Category theory Guide, Meaning , Facts, Information and Description

Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense".

See list of category theory topics for a breakdown of relevant articles.

Table of contents

1 Background 2 Historical notes 3 Categories 3.1 Definition 3.2 Examples 3.3 Types of morphisms 4 Functors 5 Natural transformations and natural isomorphisms 5.4 Definition 5.5 Examples 6 Universal constructions, limits, and colimits 7 Equivalent categories 8 Further concepts and results 9 Types of categories 10 References 11 See also 12 External link

Background

A category attempts to capture the essence of a class of related mathematical objects, for instance the class of groups. Instead of focusing on the individual objects (groups) as has been done traditionally, the morphisms, i.e. the structure-preserving maps between these objects, are emphasized. In the example of groups, these are the group homomorphisms. Then it becomes possible to relate different categories by functors, generalizations of functions which associate to every object of one category an object of another category and to every morphism in the first category a morphism in the second. Very commonly, certain "natural constructions", such as the fundamental group of a topological space, can be expressed as functors. Furthermore, different such constructions are often "naturally related" which leads to the concept of natural transformation, a way to "map" one functor to another. Throughout mathematics, one encounters "natural isomorphisms", things that are (essentially) the same in a "canonical way". This is made precise by special natural transformations, the natural isomorphisms.

Historical notes

Categories, functors and natural transformations were introduced by Samuel Eilenberg and Saunders MacLane in 1945. Initially, the notions were applied in topology, especially algebraic topology, as an important part of the transition from homology (an intuitive and geometric concept) to homology theory, an axiomatic approach. It has been claimed, for example by or on behalf of Ulam, that comparable ideas were current in the later 1930s in the Polish school.

Eilenberg/MacLane have said that their goal was to understand natural transformations; in order to do that, functors had to be defined; and to define functors one needed categories.

The subsequent development of the theory was powered first by the computational needs of homological algebra; and then by the axiomatic needs of algebraic geometry, the field most resistant to the Russell-Whitehead view of united foundations. General category theory — an updated universal algebra with many new features allowing for semantic flexibility and higher-order logic — came later; it is now applied throughout mathematics.

Special categories called topoi can even serve as an alternative to axiomatic set theory as the foundation of mathematics. These broadly-based foundational applications of category theory are contentious; but they have been worked out in quite some detail, as a commentary on or basis for constructive mathematics. One can say, in particular, that axiomatic set theory still hasn't been replaced by the category-theoretic commentary on it, in the everyday usage of mathematicians. The idea of bringing category theory into earlier, undergraduate teaching (signified by the difference between the Birkhoff-Mac Lane and later Mac Lane-Birkhoff abstract algebra texts) has hit noticeable opposition.

Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with application to the theory of functional programming and domain theory, all in a setting of a cartesian closed category as non-syntactic description of a lambda calculus. At the very least, the use of category theory language allows one to clarify what exactly these related areas have in common (in an abstract sense).

Categories

Definition

A category consists of:

• a class of objects.
• for every two objects A and B, a set Mor(A,B) of morphisms from A to B. If f is in Mor(A,B), we write f : A → B. (Some authors write Hom(A,B) for the set of morphisms.)
• for every three objects A, B and C, a binary operation Mor(A,B) × Mor(B,C) → Mor(A,C) called composition of morphisms. The composition of f : A → B and g : B → C is written as g o f or gf. (Some authors write fg.)

such that the following axioms hold:

• (associativity) if f : A → B, g : B → C and h : C → D then h o (g o f) = (h o g) o f, and
• (identity) for every object X there exists a morphism id_X : X → X called the identity morphism for X, such that for every morphism f : A → B we have id_B o f = f = f o id_A

From these axioms, one can prove that there is exactly one identity morphism for every object. Some authors use a slight variation of the definition in which each object is identified with the corresponding identity morphism.

If the class of objects is actually a set, the category is said to be small. Many important categories are not small.

The morphisms of a category are sometimes called arrows due to the influence of commutative diagrams.

Examples

Each category is presented in terms of its objects, its morphisms, and its composition of morphisms. In the following, unless stated otherwise, whenever the morphisms are functions, the composition of morphisms is given by the usual set-theoretic composition of functions.

