Functor (math)

Functors are a central basic concept of the mathematical subfield of category theory . A functor is a structure- preserving mapping between two categories. Concrete functors have a special meaning in many areas of mathematics. Functors are also called diagrams (sometimes only in certain contexts) because they represent a formal abstraction of commutative diagrams .

definition

Be categories. An assignment is called a ( covariant ) functor if and only if ${\displaystyle {\mathcal {C}},{\mathcal {D}}}$ ${\displaystyle F\colon {\mathcal {C}}\to {\mathcal {D}}}$

• Objects are mapped onto objects: ${\displaystyle F:{\rm {{Ob}({\mathcal {C}})\to {\rm {{Ob}({\mathcal {D}})}}}}}$
• it two objects and of figures are ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle F\colon \operatorname {Mor} _{\mathcal {C}}(X,Y)\to \operatorname {Mor} _{\mathcal {D}}(F(X),F(Y))}$
• for all morphisms for which is defined, and ${\displaystyle u,v\in {\mathcal {C}}}$ ${\displaystyle uv}$ ${\displaystyle F(u)F(v)=F(uv)}$
• (if one differentiates these from the objects) the identity morphism associated with an object is mapped onto the identity morphism associated with the image.

It follows that for is also . ${\displaystyle {\mathcal {C}}\ni u\colon A\to B}$ ${\displaystyle F(u)\colon F(A)\to F(B)}$

A functor from a category to itself ( ) is called an endofunctor . ${\displaystyle F\colon {\mathcal {C}}\to {\mathcal {C}}}$

A covariant functor on the dual category , is referred to as counter-functor (or Kofunktor ) referred to and may be used as imaging to be viewed by the morphisms in and identified with each other. Specifically, a mapping is a contravariant functor if and only if ${\displaystyle F\colon {\mathcal {C}}^{\operatorname {op} }\to {\mathcal {D}}}$ ${\displaystyle F\colon {\mathcal {C}}\to {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}^{\operatorname {op} }}$ ${\displaystyle F\colon {\mathcal {C}}\to {\mathcal {D}}}$

• Objects are mapped onto objects and
• it two objects and of pictures are (note the changed order) ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle F\colon \operatorname {Mor} _{\mathcal {C}}(X,Y)\to \operatorname {Mor} _{\mathcal {D}}(F(Y),F(X))}$
• for all morphisms for which is defined is also defined and (note the changed order) ${\displaystyle u,v\in {\mathcal {C}}}$ ${\displaystyle uv}$ ${\displaystyle F(v)F(u)}$ ${\displaystyle F(v)F(u)=F(uv)}$
• Identity morphisms are mapped to the matching identity morphisms as before.

A functor enables the transition from one category to another, whereby the rules mentioned apply. The existence of these rules is also called the functoriality of this transition, or it is said that the construction on which this transition is based is functorial .

Examples

• The identical functor that assigns itself to each morphism is a covariant functor. ${\displaystyle id\colon {\mathcal {C}}\to {\mathcal {C}}}$
• If the category is the vector spaces with the linear mappings as morphisms and assigns its dual space to each vector space and the dual map to each linear map , then is a contravariant functor. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle F:{\mathcal {C}}\rightarrow {\mathcal {C}}}$ ${\displaystyle V}$ ${\displaystyle V^{*}}$ ${\displaystyle f\colon V\rightarrow W}$ ${\displaystyle f^{*}\colon W^{*}\rightarrow V^{*}}$ ${\displaystyle F}$
• Forgetting functors are often encountered : For example, in the category of groups, the objects are groups, that is, sets with a link, and the morphisms are group homomorphisms , that is, certain images between these sets. The concatenation of morphisms is nothing more than the concatenation of images. The forgetting functor is now a functor in the category of sets, it “forgets” the additional structure and assigns the underlying set to each group and the corresponding mapping on this set to each group homomorphism. Corresponding forgetful functors are available for all categories of algebraic structures or for categories of topological spaces with continuous mappings, etc.
• The dual category of a category consists of the same morphisms, but the concatenation is defined in reverse. The duality functor, which assigns itself to each morphism, is therefore a contravariant functor. ${\displaystyle \omega \colon {\mathcal {C}}\to {\mathcal {C}}^{\operatorname {op} }}$
• On the category of sets the power set functor is defined , which assigns its power set to each set and assigns the archetype formation to each mapping . The power set functor is contravariant. Similar functors also appear in other categories in which only certain mappings are allowed as morphisms and, instead of the power set and mappings between them, certain associations and homomorphisms between them are considered, see for example the notation for Boolean algebras . ${\displaystyle f\colon A\to B}$ ${\displaystyle {\mathcal {P}}(B)\to {\mathcal {P}}(A),S\mapsto f^{-1}(S)}$

