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Incategory theory, a branch ofmathematics, the abstract notion of alimit captures the essential properties of universal constructions such asproducts,pullbacks andinverse limits. Thedual notion of acolimit generalizes constructions such asdisjoint unions,direct sums,coproducts,pushouts anddirect limits.

Limits and colimits, like the strongly related notions ofuniversal properties andadjoint 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.

Limits and colimits in acategory${\displaystyle C}$ are defined by means of diagrams in${\displaystyle C}$. Formally, adiagram of shape${\displaystyle J}$ in${\displaystyle C}$ is afunctor from${\displaystyle J}$ to${\displaystyle C}$:

${\displaystyle F:J\to C.}$

The category${\displaystyle J}$ is thought of as anindex category, and the diagram${\displaystyle F}$ is thought of as indexing a collection of objects andmorphisms in${\displaystyle C}$ patterned on${\displaystyle J}$.

One is most often interested in the case where the category${\displaystyle J}$ is asmall or evenfinite category. A diagram is said to besmall orfinite whenever${\displaystyle J}$ is.

Let${\displaystyle F:J\to C}$ be a diagram of shape${\displaystyle J}$ in a category${\displaystyle C}$. Acone to${\displaystyle F}$ is an object${\displaystyle N}$ of${\displaystyle C}$ together with a family${\displaystyle {\psi }_X:N\to F(X)}$ of morphisms indexed by the objects${\displaystyle X}$ of${\displaystyle J}$, such that for every morphism${\displaystyle f:X\to Y}$ in${\displaystyle J}$, we have${\displaystyle F(f)\circ {\psi }_X={\psi }_Y}$.

Alimit of the diagram${\displaystyle F:J\to C}$ is a cone${\displaystyle (L,\varphi )}$ to${\displaystyle F}$ such that for every cone${\displaystyle (N,\psi )}$ to${\displaystyle F}$ there exists aunique morphism${\displaystyle u:N\to L}$ such that${\displaystyle {\varphi }_X\circ u={\psi }_X}$ for all${\displaystyle X}$ in${\displaystyle J}$.

One says that the cone${\displaystyle (N,\psi )}$ factors through the cone${\displaystyle (L,\varphi )}$ withthe unique factorization${\displaystyle u}$. The morphism${\displaystyle u}$ is sometimes called themediating morphism.

Limits are also referred to asuniversal cones, since they are characterized by auniversal property (see below for more information). As with every universal property, the above definition describes a balanced state of generality: The limit object${\displaystyle L}$ has to be general enough to allow any cone to factor through it; on the other hand,${\displaystyle L}$ has to be sufficiently specific, so that onlyone such factorization is possible for every cone.

Limits may also be characterized asterminal objects in thecategory of cones toF.

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 uniqueup to a uniqueisomorphism. For this reason one often speaks ofthe limit ofF.

Thedual 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:

Aco-cone of a diagram${\displaystyle F:J\to C}$ is an object${\displaystyle N}$ of${\displaystyle C}$ together with a family of morphisms

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

for every object${\displaystyle X}$ of${\displaystyle J}$, such that for every morphism${\displaystyle f:X\to Y}$ in${\displaystyle J}$, we have${\displaystyle {\psi }_Y\circ F(f)={\psi }_X}$.

Acolimit of a diagram${\displaystyle F:J\to C}$ is a co-cone${\displaystyle (L,\varphi )}$ of${\displaystyle F}$ such that for any other co-cone${\displaystyle (N,\psi )}$ of${\displaystyle F}$ there exists a unique morphism${\displaystyle u:L\to N}$ such that${\displaystyle u\circ {\varphi }_X={\psi }_X}$ for all${\displaystyle X}$ in${\displaystyle J}$.

Colimits are also referred to asuniversal co-cones. They can be characterized asinitial objects in thecategory of co-cones from${\displaystyle F}$.

As with limits, if a diagram${\displaystyle F}$ has a colimit then this colimit is unique up to a unique isomorphism.

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${\displaystyle J}$). This variation, however, adds no new information. Any collection of objects and morphisms defines a (possibly large)directed graph${\displaystyle G}$. If we let${\displaystyle J}$ be thefree category generated by${\displaystyle G}$, there is a universal diagram${\displaystyle F:J\to C}$ whose image contains${\displaystyle G}$. 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 andweak 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. In the following we will consider the limit (L,φ) of a diagramF :J →C.

