**Category** theory is a mathematical theory of which we can say in principle that it deals abstractly with mathematical structures and their relationships. We say “in principle” because this theory has given rise to new fundamental unifications and visions of mathematics that would modify the very meaning of what we say to be “objective”. You can see a text that affects it and that contains a simpler version of the fundamental definition of what a category is here: Categories and foundations .

Also look at the list of topics in category theory if you need a quick look at the various concepts.

Category theory was introduced in Algebraic Topology , by Eilenberg and MacLane, in an important step for the transition from homology (an intuitive geometric concept) to Theory of Homology , an axiomatic matter . For example, it has been claimed that similar ideas existed in the Polish school of the 1930s ( Ulam ).

Subsequent developments in the theory were driven by the computational needs of homological algebra and later by the needs for axiomatics in algebraic geometry , which was the most reluctant field to go through the ring of unifying foundations “à la” Russell-Whitehead. The general theory – some updating of Universal Algebra with many new features that gave rise to some flexibility in higher-order semantics and logic – came later.

The applications of this theory in foundations are provoking a certain “fight” typical of any historical moment of incorporation and complementation – in terms of fundamental theories. Similarly, when Set Theory came to fertilize mathematics in the way we know, there were negative reactions to “the new” as fundamental and unifying.

These applications of categories in the field of fundamentals are being worked on in considerable detail and not only in mathematics. There are mathematicians like William Lawvere who work on fundamentals of physics, there are physicists working on John C. Baez n-categories, and there are even philosophers like Alain Badiou in France or Corfield in England who have been forced to put their “inquiries” under the conditions of contemporary mathematics. An idea about what is happening is revealed in some few classic textbooks in the Anglo-Saxon world: *Birkhoff-Mac Lane’s* “abstract algebra” which later becomes the same “abstract algebra” but *Mac Lane-Birkhoff’s*. The Logical Categorical is now a field well defined based on type theory for intuitionistic logic , with applications to the theory of functional programming and domain theory , all framed in a cartesian closed category as syntactical descriptions of the lambda calculus . Using the language of category theory allows one to clarify what exactly all these areas have in common.

## Summary

[ hide ]

- 1 Categories
- 1 Definition
- 2 Examples

- 2 Types of maps
- 3 Special items in a category
- 4 Founders
- 1 Definition
- 2 Examples

- 5 Universal constructions
- 6 Other concepts and results
- 7 References
- 8 Sources

## Categories

With the concept of **category we** intend to capture – emphasizing the concept of relationship, arrow, rather than element and belonging – the essence of a class of mathematical objects, which are related by arrows, the morphisms in the category in question . For example, the class of the groups . Instead of studying the individual objects (each group) as it was being done, these morphisms between them are emphasized, which are nothing more than the applications that “conserve their structure”. In the example of groups, said morphisms are group homomorphisms . So, once we have our defined “categorial universe” -that is, a category- it is possible to relate it to other categories through functors, which are a certain generalization of the concept of function in the case of categories: a functor associates each object in one category with an object in the other, and each arrow in the first with an arrow in the second. In a way, it “shapes” us, it takes us an image of the category towards the other category and with certain degrees of “refinement”. Certain “natural constructions”, like the fundamental group of a topological space , can be expressed as functors. Furthermore, such functors are very often naturally related, and this leads to the concept of natural transformation .

Historically, it was precisely the motivation to clarify certain natural transformations in Algebraic Topology that served to define the functors and the categories. And immediately, the theory was applied in Homological Algebra and in Algebraic Geometry ; it is now used in many branches of mathematics, physics, and philosophy.

Special categories like the Moles are also serving as an alternative to Set Theory as the foundation of mathematics.