Elementary properties

• The concatenation of two covariant functors is again a covariant functor.
• The concatenation of two contravariant functors is a covariant functor.
• The concatenation of a covariant with a contravariant functor is a contravariant functor.
• The image of an isomorphism under a functor is in turn an isomorphism.
• The image of a retraction or a coretraction under a covariant functor is in turn a retraction or a coretraction.
• The image of an epimorphism or a monomorphism under a covariant functor is generally not an epimorphism or monomorphism, since the ability to shorten the functor does not have to be retained because the functor is nonsurjective.
• The image of a functor is generally not a sub-category of the target category because different objects can be mapped onto the same object, so that concatenations of morphisms of the image of the functor no longer have to be in the image. Consider as a category with objects and morphisms , and a category with objects and morphisms , , . is a functor with , , , , . Then lie and in the picture of , but not . ${\displaystyle {\mathcal {C}}}$ ${\displaystyle A,B_{0},B_{1},C}$ ${\displaystyle u\colon A\to B_{0}}$ ${\displaystyle v\colon B_{1}\to C}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle A,B,C}$ ${\displaystyle f\colon A\to B}$ ${\displaystyle g\colon B\to C}$ ${\displaystyle gf\colon A\to C}$ ${\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}}$ ${\displaystyle F(A)=A}$ ${\displaystyle F(B_{0})=F(B_{1})=B}$ ${\displaystyle F(C)=C}$ ${\displaystyle F(u)=f}$ ${\displaystyle F(v)=g}$ ${\displaystyle F(u)}$ ${\displaystyle F(v)}$ ${\displaystyle F}$ ${\displaystyle F(v)F(u)=gf}$

Multifunctions

Given a family of categories with respect to a (small) set . A covariant functor from a product category to a category is now called a covariant multifunctional . Now one also looks at multifunctionals that are co-variant in some components and contravariant in some. is called the multifunctional of the variance (the show covariance, the contravariance) if and only if it is understood as a mapping of ${\displaystyle ({\mathcal {C}}_{i})_{i\in I}}$ ${\displaystyle I}$ ${\displaystyle F}$ ${\displaystyle \textstyle \prod _{i}{\mathcal {C}}_{i}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle F\colon \textstyle \prod _{i}{\mathcal {C}}_{i}\to {\mathcal {D}}}$ ${\displaystyle v\colon I\to \{0,1\}}$ ${\displaystyle 0}$ ${\displaystyle 1}$

${\displaystyle \prod _{i}{\begin{cases}{\mathcal {C}}_{i}&{\text{wenn }}v(i)=0\\{\mathcal {C}}_{i}^{\operatorname {op} }&{\text{wenn }}v(i)=1\end{cases}}}$

after is a covariant multifunctional. A multifunctional on the product of two categories is called a bi-functional . If you restrict the definition range of a multifunctional in individual components to a single object, you get a partial functor , also a multifunctional, which retains its variance in the remaining components. ${\displaystyle {\mathcal {D}}}$

comment

The variance of a functor is generally ambiguous. Trivial example: In the category that consists of only one single object with its identity morphism, the identity function is co- and contravariant. This also applies more generally in categories whose morphisms are all automorphisms, so that the automorphism groups are Abelian. An example of ambiguity in multifunctionals is a canonical projection of a product category into a component; this functor is both co- and contravariant in all other components.

