In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. Accordingly, the dual notion of a colimit, generalizes disjoint unions and direct sums. Limits and colimits have strong relationships to the categorial concepts of universal morphisms and adjoint functors. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to glue together several related objects, the precise matter of the gluing process being specified by morphisms between the objects. ... In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...

Definition

Before defining limits, it is useful to define the auxiliary notion of a cone of a functor. Cones are also perhaps more aptly called sources. Therefore consider two categories J and C and a covariant functor F : J → C. In this situation J is also called a schema. A cone of F is an object L of C, together with a family of morphisms φ_X : L → F(X), one for each object X of J, such that for every morphism f : X → Y in J, we have F(f) o φ_X = φ_Y. This situation may be depicted by a commutative diagram: For functors in computer science, see the function object article. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

The (usually infinite) collection of all these triangles can be (partially) depicted in the shape of a cone with the apex L. Image File history File links FunctorCone-01. ... A cone is a basic geometrical shape: see cone (solid). ...

A limit of a functor is just a universal cone. In detail, a cone (L, φ_X) of a functor F : J → C is a limit of that functor iff for any cone (N, ψ_X) of F, there exists precisely one morphism u : N → L such that φ_X o u = ψ_X for all X. We may say that the morphisms ψ_X factor through L with the unique factorization u. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

As for every universal property, this definition describes a balanced state of generality: The limit object L has to be general enough to allow any other cone to factor through it; on the other hand, L has to be sufficiently specific, so that only one such factorization is possible for every cone. Image File history File links FunctorCone-03. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

It is possible that a functor does not have a limit at all. However, if a functor has two limits then there exists a unique isomorphism between the respective limit objects which commutes with the respective cone maps; this isomorphism is given by the unique factorization from one limit to the other. Thus limits are unique up to isomorphism and can be denoted by lim F. Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ...

Examples

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_X) of a functor F : J → C.

• Terminal objects. If J is the empty category, then the above definitions imply that every object of C is a cone of F. The limit of F is any object that has a unique factorization through any other object. This is just the definition of a terminal object.

• Products. If J is a discrete category then the functor 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 special case where J consists of just two objects (which we will call 1 and 2) then defines a binary product. For example, assume that C is the category Set and let J be the discrete two-element category. The binary product L will then just be the cartesian product F(1) × F(2) in Set. The morphisms φ₁ and φ₂ are the projections to the respective components of the tuples from L.

  In many algebraic contexts, such as (abelian) groups, rings, boolean algebras, etc., products are just direct products, where the operations are defined pointwise. Another well-known product is the product topology in Top, the category of topological spaces and continuous maps. If a partially ordered set is viewed as a category C, then products in C are greatest lower bounds while arbitrary cones are just lower bounds. In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... In category theory, a discrete category is a category whose only morphisms are the identity morphisms. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after René Descartes... In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... The category Top has topological spaces as objects and continuous maps as morphisms. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ...

• Equalizers. If J is a two-object category with two parallel morphisms from object 1 to object 2 then the limit L is called an equalizer.

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

• Pullbacks. Let F be a functor 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 poset (considered as a small category by adding arrows i → j iff i ≤ j) and let F : J → C be a contravariant functor. The limit of F is called (confusingly) an inverse limit or projective limit but also known as a directed limit.

All of the above examples follow a common scheme for the definition of limits: in order to model a limit construction, such as a product of sets, one uses a functor that "picks out" the relevant objects (and sometimes morphisms) from the category C. Consequently, the category J is usually a small category and has fewer elements than the category C. If one considers a finite category J then the above constructions can also be specified by giving the objects and morphisms that the functor F maps to. For example one may talk about an "equalizer of two morphisms" instead of calling this limit an "equalizer of a functor that maps the only two non-trivial morphisms in J to certain values". This article is about equalisers in mathematics. ... In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. ... In category theory, a zero morphism is a special kind of trivial morphism. ... In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ... Image File history File links CategoricalPullback-01. ... In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to glue together several related objects, the precise matter of the gluing process being specified by morphisms between the objects. ... In mathematics, a directed set is a set A together with a binary relation ≤ having the following properties: a ≤ a for all a in A (reflexivity) if a ≤ b and b ≤ c, then a ≤ c (transitivity) for any two a and b in A, there exists a c in A... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... For functors in computer science, see the function object article. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

However, J may well be a large category, i.e. one that has a proper class of objects. For example, the product of all sets exists and is just the empty set (indeed, this is the only possible cone on all families of sets that contain the empty set). In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...