• Terminal objects. IfJ is the empty category there is only one diagram of shapeJ: the empty one (similar to theempty function in set theory). A cone to the empty diagram is essentially just an object ofC. The limit ofF is any object that is uniquely factored through by every other object. This is just the definition of aterminal object.
• Products. IfJ is adiscrete category then a diagramF is essentially nothing but afamily of objects ofC, indexed byJ. The limitL ofF is called theproduct of these objects. The coneφ consists of a family of morphismsφ_X :L →F(X) called theprojections of the product. In thecategory of sets, for instance, the products are given byCartesian products and the projections are just the natural projections onto the various factors.
  • Powers. A special case of a product is when the diagramF is aconstant functor to an objectX ofC. The limit of this diagram is called theJ^th power ofX and denotedX^J.
• Equalizers. IfJ is a category with two objects and two parallel morphisms from one object to the other, then a diagram of shapeJ is a pair of parallel morphisms inC. The limitL of such a diagram is called anequalizer of those morphisms.
  • Kernels. Akernel is a special case of an equalizer where one of the morphisms is azero morphism.
• Pullbacks. LetF be a diagram that picks out three objectsX,Y, andZ inC, where the only non-identity morphisms aref :X →Z andg :Y →Z. The limitL ofF is called apullback or afiber product. It can nicely be visualized as acommutative square:

• Inverse limits. LetJ be adirected set (considered as a small category by adding arrowsi →j if and only ifi ≥j) and letF :J^op →C be a diagram. The limit ofF is called aninverse limit orprojective limit.
• IfJ =1, the category with a single object and morphism, then a diagram of shapeJ is essentially just an objectX ofC. A cone to an objectX is just a morphism with codomainX. A morphismf :Y →X is a limit of the diagramX if and only iff is anisomorphism. More generally, ifJ is any category with aninitial objecti, then any diagram of shapeJ has a limit, namely any object isomorphic toF(i). Such an isomorphism uniquely determines a universal cone toF.
• Topological limits. Limits of functions are a special case oflimits of filters, which are related to categorical limits as follows. Given atopological spaceX, denote byF the set of filters onX,x ∈X a point,V(x) ∈F theneighborhood filter ofx,A ∈F a particular filter and${\displaystyle F_{x,A}=\{G\in F\mid V(x)\cup A\subset G\}}$ the set of filters finer thanA and that converge tox. The filtersF are given a small and thin category structure by adding an arrowA →B if and only ifA ⊆B. The injection${\displaystyle I_{x,A}:F_{x,A}\to F}$ becomes a functor and the following equivalence holds :

x is a topological limit ofA if and only ifA is a categorical limit of${\displaystyle I_{x,A}}$

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.

A given diagramF :J →C may or may not have a limit (or colimit) inC. Indeed, there may not even be a cone toF, let alone a universal cone.

A categoryC is said tohave limits of shapeJ if every diagram of shapeJ has a limit inC. Specifically, a categoryC is said to

• have products if it has limits of shapeJ for everysmall discrete categoryJ (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${\displaystyle \bullet \to \bullet \leftarrow \bullet }$ (i.e. every pair of morphisms with common codomain has a pullback).

Acomplete category is a category that has all small limits (i.e. all limits of shapeJ for every small categoryJ).

One can also make the dual definitions. A categoryhas colimits of shapeJ if every diagram of shapeJ has a colimit inC. Acocomplete category is one that has all small colimits.

Theexistence theorem for limits states that if a categoryC has equalizers and all products indexed by the classes Ob(J) and Hom(J), thenC has all limits of shapeJ.^{[1]: §V.2 Thm.1} In this case, the limit of a diagramF :J →C can be constructed as the equalizer of the two morphisms^{[1]: §V.2 Thm.2}

${\displaystyle s,t:\prod _{i\in \mathrm{Ob}(J)}F(i)\rightrightarrows \prod _{f\in \mathrm{Hom}(J)}F(\mathrm{cod}(f))}$

given (in component form) by

There is a dualexistence 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 shapeJ.

Limits and colimits are important special cases ofuniversal constructions.

LetC be a category and letJ be a small index category. Thefunctor categoryC^J may be thought of as the category of all diagrams of shapeJ inC. Thediagonal functor

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

is the functor that maps each objectN inC to the constant functor Δ(N) :J →C toN. That is, Δ(N)(X) =N for each objectX inJ and Δ(N)(f) = id_N for each morphismf inJ.

Given a diagramF:J →C (thought of as an object inC^J), anatural transformationψ : Δ(N) →F (which is just a morphism in the categoryC^J) is the same thing as a cone fromN toF. To see this, first note that Δ(N)(X) =N for all X implies that the components ofψ are morphismsψ_X :N →F(X), which all share the domainN. Moreover, the requirement that the cone's diagrams commute is true simply because thisψ is a natural transformation. (Dually, a natural transformationψ :F → Δ(N) is the same thing as a co-cone fromF toN.)