Examples

• A particularly important functor everywhere in category theory is the Hom functor , which is defined for each locally small category on the product as a bifunctional of the variance in the category of sets: For two objects in the category, let first be the set of all morphisms from to Are defined. For two morphisms in is ${\displaystyle \operatorname {Hom} _{\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}\times {\mathcal {C}}}$ ${\displaystyle (1,0)}$ ${\displaystyle A,B}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \operatorname {Hom} _{\mathcal {C}}(A,B)}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle f\colon A\to B,g\colon C\to D}$ ${\displaystyle C}$

${\displaystyle \operatorname {Hom} _{\mathcal {C}}(f,g)\colon \operatorname {Hom} _{\mathcal {C}}(B,C)\to \operatorname {Hom} _{\mathcal {C}}(A,D),m\mapsto gmf}$

Are defined. For every object are the partial Hom-functors or co- and contravariant functors. ${\displaystyle A}$ ${\displaystyle \operatorname {Hom} (A,-)}$ ${\displaystyle \operatorname {Hom} (-,A)}$

• The Kronecker product is a bifunctional of the variance in the category of matrices (this also applies more generally to tensor products ). ${\displaystyle (0,0)}$
• In homological algebra , the Ext functor and the Tor functor play a special role.

Properties of functors

As is usual with most mathematical structures, it makes sense to consider injective , surjective and bijective functors. The inverse function of a bijective functor is, like all algebraic structures , a functor, so in this case one speaks of an isomorphism between categories. However, this concept of isomorphism is in a certain sense unnatural for category theory: For the structure of a category it is essentially irrelevant whether there are other isomorphic objects for an object. The morphisms of two isomorphic objects to any one object correspond perfectly to one another, and vice versa. For an isomorphism in the above sense, however, there is a difference in how many (assuming you are moving in a small category so that you can speak of numbers ) isomorphic objects are present, a property that is generally irrelevant for category-theoretical considerations . Such numbers can depend on completely irrelevant details in the construction of a category - do you define differentiable manifolds as subsets of the (in this case there is a set of all manifolds) or as arbitrary sets with a differentiable structure (these form a real class )? Are two zero-dimensional vector spaces the same (according to the way of speaking, the zero vector space ) or only isomorphic? etc. Therefore, one defines certain properties of functors that are "insensitive" to adding or removing isomorphic objects: ${\displaystyle \mathbb {R} ^{n}}$

A functor is called true if no two different morphisms between the same objects are mapped onto the same morphism, i.e. i.e., it is injective on any class of morphisms between and . Similarly, it is called full if it is surjective in every class . A fully faithful functor is a functor that is full and faithful. An essentially surjective functor is now a functor, so that for every object in there exists an isomorphic object that lies in the image of . An equivalence is now a functor that is completely faithful and essentially surjective. In a certain sense, this represents a more natural isomorphic term for categories. An equivalence does not have an inverse function in the literal sense, but something similar in the form of an equivalence of to , so that when the two equivalences are linked, objects are mapped onto isomorphic objects. If one only looks at the skeletons of categories instead of categories, the concept of equivalence corresponds to that of isomorphism. ${\displaystyle F\colon {\mathcal {C}}\to {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}(A,B)}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle {\mathcal {C}}(A,B)}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle F}$ ${\displaystyle D}$ ${\displaystyle C}$

Natural transformations

→ Main article : Natural transformation

Functors can not only be understood as morphisms in categories of categories, but can also be viewed as objects of categories. Natural transformations are usually considered as morphisms between functors .

Diagrams and Limits

→ Main article : Limes (category theory)

Many terms in mathematics are defined using commutative diagrams . For example, the inverse of a morphism in a category can be defined so that the following diagram commutes : ${\displaystyle f^{-1}}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {C}}}$

This can be formalized in such a way that a functor of a category with two objects and two non-identical morphisms between them (according to the shape of the diagram) exists in the category so that the image of one is non-identical morphism and that of the other . This functor is then also called a diagram. As a generalization of typical definitions of universal properties , the term Limes of a functor emerges . ${\displaystyle {\mathcal {C}}}$ ${\displaystyle f}$ ${\displaystyle f^{-1}}$

See also

• Adjunction (category theory)

literature

• Jiří Adámek, Horst Herrlich, George E. Strecker: Abstract and Concrete Categories: The Joy of Cats . John Wiley & Sons, New York 1990, ISBN 0-471-60922-6 (English, uni-bremen.de [PDF; 4.2 MB ]).
• Dieter Pumplün: Elements of the category theory . 1st edition. Spektrum Akademischer Verlag , Heidelberg 1999, ISBN 3-86025-676-9 .

Web links

• principle of equivalence , explanation in the nLab , accessed on August 22, 2012