• The category Set of all sets together with functions between sets.
  • The category Ord of all preordered sets with monotonic functions.
  • The category Mag consisting of all magmas together with their homomorphisms.
  • The category Med consisting of all medial magmas together with their homomorphisms.
  • The category Grp consisting of all groups together with their group homomorphisms.
  • The category Ab consisting of all abelian groups together with their group homomorphisms.
  • The category Vect_K of all vector spaces over the field K (which is held fixed) together with their K-linear maps.
  • The category Top of all topological spaces with continuous functions.
  • The category Met of all metric spaces with short maps.
  • The category Uni of all uniform spaces with uniformly continuous functionss.
• Any preordered set (P, ≤) forms a small category, where the objects are the members of P, and the morphisms are arrows pointing from x to y when x ≤ y.
• Any monoid forms a small category with a single object x. (Here, x is any fixed set.) The morphisms from x to x are precisely the elements of the monoid, and the categorical composition of morphisms is given by the monoid operation. In fact, one may view categories as generalizations of monoids; several definitions and theorems about monoids may be generalized for categories.
• Any directed graph generates a small category: the objects are the vertices of the graph and the morphisms are the paths in the graph. Composition of morphisms is concatenation of paths. This is called the free category generated by the graph.
• If I is a set, the discrete category on I is the small category which has the elements of I as objects and only the identity morphisms as morphisms.
• Any category C can itself be considered as a new category in a different way: the objects are the same as those in the original category but the arrows are those of the original category reversed. This is called the dual or opposite category and is denoted by C^op.
• If C and D are categories, one can form the product category C × D: the objects are pairs consisting of one object from C and one from D, and the morphisms are also pairs, consisting of one morphism in C and one in D. Such pairs can be composed componentwise.

Types of morphisms

A morphism f : A → B is called a

• monomorphism if fg₁ = fg₂ implies g₁ = g₂ for all morphisms g₁, g₂ : X → A.
• epimorphism if g₁f = g₂f implies g₁ = g₂ for all morphisms g₁, g₂ : B → X.
• isomorphism if there exists a morphism g : B → A with fg = id_B and gf = id_A.
• automorphism if f is an isomorphism and A = B.
• endomorphism if A = B.

Relations among morphisms (such as fg = h) can most conveniently be represented with commutative diagrams, where the objects are represented as points and the morphisms as arrows.

Functors

Main article: functor.

Functors are structure-preserving maps between categories. They can be thought of as morphisms in the category of all (small) categories.

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

• associates to each object X in C an object F(X) in D;
• associates to each morphism f : X → Y a morphism F(f) : F(X) → F(Y)

such that the following two properties hold:

• F(id_X) = id_F(X) for every object X in C.
• F(g o f) = F(g) o F(f) for all morphisms f : X → Y and g : Y → Z.

A contravariant functor F from C to D is a functor that "turns morphisms around" (i.e. "reverses all the arrows"). Specifically, F is contravariant if whenever f : X → Y is a morphism in C, then F(f) : F(Y) → F(X)). The quickest way to define a contravariant functor is as a covariant functor between C^op and D.

For examples and properties of functors see the functor article.

Natural transformations and natural isomorphisms

Main article: natural transformation.

A natural transformation is a relation between two functors. Functors often describe "natural constructions" and natural transformations then describe "natural homomorphisms" between two such constructions. Sometimes two quite different constructions yield "the same" result; this is expressed by a natural isomorphism between the two functors.

Definition

If F and G are (covariant) functors between the categories C and D, then a natural transformation from F to G associates to every object X in C a morphism η_X : F(X) → G(X) in D such that for every morphism f : X → Y in C we have η_Y o F(f) = G(f) o η_X; this means that the following diagram is commutative:

The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that η_X is an isomorphism for every object X in C.

Examples

If K is a field, then for every vector space V over K we have a "natural" injective linear map V → V^** from the vector space into its double dual. These maps are "natural" in the following sense: the double dual operation is a functor, and the maps form a natural transformation from the identity functor to the double dual functor. If we restrict to finite-dimensional vector spaces, we even obtain a natural isomorphism: "Every finite-dimensional vector space is naturally isomorphic to its double dual."

