Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". Categories appear in most branches of mathematics, in some areas of theoretical computer science and mathematical physics, and have been a unifying notion. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1945, in connection with algebraic topology.

See Category theory topics for a breakdown of relevant articles.

Background

The study of categories is an attempt to capture what is commonly found in various classes of related mathematical structures.

Consider the following example. The class Grp of groups consists of all objects having a "group structure". More precisely, Grp consists of all sets G endowed with a binary operation satisfying a certain set of axioms. One can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proved from the axioms that the identity element of a group is unique.

Instead of focusing merely on the individual objects (groups) possessing a given structure, as mathematical theories have traditionally done, category theory emphasizes the morphisms — the structure-preserving processes — between these objects. It turns out that by studying these morphisms, we are able to learn more about the structure of the 'objects' (groups). Here the morphisms are the group homomorphisms. A group homomorphism between two groups "preserves the group structure" in a very precise way — it is a "process" taking one group to another, in a way that carries along information about the structure of the first group into the second group. The study of group homomorphisms then provides a tool for studying general properties of groups and consequences of the group axioms.

A similar type of investigation occurs in many mathematical theories. A category is an axiomatic formulation of this idea of relating mathematical structures to the structure-preserving functions between them. A systematic study of categories then allows us to prove general results from the axioms of a category.

A category is itself a type of mathematical structure, so we can look for 'processes' which preserve this structure in some sense. Such a process is called a functor. It associates to every object of one category an object of another category; and to every morphism in the first category a morphism in the second. By studying categories and functors, we are not just studying a class of mathematical structures and the morphisms between them, we are studying the relationships between various classes of mathematical structure. This is a fundamental idea, which first surfaced in algebraic topology. Difficult topological questions can be translated into algebraic questions which are much easier to solve. Basic constructions, such as the fundamental group of a topological space, can be expressed as functors in this way.

Constructions are often "naturally related", a vague notion at first sight. This leads to the clarifying concept of natural transformation, a way to "map" one functor to another. Many important constructions in mathematics can be studied in this context. 'Naturality' is a principle, like general covariance in physics, that cuts deeper than is initially apparent.

Historical notes

Categories, functors and natural transformations were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1945 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 Stanislaw Ulam, that comparable ideas were current in the late 1930s in the Polish school. These ideas were in some ways a continuation of the contributions of Emmy Noether in formalizing abstract processes in the first half of the 20th-century. Noether realized that in order to understand a type of mathematical structure, one really needs to understand the processes preserving this structure. Eilenberg and Mac Lane gave an axiomatic formalization of this relation between structures and the processes preserving them.

Eilenberg/Mac Lane 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 (singular topos) 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, objects, and morphisms

A category C consists of

a class ob(C) of objects:
a class hom(C) of morphisms. Each morphism f has a unique source object a and target object b. We write f: a → b, and we say "f is a morphism from a to b". We write hom(a, b) Hom(a, b), or hom_C(a, b) to denote the hom-class of all morphisms from a to b. (Some authors write Mor(a, b) or C(a, b).)
for every three objects a, b and c, a binary operation ○: hom(a, b) × hom(b, c) → hom(a, c) called composition of morphisms; the composition of f : a → b and g : b → c is written as g ○ 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 ○ (g ○ f) = (h ○ g) ○ f, and
(identity) for every object x, there exists a morphism 1_x : x → x called the identity morphism for x, such that for every morphism f : a → b, we have 1_b ○ f = f = f ○ 1_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.

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

Some properties of morphisms

A morphism f : a → b is called

a monomorphism (or monic) if fg₁ = fg₂ implies g₁ = g₂ for all morphisms g₁, g₂ : x → a.
an epimorphism (or epic) if g₁f = g₂f implies g₁ = g₂ for all morphisms g₁, g₂ : b → x.
an isomorphism if there exists a morphism g : b → a with fg = 1_b and gf = 1_a.Note that a morphism that is both epic and monic is not necessarily an isomorphism! For example, in the category consisting of two objects A and B, the identity morphisms, and a single morphism f from A to B, f is both epic and monic but is not an isomorphism.
an endomorphism if a = b. The class of endomorphisms of a is denoted end(a).
an automorphism if f is both an endomorphism and an isomorphism. The class of automorphisms of a is denoted aut(a).

Functors

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(1_x) = 1_F(x) for every object x in C.
F(g ○ f) = F(g) ○ 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" ("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 from the opposite category C^op to D.

Natural transformations and isomorphisms

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.

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 ○ F(f) = G(f) ○ 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.

Universal constructions, limits, and colimits

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

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.

Higher-dimensional categories

Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can be situated into the context of higher-dimensional categories. Briefly, if we consider a morphism between two objects as a "process taking us from one object to another", then higher-dimensional categories allow us to profitably generalize this by considering "higher-dimensional processes".

For example, a (strict) 2-category is a category together with "morphisms between morphisms", i.e. processes which allow us to transform one morphism into another. We can then "compose" these "bimorphisms" both horizontally and vertically, and we require a 2-dimensional "exchange law" to hold, relating the two composition laws. In this context, the standard example is Cat, the 2-category of all (small) categories, and in this example, bimorphisms of morphisms are simply natural transformations of morphisms in the usual sense. Another basic example is to consider a 2-category with a single object—these are essentially monoidal categories. Bicategories are a weaker notion of 2-dimensional categories where the composition of morphisms is not strictly associative, but only associative "up to" an isomorphism.

This process can be extended for all natural numbers n, and these are called n-categories. There is even a notion of ω-category corresponding to the ordinal number ω. For a conversational introduction to these ideas, see Baez (1996).

Notes

References

Available online for free:

Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990) Abstract and concrete categories. John Wiley & Sons. ISBN 0-471-60922-6.
Barr, Michael, & Wells, Charles (2002) Toposes, triples and theories. Revised and corrected translation of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, 1983).
Leinster, Tom (2004) Higher operads, higher categories (London Math. Society Lecture Note Series 298). Cambridge Univ. Press.

Other:

Borceux, Francis (1994) Handbook of categorical algebra (Encyclopedia of Mathematics and its Applications 50-52). Cambridge Univ. Press.
Freyd, Peter J. & Scedrov, Andre, (1990) Categories, allegories (North Holland Mathematical Library 39). North Holland.
Lawvere, William, & Schanuel, Steve (1997) Conceptual mathematics: a first introduction to categories. Cambridge University Press.
Mac Lane, Saunders (1998) Categories for the Working Mathematician. 2nd ed. (Graduate Texts in Mathematics 5). Springer-Verlag.
Pedicchio, Maria Cristina & Tholen, Walter (2004) Categorical foundations (Encyclopedia of Mathematics and its Applications 97). Cambridge Univ. Press.
Taylor, Paul, 1999. Practical Foundations of Mathematics. Cambridge University Press.

External links

Stanford Encyclopedia of Philosophy: ""Category Theory" by Jean-Pierre Marquis.
Homepage of the Categories mailing list, with extensive list of resources
Baez, John, 1996,"The Tale of n-categories." An informal introduction to higher order categories.

Category theory | Higher category theory

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