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Diagram (category theory)

 

 
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Diagram (category theory)
 
 
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In category theory
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
, a branch of mathematics, a diagram is the categorical analogue of a indexed family
Indexed family

In mathematics, an indexed family of sets is defined in stages, beginning with the more general concept of an indexed family of elements, which an alternative way of conceptualizing the range of a function or mapping....
 in set theory
Set theory

Set theory is the branch of mathematics that studies Set , which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics....
. The primary difference is that in the categorical setting one has morphism
Morphism

In mathematics, a morphism is an Abstraction derived from structure-preserving map between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory....
s as well: an indexed family of sets is a collection of sets, indexed by a fixed set (equivalently, a function from a fixed index set to the class of sets), while a diagram is a collection of objects and morphisms, indexed by a fixed category (equivalently, a functor from a fixed index category to some category). Formally, a diagram is an element of a functor category
Functor category

In category theory, a branch of mathematics, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors....
.

Diagrams are used in the definition of limit and colimits
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 product and inverse limits....
 and the related notion of cone
Cone (category theory)

In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well....
s.

D : J → C
The category J is called the index category or the scheme of the diagram D.






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In category theory
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
, a branch of mathematics, a diagram is the categorical analogue of a indexed family
Indexed family

In mathematics, an indexed family of sets is defined in stages, beginning with the more general concept of an indexed family of elements, which an alternative way of conceptualizing the range of a function or mapping....
 in set theory
Set theory

Set theory is the branch of mathematics that studies Set , which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics....
. The primary difference is that in the categorical setting one has morphism
Morphism

In mathematics, a morphism is an Abstraction derived from structure-preserving map between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory....
s as well: an indexed family of sets is a collection of sets, indexed by a fixed set (equivalently, a function from a fixed index set to the class of sets), while a diagram is a collection of objects and morphisms, indexed by a fixed category (equivalently, a functor from a fixed index category to some category). Formally, a diagram is an element of a functor category
Functor category

In category theory, a branch of mathematics, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors....
.

Diagrams are used in the definition of limit and colimits
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 product and inverse limits....
 and the related notion of cone
Cone (category theory)

In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well....
s.

Definition


Formally, a diagram of type J in a category
Category (mathematics)

In mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships. A category is composed of a collection of abstract "objects" of any kind, linked together by a collection of abstract "morphism" of any kind that have a few basic properties ....
 C is a functor
Functor

In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of small categories....
D : J → C
The category J is called the index category or the scheme of the diagram D. The actual objects and morphisms in J are largely irrelevant, only the way in which they are interrelated matters. The diagram D is thought of as indexing a collection of objects and morphisms in C patterned on J.

Although, technically, there is no difference between an individual diagram and a functor or between a scheme and a category, the change in terminology reflects a change in perspective, just as in the set theoretic case: one fixes the index category, and allows the functor (and, secondarily, the target category) to vary.

One is most often interested in the case where the scheme J is a small or even finite
Finite

Finite is the opposite of infinite. It may refer to:* Having a finite number of elements: finite set* Being a finite number, so not equal to ; all real numbers are finite...
. A diagram is said to be small or finite whenever J is.

One may defined a morphism of diagrams of type J in a category C as a natural transformation
Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved....
 between functors. One can then interpret the category of diagrams of type J in C as the functor category
Functor category

In category theory, a branch of mathematics, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors....
 CJ, and a diagram is then an object in this category.

Examples


  • If J is a (small) discrete category
    Discrete category

    In mathematics, especially category theory, a discrete category is a category whose only morphisms are the identity morphisms. It is the simplest kind of category....
    , then a diagram of type J is essentially just an indexed family of objects in C (indexed by J).


  • If J is a poset category then a diagram of type J is a family of objects Di together with a unique morphism fij : Di ? Dj whenever i = j. If J is directed
    Directed set

    In mathematics, a directed set is a nonempty Set A together with a reflexive relation and transitive relation binary relation = , with the additional property that every pair of elements has an upper bound....
     then a diagram of type J is called a direct system of objects and morphisms. If the diagram is contravariant then it is called an inverse system
    Inverse system

    In mathematics, an inverse system in a category C is a functor from a small filtered category I to C. An inverse system is sometimes called a pro-object in C....
    .


  • If , then a diagram of type J is called "two parallel morphisms": its limit is an equalizer, and its colimit is a coequalizer
    Coequalizer

    In mathematics, a coequalizer is a generalization of a quotient set by an equivalence relation to objects in an arbitrary category . It is the categorical construction dual to the equaliser ....
    .


  • If J = -1 ? 0 ? +1, then a diagram of type J (A ? B ? C) is a span
    Span (category theory)

    A span, in category theory, is a generalization of the notion of binary relation between two objects of a category. When the category has all Pullback , spans can be considered as morphisms in a Localization of a category....
    , and its colimit is a pushout
    Pushout (category theory)

    In category theory, a branch of mathematics, a pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain: it is the colimit of the Span ....
    .


  • If J = -1 ? 0 ? +1, then a diagram of type J (A ? B ? C) is a cospan, and its limit is a pullback
    Pullback (category theory)

    In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X ? Z and g : Y ? Z with a common codomain; it is the limit of the cospan ....
    .


Cones and limits


A cone
Cone (category theory)

In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well....
 of a diagram D : J ? C is a morphism from the constant diagram ?(N) to D. The constant diagram is the diagram which sends every object of J to an object N of C (and every morphism to the identity morphism on N).

The limit
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 product and inverse limits....
 of a diagram D is a universal cone to D. That is, a cone through which all other cones uniquely factor. If the limit exists in a category C for all diagrams of type J one obtains a functor
lim : CJ → C
which sends each diagram to its limit.

Dually, the colimit of diagram D is a universal cone from D. If the colimit exists for all diagrams of type J one has a functor
colim : CJ → C
which sends each diagram to its colimit.

Commutative diagrams


Diagrams and functor categories are often visualized by commutative diagrams, particularly if the index category is a finite poset category with few elements: one draws a commutative diagram with a node for every object in the index category, and an arrow for a generating set of morphisms, omitting identity maps and morphisms that can be expressed as compositions. The commutativity corresponds to the uniqueness of a map between two objects in a poset category. Conversely, every commutative diagram represents as diagram (a functor from a poset index category) in this way.

Not every diagram commutes, as not every index category is a poset category: most simply, the diagram of a single object with an endomorphism , or with two parallel arrows (; ) need not commute. Further, diagrams may be impossible (because infinite) or messy (because many objects or morphisms) to draw; however, schematic commutative diagrams (for subcategories of the index category, or with ellipses, such as for a directed system) are used to clarify such complex diagrams.

See also


  • Commutative diagram
    Commutative diagram

    In mathematics, and especially in category theory a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition....
  • Functor category
    Functor category

    In category theory, a branch of mathematics, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors....


Limits

  • Colimit
  • Cone (category theory)
    Cone (category theory)

    In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well....
  • Limit (category theory)
    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 product and inverse limits....


Examples

  • Indexed family
    Indexed family

    In mathematics, an indexed family of sets is defined in stages, beginning with the more general concept of an indexed family of elements, which an alternative way of conceptualizing the range of a function or mapping....
  • Direct system
  • Inverse system
    Inverse system

    In mathematics, an inverse system in a category C is a functor from a small filtered category I to C. An inverse system is sometimes called a pro-object in C....
  • Span
    Span (category theory)

    A span, in category theory, is a generalization of the notion of binary relation between two objects of a category. When the category has all Pullback , spans can be considered as morphisms in a Localization of a category....
  • Cospan