Category theory

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The category theory or categorical algebra is a branch of mathematics , the beginning of 1940 years first as part of the topology has been developed; Saunders MacLane calls his "General Theory of Natural Equivalences" (in Trans. Amer. Math. Soc. 58, 1945), which he wrote in collaboration with Samuel Eilenberg in 1945, as the first explicitly category-theoretical work. The basic concepts of this theory are category , functor and natural transformation . In order to clarify the latter term, the first two were originally introduced.

The category theory, like the universal algebra , can be understood as a general theory of mathematical structures (classical structures are e.g. groups , rings , modules and topological spaces ). The properties of mathematical structures are not defined via relations between elements of the carrier set (s), but rather using morphisms and functors, as it were, via comparisons both within and between categories.


This type of abstraction not only leads to a clarification of fundamental, cross-theoretical terms, it also enables successful methods and concepts of a special mathematical theory to be transferred to other areas and object classes.
An illustrative example is provided by the history of homological algebra , the methods of which were first restricted to Abelian groups , then generalized to modules over rings and finally, as the theory of Abelian categories , transferred to Abelian sheaves .

The category theory is also relevant for basic questions. Thus forming Topoi , category theory extracts the category of amounts in the important properties of sets purely theoretically arrow (ie morphisms be formulated), an alternative to axiomatic set-theoretic construction of mathematics. In addition, category theory plays a role in logic , theoretical computer science ( semantics of programming languages , domain theory , graph grammars ) and mathematical physics ( topological quantum field theory ).

Because of its high level of abstraction, category theory is sometimes referred to as general nonsense , even by the mathematicians who developed it .



A category consists of the following:

  • A class of objects .
  • A class of so-called arrows or morphisms . A morphism is a member of a class which to each pair of objects are (with , , or indicated). These classes are pairwise disjoint ; H. no morphism , also written, is an element of another class of morphisms. is the source of a morphism and is also referred to with (from English domain ), the target with (from co-domain ).
  • Link images
which are associative in the obvious sense:
provided and .
(Occasionally, the omitted, and as written.)
  • an identity morphism for each object , which is a neutral element for the connection with morphisms with source or target , d. H. it applies if is and if . Instead , the shape is also used .

The class of all morphisms is also referred to as or (from English arrow , French flèche , German arrow ).


A sub-category of a category is a category such that a sub- class is of and is a subset of for every two objects and in the set of morphisms . If the morphism sets are equal to those of , is a full subcategory. A full sub-category is already determined by specifying the objects.

Dual category

The dual category for a category is the category with and


The linkage maps and identity morphisms are the same as in . To put it simply, all arrows point in the other direction. The category is the same .

Product category

The product category to two categories and is the category whose objects are exactly the pairs with and and whose morphisms are given by


The linking of morphisms is done component-wise, i. H. , and it is .


A (covariant) functor is a structurally compatible mapping between categories. A functor from a category to a category consists of the following data:

  • an assignment
  • Illustrations for two objects each , from .

The mappings between the morphism sets must have the following properties:

  • They are compatible with shortcuts; H. .
  • You get Identitätsmorphismen: .

A contravariant functor (or cofunctor ) from to is a functor . The description as above is equivalent to this, with the following differences:

  • The images on the morphism sets go from to .
  • The compatibility with the shortcuts is .

A functor from a category into itself is called an endofunctor .

If categories and as well as co- or contravariant functors are, then the concatenation is formalized by

for objects and morphisms is a functor . is covariant if and only if and both are co- or both are contravariant, otherwise contravariant.

Natural transformation

Natural transformations are a kind of mapping between "parallel" functors. We assume functors and that both go from the same category into the same category . For every object of, a natural transformation from to contains a morphism called a component of at . The following diagram must commute for each morphism between objects of :

As a formula, this means: .

Of course , two functors and from to are equivalent if there are natural transformations and such that and are each the identity. In other words: Natural equivalence is the isomorphism concept in the functor category . A natural transformation is a natural equivalence if and only if every component is an isomorphism.

Equivalence of categories : A functor is called an equivalence of categories if there is a functor such that and are each of course equivalent to the identity of or . One can show that equivalences of categories are exactly the fully faithful , essentially surjective functors.



Note: The names for special categories are extremely inconsistent in the literature. Often a description of the category is put in round or curly brackets, e.g. B. (Groups), or underlined.

  • The category Set , Ens or Me (from English set , French ensemble , German quantity ) is the category of the quantities . The category consists of the class that contains all sets, and the set of morphisms contains exactly the mappings from to , ie the link between two morphisms is the concatenation of the mappings.
  • PoSet or Pos is called the category of semi-ordered sets (objects) and monotonous mappings (morphisms).
  • Top designates the category of topological spaces (objects) and continuous mappings (morphisms). An interesting sub-category is, for example, the full KHaus sub-category of the compact Hausdorff rooms .
  • the category NLinSp of the normalized linear spaces with the continuous (= restricted) linear mappings. Subcategories are e.g. B. the Banach spaces with continuous linear mappings ( BanSp 1 ), the Banach spaces with continuous norm-reducing maps ( BanSp 2 ), or commutative complex Banach algebras with unity and norm-reducing algebra homomorphisms ( CBanAlg ).
  • The category of the small categories Cat or Kat : A category is called small if the class of its morphisms is a lot. Cat's objects are the small categories and the morphisms are the functors. (The restriction to small categories is necessary for reasons of set theory .)
  • A set with a partial order defines a category: Objects are the elements of the set, and have exactly one element (e.g. the ordered pair ) if , and otherwise be empty.
  • If this is empty, the result is a category without any objects or morphisms. It is labeled with and is called the initial or empty category. The naming comes from the fact that the initial object is in Cat .
  • If, on the other hand, there is one element, a category results which consists of exactly one object and its identity morphism. It is called the final or terminal category, which is motivated by the fact that the final object is in Cat .
  • Are and categories so you can see the functor form: objects are functors of after , morphisms are natural transformations.
  • Is a category, and an object of , the category of the objects on defined as follows: objects are morphisms in with target , and morphisms are morphisms of which with the " Strukturmorphismen " to be compatible, i. H. are , and two objects of so are morphisms of after in the morphisms of to for which applies.
  • Conversely, let * be a fixed one-point topological space. Then the category of the topological spaces under * isomorphic to the category Top * of the dotted topological spaces .