### Definition

One category consists of:

- a classof things called objects. (A class is “more” than a set, we do not allow a class of its own to be an element of itself).
- and for each pair of objects
*A*and*B*one set Mor (*A*,*B*) of things called*morphisms**of A to B*. If*f*is in said set Mor (*A*,*B*), write*f*:*A*->*B*. - for every three objects
*A*,*B*and*C*there is a binary operation Mor (*A*,*B*) x Mor (*B*,*C*) -> Mor (*A*,*C*) called*composition of morphisms*. The composition of*f*:*A*->*B*and*g*:*B*->*C*is written like this:*g*or*f*or*gf*. (And some authors:*fg*.)

such that all follow the following axioms:

- (associativity) if
*f*:*A*->*B*,*g*:*B*->*C*and*h*:*C*->*D*then*h*o (*g*or*f*) = (*h*or*g*) or*f*, and - (identity) for each object
*X*there is a morphism**id**:_{X}*X*->*X*called*identity morphism in X*, such that for every morphism*f*:*A*->*B*we have**id**or_{B}*f*=*f*=*f*or**id**_{A}

From these axioms it can be proved that there is only one identity morphism for each object.

If the class of objects is only a set, not a class, the category is said to be “small”. There are important categories that are not.

### Examples

Each category is presented in terms of its objects and morphisms.

- The
**Mag**category of all magmas along with their homomorphisms .- The
**Med**category of all medial magmas along with their homomorphisms .

- The
- The
**Grp**category consisting of all groups along with their group homomorphisms . - The
**Vect**category of all vector spaces on the_{K}*K*body along with its*K*– linear applications . - The
**Met**category of all metric spaces together with short functions . - The
**Uni**category of all uniform spaces together with unimorphisms . - The
**Top**category of all topological spaces together with continuous functions .- The
**Ord**category of all preordered sets along with increasing functions .- The
**Set**category of all sets along with the functions between sets.

- The
- A partially ordered set(
*P*, ≤) is a small category, the objects of which are the members of*P*, and the maps are the arrows from*x*to*y*when*x*≤*y*occurs . - A monoidis a small category with a single object
*x*, y in which every element of the monoid is viewed as a morphism from ‘x*to*x*(the operation given on the monoid enforces the categorical rule of composition). In fact the categories can be seen as generalizations of the monoids; Various definitions and theorems about monoids can be generalized for categories.* - The graphcan be considered as a small category: the objects would be the vertices of the graph and the morphisms would be the paths in the graph. The composition of morphisms is the concatenation of paths.
- If
*I*is a set , the category:*discrete category over I*is the small category that has the elements of*I*as objects and only the identity morphisms (which exist in every category, as you will remember). - Any category
*C*can be considered a new category looking at it in a certain different way: the objects would be the same as those of the original but the arrows are those of the original but with the arrows “backwards”, that is, they change the direction. It is called the*dual*category or*opposite category*and is denoted by*C*^{op}. - If
*C*and*D*are categories, a certain product can be formed, the*product of categories C*x*D*: the objects are the pairs consisting of one object of each, one of*C*and the other of*D*, and the morphisms would also be the pairs , consisting of a morphism*C*and another*D*. Such pairs can be compound component by component.

- The

## Types of maps

A morphism *f* : *A *→ *B* is called

*monomorphism*if*fg*=_{1}*fg*implies_{2}*g*=_{1}*g*for all morphisms_{2}*g*,_{1}*g*:_{2}*X*→*A*.*epimorphism*if*g*_{1}*f*=*g*_{2}*f*implies*g*=_{1}*g*for all morphisms_{2}*g*,_{1}*g*:_{2}*B*->*X*.*isomorphism*if a morphism*g*:*B*→*A*with*fg*=**id**and_{B}*gf*=**id**. A morphism (bijective) that is, injective and over is called isomorphism: it is both monomorphism and epimorphism_{A}^{ [1]}*automorphism*if*f*is an isomorphism and*A*=*B*.*endomorphism*if f is a morphism and*A*=*B*.