Complete categories

A category C is called complete iff every functor F : J → C, where J is any small category, has a limit; i.e. "all small limits in C exist". Similarly, if every such functor with J finite has a limit, then C is said to have finite limits. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

Many important categories are complete: groups, abelian groups, sets, modules over some ring, topological spaces and compact Hausdorff spaces. Typical examples of categories that are not complete are categories with some "size restriction": the category of finite groups or the category of finite-dimensional vector spaces over a fixed field. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

The Existence Theorem for Limits states that a category is complete iff it has equalizers and products over arbitrary sets of objects. Note that one doesn't require products of proper classes of objects to exist. It turns out that the property of having all (even large) limits is too strong to be practically relevant. Any category with this property necessarily is of a very restricted form: for any two objects there can be at most one morphism from one object to the other. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... This article is about equalisers in mathematics. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ...

If J is a particular small category and every functor from J to C has a limit, then the limit operation forms a functor from the functor category 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 J-indexed 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 J indexed system of abelian groups its corresponding inverse limit. In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ... In category theory, a discrete category is a category whose only morphisms are the identity morphisms. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ... In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ... In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to glue together several related objects, the precise matter of the gluing process being specified by morphisms between the objects. ...

Continuous functors

It is a natural question to ask, which functors are compatible with the construction of limits in the sense that they map limits to limits. These functors are called continuous or limit preserving. Formally, a functor G : C → D is continuous iff, for every small category I and every functor F : I → C that has a limit (L,φ_X) in C, the functor GF : I → D has the limit (G(L), G(φ_X) ). Since the Existence Theorem for Limits mentioned above shows that all limits can be expressed by products and equalizers, it is sufficient for continuity if G preserves these special limits.

Important examples of continuous functors are given by representable ones: if U is some object of C, then the functor G_U : C → Set with G_U(V) = Mor_C(U, V) for all objects V in C is continuous. In category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. ...

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. This illustrates that one may also say that a continuous functor commutes with the formation of limits. The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ...

Being a universal construction, limits also have other strong relationships to adjoint functors. 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. In particular, limit functors are continuous; intuitively, this means that the order in which two limits are taken doesn't matter. The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...

Colimits

The dual notion of limits and cones are colimits and co-cones. Co-cones are also known as sinks. Although it is straightforward to obtain these definitions by inverting all morphisms in the above definitions, we will explicitly state them here: In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...

Consider two categories J and C and a covariant functor F : J → C. A co-cone of F is an object L of C, together with a family of 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 o F(f)= φ_X. Again, the commutative diagram for this situation resembles a cone (this time pointing downwards): In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

A colimit of a functor is a universal co-cone: a co-cone (L, φ_X) of a functor F : J → C is a colimit of F iff for any co-cone (N, ψ_X) of F, there exists precisely one morphism u : L → N such that u o φ_X = ψ_X for all X. Image File history File links FunctorCone-02. ...

If it exists, the colimit of F is unique up to a unique isomorphism and is denoted by colim F. Image File history File links FunctorCone-04. ...

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

• Initial objects
• Coproducts
• Coequalizers
• Cokernels
• Pushouts
• Direct limits

The category C is called co-complete if every functor F : J → C with small J has a colimit. Similarly, a category is said to have finite colimits if every such functor with finite J has a colimit. In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ... In mathematics, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. ... In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f. ... In category theory, a branch of mathematics, the pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain. ... In mathematics, the direct limit (also called the inductive limit) is a general method of taking limits of directed families of objects. We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category. ...

The following categories are co-complete: sets, groups, abelian groups, modules over some ring and topological spaces.

A covariant functor that commutes with the construction of colimits is said to be cocontinuous or colimit preserving. Every functor which has a right adjoint (and hence is a left adjoint) is cocontinuous. The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...

As an example in the category of groups, Grp, the functor F : Set → Grp which assigns to every set S the free group over S has a right adjoint (the forgetful 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. In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. ... The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... A forgetful functor is a type of functor in mathematics. ... In abstract algebra, the free product of groups constructs a group from two or more given ones. ...

Limits and colimits are related as follows: A functor F : J → C has a colimit iff for every object N of C, the contravariant functor G : J → Set defined by G(X) = Mor_C(F(X), N) has a limit. If that is the case, then ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

Mor_C(colim F, N) = lim G

for every object N of C.

Creation of Limits and Co-Limits

Limits for a diagram D: J → X are said to be created by a functor F: X → Y if every limit cone of F o D has a unique pre-image cone under F in X, and additionally this unique pre-image is a limit cone of D.

For example, the forgetful functor from Grp to Set creates limits, that is to say, for example that the product of groups is determined by their underlying sets. This is in a way characteristic of algebraic situations.

There is a dual notion for co-limits. In the case of groups the aforementioned forgetful functor creates only filtered co-limits. In category theory, filtered categories generalize the notion of directed set. ...