Functor

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

Definition

Be categories. An assignment is said to be ( covariant ) functor if

• Objects are mapped to objects:
• There is for every two objects and pictures
• Morphisms all, for which is defined, and is also defined and
• (if this is different from the objects ) of the corresponding object to a Identitätsmorphismus is imaged on the image corresponding to Identitätsmorphismus.

It follows that even for.

A covariant functor on the dual category is called the contravariant functor (or Kofunktor ) and can be regarded as a representation of by, by identifying the morphisms in and with each other. Specifically, a picture is just a contravariant functor then when

• Is defined for all morphisms, for which is also defined and
• Objects are mapped to objects and
• Identitätsmorphismen as previously mapped to the appropriate Identitätsmorphismen.

Examples

• The identical functor which assigns to each morphism itself, is a covariant functor.
• Frequently encountered are forget- functors: For example, in the category of groups the objects groups, amounts thus provided with a link, and the morphisms group homomorphisms, so certain mappings between these levels. The concatenation of morphisms is nothing but the concatenation of pictures. The forgetful is now a functor in the category of sets, he "forgets" the additional structure and assigns each group the underlying amount and any group homomorphism corresponding picture on this lot too. Corresponding Forget functors are available for all categories of algebraic structures or for categories of topological spaces with continuous maps, etc.
• Dual category consists of the same category of a morphism, except that the linkage is inversely defined. The Dualitätsfunktor that associates itself every morphism, so is a contravariant functor.
• In the category of sets we define the Potenzmengenfunktor, of any amount allocates its power set and each figure maps the preimage education. The Potenzmengenfunktor is contravariant. Similar functors also appear in other categories, where you allow only certain pictures as morphisms, and instead of the power set and figures considered certain associations and homomorphisms between them between them, behold some representation theorem for Boolean algebras.

Elementary Properties

• The concatenation of two covariant functors is a covariant functor again.
• 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 again an isomorphism.
• The image of a retraction or a Koretraktion under a covariant functor is again a retraction or a Koretraktion.
• The image of an epimorphism or a monomorphism under a covariant functor is generally no epimorphism or monomorphism, since the Kürzbarkeit does not have to be preserved by a Nichtsurjektivität the functor.

Multi functors

Be a family of categories with respect to a ( small ) amount given. A covariant functor from the category of products in a category now called covariant Multifunktor. Now we consider also multi- functors which are in some components contravariant co- and in some. exactly it means Multifunktor of variance ( see the covariance contravariance up) when he construed as a representation of

After a covariant Multifunktor. A Multifunktor on the product of two categories ie Bifunktor. Restricting the domain of an Multifunktors in individual components on a single object a, we obtain a partial functor, also a Multifunktor, which retains its variance in the remaining components.

Remark

The variance of a functor is not unique in general. Trivial example: On the category that consists only of a single object with its Identitätsmorphismus, the Identitätsfunktor is co- and contravariant. This is also true in general categories whose morphisms are all automorphisms, so that the automorphism groups are abelian. Example of ambiguity in multi- functors is a canonical projection of a product category in a component, this functor is co- and contravariant in all other components of both.

Examples

• A everywhere in the category theory particularly important functor is the Hom - functor that for each locally small category on the product defined as Bifunktor of the variance in the category of sets: first is for two objects in the category as the set of all morphisms from to defined. Was for two morphisms in

• The Kronecker product is a Bifunktor of the variance in the category of matrices ( this is also true for general tensor ).
• In the homological algebra of Ext- functor and the Tor functor play a special role.

Properties of functors

As with most mathematical structures common, it is natural to consider injective, surjective and bijective functors. The inverse function of a bijective functor, as with all algebraic structures in turn a functor, one speaks therefore in this case by a isomorphism between categories. This is Isomorphismenbegriff but unnatural for the category theory in a certain sense: For the structure of a category, it does, in essence, no matter if more objects exist isomorphic to an object. The morphisms of two isomorphic objects to any object correspond to each other perfectly, and vice versa. However, for an isomorphism in the above sense, it makes a difference how many ( assumed to move in a small category, so that one can speak of numbers ) isomorphic objects are each available, a property that does not matter for category theoretical considerations generally. Such numbers may depend about from totally inconsequential details in the construction of a category - defined to differentiable manifolds as subsets of (in the case there is a set of all manifolds ) or as any quantities with a differentiable structure ( these form a proper class )? Are any two zero-dimensional vector spaces equal to ( according to the manner of speech of the zero vector space ) or just isomorphic? etc. Therefore, we define some properties of functors that are " insensitive" Adding or removing isomorphic objects:

A functor is called faithful if any two different morphisms are mapped between the same objects on the same morphism, that is, it is injective on each class of morphisms between and. Similarly, it is called full if it is surjective on each class. A fully faithful functor is a functor which is full and faithful. A much surjective functor is now a functor such that for each object exists in an isomorphic object that is in the picture. An equivalence is now a functor is fully faithful and essentially surjective. This is in some ways a more natural Isomorphiebegriff for categories dar. Although an equivalence has no inverse function in the literal sense, but rather something similar in the form of an equivalence from to, so that when concatenation of the two equivalences objects are mapped to isomorphic objects. If one considers instead of categories only skeletons of categories, so the concept of equivalence with the isomorphism agrees.

Natural transformations

Functors can not be understood only as morphisms in categories of categories, but can also be regarded as objects of categories. As morphisms between functors one usually considered in that case natural transformations.

Diagrams and Limites

Many terms are defined in mathematics over commutative diagrams. For example, can the inverse of a morphism in a category defined such that the following diagram commutes:

This can be formalized so that a functor from a category with two objects and two non-identical morphisms between them exists ( according to the shape of the diagram) in the category, so that the image is of a non-identical morphism and that of others. This functor is then also called chart. As a generalization of typical definitions of universal properties, there is the concept of the limit of a functor.