## Special objects in a category

An object *A* of category *C* is called

- Initial, if for every object
*B*of the category exists one and only one morphism to him*A*->*B*. For example, the Empty Set is an initial object in the sets category. - terminal, or end, if for each object
*B*exists one and only one morphism*B*->*A*. For example, each singletton (set with a single element) is a terminal object in the sets category. - zero, if it is initial and terminal at the same time.

## Founders

Functors are cross-category applications that preserve structure.

### Definition

A *functor* (covariant) *F* from category *C* to category *D*

- associate each object
*X*in*C with*an object*F*(*X*) in*D*; - associates each morphism
*f*:*X*->*Y with*a morphism*F*(*f*):*F*(*X*) ->*F*(*Y*)

fulfilling the following pair of properties

*F*(**id**) =_{X}**id**_{F}_{ ( X )}for every object*X*in*C*.*F*(*g*or*f*) =*F*(*g*) or*F*(*f*) for all maps*f*:*X*->*Y*and*g*:*Y*->*Z*.

A *counter variant functor F* from *C* to *D* is a functor that “turns the morphisms *(that is, if* f *:* X *->* Y *is a morphism in* C *, then* F *(* f *):* F *(* Y *) ->* F *(* X *)), the fastest way to define a contravariant functor is to covariant functor between* C ^{p}* and* D *.*

An important consequence of the axioms for functor is this: if *f* is an isomorphism in *C* , then *F* ( *f* ) is also in *D* .

### Examples

**Dual vector space:** An example of a counter variant functor from the category of all real vector spaces to the category of all real vector spaces is given by the assignment to each object (each real vector space) of an object called dual space and to each morphism that is, to each linear application ) its dual or transpose.

**Algebra of continuous functions:** a counter variant functor from the category of topological spaces (whose morphisms are continuous applications) to the category of real associative algebras , is given by assigning to each topological space *X* the algebra C ( *X* ) of all continuous real functions on such space. Each continuous application *f* : *X *-> *Y* (morphism in the category of topological spaces) induces a homomorphism of algebras C ( *f* ): C ( *Y* ) -> C ( *X* ) by means of the rule C ( *f* ) (φ) = φ or *f*for all φ in C ( *Y* ).

**Homomorphism:** each pair *A* , *B* of abelian groups can be assigned the abelian group Hom ( *A* , *B* ) consisting of all homomorphisms from *A* to *B* . This is a functor that is counter variant in the first argument and covariant in the second, that is, it is a functor **Ab **^{op} x **Ab **-> **Ab** (where **Ab** denotes the category of abelian groups with group homomorphisms). If *f* : *A *_{1 }-> *A *_{2} and*g* : *B *_{1 }-> *B *_{2} are morphisms in **Ab** , so we have this homomorphism Hom ( *f* , *g* ): Hom ( *A *_{2} , *B *_{1} ) -> Hom ( *A *_{1} , *B *_{2} ) given by φ | -> *g* or φ or *f* .

**Forgetful functors:** the functor *F* : **Ring** -> **Ab** that applies a ring ring towards its underlying abelian group is a “forgetful” functor, who forgets, who creates an image of something “richer” in a poorer object, with less structure. The morphisms in the category of **Rings** (ring homomorphisms) become morphisms in **Ab** (the category of abelian groups and their homomorganisms).

**Tensor products:** If *C* denotes the category of vector spaces on a fixed body, with linear applications as morphisms, then the tensor product *V *File: DirectProduct.png *W* defines a functor *C* x *C *-> *C* that is covariant in both arguments .

**Lie Algebras:** Each real or complex Lie group is assigned its real (or complex) Lie Algebra , thereby defining a functor.

**Fundamental group: It** considers the category of topological spaces with “base points”, with “distinguished points”. Objects are the pairs ( *X* , *x* ), where *X* is a topological space and *x* is an element of *X* . A map from ( *X* , *x* ) to ( *Y* , *y* ) is given by a continuous application *f* : *X *-> *Y* such that *f* ( *x* ) = *y* .