Most of the examples mentioned above are of such a nature (or can easily be adapted) that the objects are sets together with an additional structure, the morphisms are images that are compatible with this structure, and the linking of morphisms is the sequential execution of images. In this case one speaks of a specific category . However, not every category is concrete or even equivalent to a concrete category (ie can be made concrete ). For example, the following cannot be specified (without proof):

  • The category of the small categories, but with the natural equivalence classes of functors as morphisms.


Usually one only gives the assignment of the objects for functors if the mappings on the morphism sets can be easily seen from it.

  • For an object of a category is the assignment
a (covariant) functor . The functor
is contravariant. See also Hom functor .
  • It is a body and the category of vector spaces over with - linear maps as morphisms. Let it now be a contravariant functor
defined as follows:
  • For an object is the dual space of
  • For a linear mapping is
It is easy to check that and apply.
  • : assigns its group of units to a unitary ring . More generally :: assigns the group of invertible matrices to a ring .
  • The fundamental group is a functor , from the category of dotted topological spaces (the dotting indicates the base point) to the category of groups; the higher homotopy groups are functors ; the homology groups are functors ; the cohomology groups are contravariant functors .
  • Forget functors : There are obvious functors , , etc., simply "forgotten" part of the structure of an Abelian group that is the underlying amount of an Abelian group itself (but without the information that it is abelian), a topological space the Assign underlying quantity etc.
  • " Free " constructions, here free Abelian group : The Abelian group can be assigned to each set (with pointwise addition). Together with obvious mappings, namely , a functor of to results . There is then a canonical isomorphism , where is the forget function. It is said that (left-) adjoint functor is closed . Similar constructs exist for many forgetful functors.
  • Functors between categories, which are determined by semi-ordered sets (see above), are precisely monotonic mappings .

Natural transformations

  • The designations are as in the example of the functor “dual space” above. The illustrations
of a vector space into its dual space form a natural transformation
On the full subcategory of finite-dimensional vector spaces is a natural equivalence.
  • : For a ring , the group homomorphism is the determinant .
  • The Hurewicz figure

Yoneda lemma and universal constructions

Universal constructions transfer simple terms from the category of quantities to any category.

The Yoneda lemma

It is a category. The functor

the functor of an object

assigns is completely faithful . More generally applies to objects by and from :


a natural transformation is assigned (note ).

Structure transfer

The Yoneda lemma allows terms that are familiar from the category of sets to be transferred to any category. For example, a product of objects can be defined as an object for which object-wise the Cartesian product is, ie that

applies; here means a natural equivalence of functors in . This equivalence also provides morphisms for as the equivalent of . The Yoneda lemma then shows that except for canonical isomorphism it is uniquely determined: are and via are naturally equivalent functors, so and via are isomorphic.

This categorical product is “universal” in the following sense: whenever one has given images , these come from the universal images , ie there is an image , so that applies.

In addition, for each construction obtained in this way, one can form the dual construction (usually indicated by the prefix “Ko”) by moving to the dual category. For example, the co- product of objects in one category is the same as the product of the same objects in the dual category .

Correspondingly, properties of set maps can also be transferred to any categories: for example, a morphism is a monomorphism if it is object-wise injective.

Special universal constructions or terms

See also



Classic textbooks:

  • J. Adámek, H. Herrlich, GE Strecker: Abstract and concrete categories. The Joy of Cats. John Wiley, 1990.
  • Horst Herrlich, George E. Strecker: Category Theory: An Introduction. Boston 1973.
  • Saunders MacLane : Categories: Conceptual Language and Mathematical Theory. Berlin 1972, ISBN 3-540-05634-3 .
  • Saunders MacLane: Categories for the Working Mathematician. 2nd Edition. Springer, 1998, ISBN 0-387-98403-8 .
  • Bodo Pareigis : Categories and Functors. BG Teubner, Stuttgart 1969.
  • Horst Schubert : Categories I / II. Springer, 1970.

A reference work:

  • Francis Borceux: Handbook of categorical algebra. 3 vol (1: Basic category theory; 2: Categories and structures; 3: Categories of sheaves). - Cambridge 1994. (Encyclopedia of Mathematics and its Applications, 50/52) ISBN 0-521-44178-1 , ISBN 0-521-44179-X , ISBN 0-521-44180-3 .

An anthology:

Individual evidence

  1. ^ Serge Lang : Algebra . Springer, 2002, ISBN 0-387-95385-X , p. 759 .
  2. ^ Theodor Bröcker : Linear Algebra and Analytical Geometry . Springer, 2004, ISBN 3-0348-8962-3 , pp. 212 .
  3. Bodo Pareigis: Categories and Functors . Teubner, Stuttgart 1969, ISBN 3-663-12190-9 , pp. 8 , doi : 10.1007 / 978-3-663-12190-9 .

Web links