Consider the category Ab of abelian groups and group homomorphisms. For all abelian groups X, Y and Z we have a group isomorphism

Mor(X, Mor(Y, Z)) → Mor(XY, Z).

These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors Ab^op × Ab^op × Ab → Ab.

Universal constructions, limits, and colimits

Main articles: universal property, limit (category theory).

Using the language of category theory, many areas of mathematical study can be cast into appropriate categories, such as the categories of all sets, groups, topologies, and so on. These categories surely have some objects that are "special" in a certain way, such as the empty set or the product of two topologies. Yet, in the definition of a category, objects are considered to be atomic, i.e. we do not know, whether an object A is a set, a topology, or any other abstract concept. Hence, the challenge is to define special objects without referring to the internal structure of these objects. But how can we define the empty set without referring to elements, or the product topology without referring to open sets?

The solution is to characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories. Thus the task is to find universal properties that uniquely determine the objects of interest. Indeed, it turns out that numerous important constructions can be described in a purely categorical way. The central concept which is needed for this purpose is called (categorical) limit, and can be dualized to yield the notion of a colimit.

Equivalent categories

Main articles: equivalence of categories, isomorphism of categories.

It is a natural question to ask, under which conditions two categories can be considered to be "essentially the same", in the sense that theorems about one category can readily be transformed into theorems about the other category. The major tool one employs to describe such a situation is called equivalence of categories. It is given by appropriate functors between two categories. Categorical equivalence has found numerous applications in mathematics.

Further concepts and results

The definitions of categories and functors provide only the very basics of categorical algebra. Additional important topics are listed below. Although there are strong interrelations between all of these topics, the given order can be considered as a guideline for further reading.

• The functor category D^C has as objects the functors from C to D and as morphisms the natural transformations of such functors. The Yoneda lemma is one of the most famous basic results of category theory; it describes representable functors in functor categories.
• Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by "reversing all the arrows". If one statement is true in a category C then its dual will be true in the dual category C^op. This duality, which is transparent at the level of category theory, is often obscured in applications and can lead to surprising relationships.
• Adjoint functors: A functor can be left (or right) adjoint to another functor that maps in the opposite direction. Such a pair of adjoint functors typically arises from a construction defined by a universal property; it can be seen as a more abstract and powerful view on universal properties.

Types of categories

• In many categories, the morphism sets Mor(A,B) are not just sets but actually abelian groups, and the composition of morphisms is compatible with these group structures, i.e. is bilinear. Such a category is called preadditive. If furthermore the category has all finite products and coproducts, it is called an additive category. If all morphisms have a kernel and a cokernel, and all epimorphism are cokernels and all monomorphisms are kernels, then we speak of an abelian category. A typical example of an abelian category is the category of abelian groups.
• A category is called complete if all limits in it exist. The categories of sets, abelian groups and topological spaces are complete.
• A category is called cartesian closed if it has finite direct products and a morphism defined on a finite product can always be represented by a morphism defined on just one of the factors.
• A topos is a certain type of cartesian closed category in which all of mathematics can be formulated (just like classically all of mathematics is formulated in the category of sets). A topos can also be used to represent a logical theory.
• A groupoid is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups, group actions and equivalence relations.

References

• Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990). Abstract and Concrete Categories. Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition)
• Barr, Michael, & Wells, Charles (2002). Toposes, Triples and Theories. (revised and corrected free online version of Grundlehren der mathematischen Wissenschaften (278). Springer-Verlag,1983)
• Borceux, Francis (1994). Handbook of Categorical Algebra.. Vols. 50-52 of Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press.
• Lawvere, William, & Schanuel, Steve. (1997). Conceptual Mathematics: A First Introduction to Categories.. Cambridge: Cambridge University Press.
• Mac Lane, Saunders (1998). Categories for the Working Mathematician (2nd ed.). Graduate Texts in Mathematics 5. Springer. ISBN 0-387-98403-8.

See also

• — wikibook link
• Important publications in category theory

External link

• "Category Theory" in Stanford Encyclopedia of Philosophy
• Homepage of the Categories mailing list, with extensive list of resources;
• Category Theory section of Alexandre Stefanov's list of free online mathematics resources.

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