For each topological space with base point ( *X* , *x* ), we will define a fundamental group . Which is going to be a functor from the category of topological spaces with base points towards the category of groups.

Let *f be* a continuous function from the unit interval [0,1] to *X* such that *f* (0) = *f* (1) = *x* . (This is equivalent to that *f* is continuous from an application circle unit in the complex plane such that *f* (1) = *x* .) We call such a loop function in *X* . If *f* and *g* are loops in *X* , we can glue them one after the other by defining *h* ( *t* ) = *f* (2 *t* ) when *t*go through [0,0.5] and *h* ( *t* ) = *g* (2 ( *t* – 0.5)) when *t goes* through [0.5,1]. It is easy to verify that this *h* is also a loop. If there is a continuous application *F* ( *x* , *t* ) from [0,1] × [0.1] to *X* such that *f* ( *t* ) = *F* (0, *t* ) is a loop and *g* ( *t* ) = *F* (1 , *t* ) is also a loop so it is said that *f* and *g*they are equivalent. It can be proved that this defines an equivalence relation . Our composition rule ensures that everything goes well. Now, in addition, we can see that there is a neutral element *e* ( *t* ) = *x* (a constant application) and that each loop has an inverse loop. In fact, if *f* ( *t* ) is a loop then *f* (1 – *t* ) is its inverse. The set of loop equivalency classes then forms a group (the fundamental group of *X*). It can be verified that the application from the category of topological spaces with base point to the category of groups is functional: a (homo / iso) topological morphism will naturally correspond to a (homo / iso) morphism of groups.

**Pre-Beams**** :** If* X* is a topological space , then the open sets in* X* can be considered as the objects of a category* C*X; having a morphism from *U* to *V* if and only if *U* is a subset of *V* . In itself, this category is not very exciting, but the functors from *C*X^{op} to other categories, called *pre-beams on X* , are interesting. For example, assigning each open set *U* the associative algebra of the real functions of *U* , a pre-beam is obtained algebras over *X* .

This example of motivation is generalized by considering pre-beams on arbitrary categories: a pre-beam on *C* is a functor defined on *C *^{op} . The lemma Yoneda realizes that often a category *C* can be extended by considering the category of pre-beams onto *C* .

**The Category of the small categories:** The **Cat** category has as objects all the small categories, and as morphisms the functors among them.

## Universal constructions

Functors are often defined by universal properties ; as examples we have the tensor products above, the direct sum and the direct product of groups or vector spaces, the construction of free groups, modules, and direct and inverse limits . The concepts of limit and collimit generalize multiple concepts. Universal constructions often give rise to pairs of adjunct functors .

## Other concepts and results

The definitions of categories and functors provide us with only the initial basis for categorical algebra. The topics listed below are very important. Although there are strong interrelationships between all of them, the order in which we give them can be considered a guide for further reading.

- natural transformation: While functors give one way to go, printing one category on another, natural transformations provide us with a similar relationship between functors.
- The motto of Yonedais one of the most famous results of the theory of categories.
- Limits and collimits: In order to introduce certain constructions such as products (of sets, of topologies, of partial orders, …), in theory, limits and collimits are helpful.
- adjunct functors: A functor may be the adjutant on the left (or on the right) of another functor going in the opposite direction. However, when we compare them with the classic relations of the applications that preserve the structures (inverses …), the concept of adjunct of functors appears to be quite abstract and general. It is still very useful and is related to many other important concepts, such as in the construction of limits.
- equivalence of categories: In order to obtain an adequate criterion to discern whether or not two categories can be considered similar, it is necessary to find a more general notion than the classical concept of isomorphism . The equivalences of categories are closely related to related with
**duality of categories**. - Commutative diagrams: Since category theory usually deals with objects and arrows, it is convenient to express identities by diagrams.