Therefore, the definitions of limits and colimits can then be restated in the form:

• A limit ofF is a universal morphism from Δ toF.
• A colimit ofF is a universal morphism fromF to Δ.

Like all universal constructions, the formation of limits and colimits is functorial in nature. In other words, if every diagram of shapeJ has a limit inC (forJ small) there exists alimit functor

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

which assigns each diagram its limit and eachnatural transformation η :F →G the unique morphism lim η : limF → limG commuting with the corresponding universal cones. This functor isright adjoint to the diagonal functor Δ :C →C^J.This adjunction gives a bijection between the set of all morphisms fromN to limF and the set of all cones fromN toF

${\displaystyle \mathrm{Hom}(N,limF)\cong \mathrm{Cone}(N,F)}$

which is natural in the variablesN andF. The counit of this adjunction is simply the universal cone from limF toF. If the index categoryJ isconnected (and nonempty) then the unit of the adjunction is an isomorphism so that lim is a left inverse of Δ. This fails ifJ is not connected. For example, ifJ is a discrete category, the components of the unit are thediagonal morphisms δ :N →N^J.

Dually, if every diagram of shapeJ has a colimit inC (forJ small) there exists acolimit functor

${\displaystyle \mathrm{colim}:{\mathcal{C}}^{\mathcal{J}}\to \mathcal{C}}$

which assigns each diagram its colimit. This functor isleft adjoint to the diagonal functor Δ :C →C^J, and one has a natural isomorphism

${\displaystyle \mathrm{Hom}(\mathrm{colim}F,N)\cong \mathrm{Cocone}(F,N).}$

The unit of this adjunction is the universal cocone fromF to colimF. IfJ 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 arecovariant functors.

One can useHom functors to relate limits and colimits in a categoryC to limits inSet, thecategory of sets. This follows, in part, from the fact the covariant Hom functor Hom(N, –) :C →Setpreserves all limits inC. By duality, the contravariant Hom functor must take colimits to limits.

If a diagramF :J →C has a limit inC, denoted by limF, there is acanonical isomorphism

${\displaystyle \mathrm{Hom}(N,limF)\cong lim\mathrm{Hom}(N,F-)}$

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

One can use the above relationship to define the limit ofF inC. The first step is to observe that the limit of the functor Hom(N,F–) can be identified with the set of all cones fromN toF:

${\displaystyle lim\mathrm{Hom}(N,F-)=\mathrm{Cone}(N,F).}$

The limiting cone is given by the family of maps π_X : Cone(N,F) → Hom(N,FX) whereπ_X(ψ) =ψ_X. If one is given an objectL ofC together with anatural isomorphismΦ : Hom(L, –) → Cone(–,F), the objectL will be a limit ofF with the limiting cone given byΦ_L(id_L). In fancy language, this amounts to saying that a limit ofF is arepresentation of the functor Cone(–,F) :C →Set.

Dually, if a diagramF :J →C has a colimit inC, denoted colimF, there is a unique canonical isomorphism

${\displaystyle \mathrm{Hom}(\mathrm{colim}F,N)\cong lim\mathrm{Hom}(F-,N)}$

which is natural in the variableN 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 diagramF as a representation of the functor Cocone(F, –).

LetI be a finite category andJ be a smallfiltered category. For anybifunctor

${\displaystyle F:I\times J\to \mathbf{S}\mathbf{e}\mathbf{t},}$

there is anatural isomorphism

${\displaystyle {\mathrm{colim}}_J\underset{I}{lim}F(i,j)\to \underset{I}{lim}{\mathrm{colim}}_{J}F(i,j).}$

In words, filtered colimits inSet commute with finite limits. It also holds that small colimits commute with small limits.^[2]

IfF :J →C is a diagram inC andG :C →D is afunctor then by composition (recall that a diagram is just a functor) one obtains a diagramGF :J →D. A natural question is then:

“How are the limits ofGF related to those ofF?”

A functorG :C →D induces a map from Cone(F) to Cone(GF): ifΨ is a cone fromN toF thenGΨ is a cone fromGN toGF. The functorG is said topreserve the limits ofF if (GL,Gφ) is a limit ofGF whenever (L,φ) is a limit ofF. (Note that if the limit ofF does not exist, thenGvacuously preserves the limits ofF.)

A functorG is said topreserve all limits of shapeJ if it preserves the limits of all diagramsF :J →C. For example, one can say thatG preserves products, equalizers, pullbacks, etc. Acontinuous functor is one that preserves allsmall limits.

One can make analogous definitions for colimits. For instance, a functorG preserves the colimits ofF ifG(L,φ) is a colimit ofGF whenever (L,φ) is a colimit ofF. Acocontinuous functor is one that preserves allsmall colimits.

