only one diagram of shape J: the empty one (similar to the empty function in set theory). A cone to the empty diagram is essentially just an object of C. The limit of F is any object that is uniquely factored through by every other object. This is just the definition of a terminal object.

Products. If J is a discrete category then a diagram F is essentially nothing but a family of objects of C, indexed by J. The limit L of F is called the product of these objects. The cone φ consists of a family of morphisms φX : L → F(X) called the projections of the product. In the category of sets, for instance, the products are given by Cartesian products and the projections are just the natural projections onto the various factors.

Powers. A special case of a product is when the diagram F is a constant functor to an object X of C. The limit of this diagram is called the Jth power of X and denoted XJ.

Equalizers. If J is a category with two objects and two parallel morphisms from one object to the other, then a diagram of shape J is a pair of parallel morphisms in C. The limit L of such a diagram is called an equalizer of those morphisms.

Kernels. A kernel is a special case of an equalizer where one of the morphisms is a zero morphism.

Pullbacks. Let F be a diagram that picks out three objects X, Y, and Z in C, where the only non-identity morphisms are f : X → Z and g : Y → Z. The limit L of F is called a pullback or a fiber product. It can nicely be visualized as a commutative square:

Inverse limits. Let J be a directed set (considered as a small category by adding arrows i → j if and only if i ≤ j) and let F : Jop → C be a diagram. The limit of F is called an inverse limit or projective limit.

Colimits

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

Initial objects are colimits of empty diagrams.

Coproducts are colimits of diagrams indexed by discrete categories.

Copowers are colimits of constant diagrams from discrete categories.

Coequalizers are colimits of a parallel pair of morphisms.

Cokernels are coequalizers of a morphism and a parallel zero morphism.

Pushouts are colimits of a pair of morphisms with common domain.

Direct limits are colimits of diagrams indexed by directed sets.

Properties

Existence of limits

A given diagram F : J → C 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

have products if it has limits of shape J for every small discrete category J (it need not have large products),

have equalizers if it has limits of shape

∙

⇉

∙

{\displaystyle \bullet \rightrightarrows \bullet }

(i.e. every parallel pair of morphisms has an equalizer),

have pullbacks if it has limits of shape

∙

→

∙

←

∙

{\displaystyle \bullet \rightarrow \bullet \leftarrow \bullet }

(i.e. every pair of morphisms with common codomain has a pullback).

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.

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

Δ

:

C

→

C

J

{\displaystyle \Delta :{\mathcal {C}}\to {\mathcal {C}}^{\mathcal {J}}}

is the functor that maps each object N in C to the constant functor Δ(N) : J → C that maps all objects in J 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: 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:

A limit of F is a universal morphism from Δ to F.

A colimit of F is a universal morphism from F to Δ.

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

lim

:

C

J

→

C

{\displaystyle \lim :{\mathcal {C}}^{\mathcal {J}}\to {\mathcal {C}}}

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 functor is right adjoint to the diagonal functor Δ : C → CJ.

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

Hom

⁡

(

N

,

lim

F

)

≅

Cone

⁡

(

N

,

F

)

{\displaystyle \operatorname {Hom} (N,\lim F)\cong \operatorname {Cone} (N,F)}

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, –) : C → Set preserves all limits in C. By duality, the contravariant Hom functor must take colimits to limits.

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

Hom

⁡

(

N

,

lim

F

)

≅

lim

Hom

⁡

(

N

,

F

−

)

{\displaystyle \operatorname {Hom} (N,\lim F)\cong \lim \operatorname {Hom} (N,F-)}

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:

Interchange of limits and colimits of sets

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

F

:

I

×

J

→

S

e

t

,

{\displaystyle F:I\times J\to \mathbf {Set} ,}

there is a natural isomorphism

colim

J

⁡

lim

I

F

(

i

,

j

)

→

lim

I

colim

J

⁡

F

(

i

,

Functors and limits

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

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

Preservation of limits

A functor G : C → D induces a map from Cone(F) to Cone(GF): if Ψ is a cone from N to F then GΨ is a cone from GN to GF. The functor G is said to preserve the limits of F if (GL, Gφ) 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 : J → C. 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 : 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.

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 : J → C and functor G : C → D, if both F and GF have specified limits there is a unique canonical morphism

τ

F

:

G

lim

F

→

lim

G

Lifting of limits

A functor G : C → D is said to lift limits for a diagram F : J → C 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 G lifts 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 G lifts limits if it lifts all limits. There are dual definitions for the lifting of colimits.

A functor G lifts 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:

If G lifts limits of all shape J and D has all limits of shape J, then C also has all limits of shape J and G preserves these limits.

If G lifts all small limits and D is complete, then C is also complete and G is continuous.

The dual statements for colimits are equally valid.

Creation and reflection of limits

Let F : J → C be a diagram. A functor G : C → D is said to

create limits for F if whenever (L, φ) is a limit of GF there exists a unique cone (L′, φ′) to F such that G(L′, φ′) = (L, φ), and furthermore, this cone is a limit of F.

reflect limits for F if each cone to F whose image under G is a limit of GF is already a limit of F.

Dually, one can define creation and reflection of colimits.

The following statements are easily seen to be equivalent:

The functor G creates limits.

The functor G lifts limits uniquely and reflects limits.

There are examples of functors which lift limits uniquely but neither create nor reflect them.

Examples

Every representable functor C → Set preserves limits (but not necessarily colimits). In particular, for any object A of C, this is true of the covariant Hom functor Hom(A,–) : C → Set.

The forgetful functor U : Grp → Set creates (and preserves) all small limits and filtered colimits; however, U does not preserve coproducts. This situation is typical of algebraic forgetful functors.

The free functor F : Set → Grp (which assigns to every set S the free group over S) is left adjoint to forgetful functor U and is, therefore, cocontinuous. This explains why the free product of two free groups G and H is the free group generated by the disjoint union of the generators of G and H.

The inclusion functor Ab → Grp creates limits but does not preserve coproducts (the coproduct of two abelian groups being the direct sum).

The forgetful functor Top → Set lifts limits and colimits uniquely but creates neither.

Let Metc be the category of metric spaces with continuous functions for morphisms. The forgetful functor Metc → Set lifts finite limits but does not lift them uniquely.

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,

cokernels,

coproducts,

coequalizers, and

codomains

are types of colimits, whereas

kernels,

products

equalizers, and

domains

are types of limits. Second, the prefix "co" implies "first variable of the

Hom

{\displaystyle \operatorname {Hom} }

". Terms like "cohomology" and "cofibration" all have a slightly stronger association with the first variable, i.e., the contravariant variable, of the

Hom

{\displaystyle \operatorname {Hom} }

bifunctor.