IfC is acomplete category, then, by the above existence theorem for limits, a functorG :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 ofadjoint 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 diagramF :J →C and functorG :C →D, if bothF andGF have specified limits there is a unique canonical morphism

${\displaystyle {\tau }_F:GlimF\to limGF}$

which respects the corresponding limit cones. The functorG preserves the limits ofF if and only if this map is an isomorphism. If the categoriesC andD have all limits of shapeJ then lim is a functor and the morphisms τ_F form the components of anatural transformation

${\displaystyle \tau :Glim\to lim{G}^J.}$

The functorG preserves all limits of shapeJ if and only if τ is a natural isomorphism. In this sense, the functorG can be said tocommute with limits (up to a canonical natural isomorphism).

Preservation of limits and colimits is a concept that only applies tocovariant functors. Forcontravariant functors the corresponding notions would be a functor that takes colimits to limits, or one that takes limits to colimits.

A functorG :C →D is said tolift limits for a diagramF :J →C if whenever (L,φ) is a limit ofGF there exists a limit (L′,φ′) ofF such thatG(L′,φ′) = (L,φ). A functorGlifts limits of shapeJ if it lifts limits for all diagrams of shapeJ. One can therefore talk about lifting products, equalizers, pullbacks, etc. Finally, one says thatGlifts limits if it lifts all limits. There are dual definitions for the lifting of colimits.

A functorGlifts limits uniquely for a diagramF if there is a unique preimage cone (L′,φ′) such that (L′,φ′) is a limit ofF andG(L′,φ′) = (L,φ). One can show thatG lifts limits uniquely if and only if it lifts limits and isamnestic.

Lifting of limits is clearly related to preservation of limits. IfG lifts limits for a diagramF andGF has a limit, thenF also has a limit andG preserves the limits ofF. It follows that:

• IfG lifts limits of all shapeJ andD has all limits of shapeJ, thenC also has all limits of shapeJ andG preserves these limits.
• IfG lifts all small limits andD is complete, thenC is also complete andG is continuous.

The dual statements for colimits are equally valid.

LetF :J →C be a diagram. A functorG :C →D is said to

• create limits forF if whenever (L,φ) is a limit ofGF there exists a unique cone (L′,φ′) toF such thatG(L′,φ′) = (L,φ), and furthermore, this cone is a limit ofF.
• reflect limits forF if each cone toF whose image underG is a limit ofGF is already a limit ofF.

Dually, one can define creation and reflection of colimits.

The following statements are easily seen to be equivalent:

• The functorG creates limits.
• The functorG lifts limits uniquely and reflects limits.

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

• Everyrepresentable functorC →Set preserves limits (but not necessarily colimits). In particular, for any objectA ofC, this is true of the covariantHom functor Hom(A,–) :C →Set.
• Theforgetful functorU :Grp →Set creates (and preserves) all small limits andfiltered colimits; however,U does not preserve coproducts. This situation is typical of algebraic forgetful functors.
• Thefree functorF :Set →Grp (which assigns to every setS thefree group overS) is left adjoint to forgetful functorU and is, therefore, cocontinuous. This explains why thefree product of two free groupsG andH is the free group generated by thedisjoint union of the generators ofG andH.
• The inclusion functorAb →Grp creates limits but does not preserve coproducts (the coproduct of two abelian groups being thedirect sum).
• The forgetful functorTop →Set lifts limits and colimits uniquely but creates neither.
• LetMet_c be the category ofmetric spaces withcontinuous functions for morphisms. The forgetful functorMet_c →Set lifts finite limits but does not lift them uniquely.

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${\displaystyle \mathrm{Hom}}$". Terms like "cohomology" and "cofibration" all have a slightly stronger association with the first variable, i.e., the contravariant variable, of the${\displaystyle \mathrm{Hom}}$ bifunctor.

• Cartesian closed category – Type of category in category theory
• Limits and colimits in an ∞-category

• Adámek, Jiří; Horst Herrlich; George E. Strecker (1990).Abstract and Concrete Categories(PDF). John Wiley & Sons.ISBN 0-471-60922-6.
• Mac Lane, Saunders (1998).Categories for the Working Mathematician.Graduate Texts in Mathematics. Vol. 5 (2nd ed.).Springer-Verlag.ISBN 0-387-98403-8.Zbl 0906.18001.
• Borceux, Francis (1994). "Limits".Handbook of categorical algebra. Encyclopedia of mathematics and its applications 50-51, 53 [i.e. 52]. Vol. 1. Cambridge University Press.ISBN 0-521-44178-1.

• Interactive Web page which generates examples of limits and colimits in the category of finite sets. Written byJocelyn Paine.
• Limit at thenLab

• Limits (category theory)

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