category theory

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". The phrase originated at a time when category theory was novel and some considered it too abstract to be useful or interesting. This derogatory designation was then co-opted by advocates of the new theory. In any case, the term should not be interpreted that category theory is nonsensical or non-rigourous; indeed, an acquaintance with the basic notions of category theory is now considered part of the standard education of a mathematician, and category theory has even found applications in modern physics and theoretical computer science. See list of category theory topics for a breakdown of relevant articles.

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 mathematical theories have traditionally done, category theory emphasizes the morphisms the structure-preserving maps between these objects. 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. The non-mathematician might think of a category as a road map. Just as a road map gives little information about individual towns and cities, a category theorist is in some sense "unconcerned" about the actual objects in the category. What is important, in both cases, is how we get from one place (or object) to another. Thus, the roads on a map are analogous to the morphisms of the category. Just as some roads are more important than others, some morphisms are more interesting than others. The isomorphisms, in particular, are those morphisms that are "two-way", or "reversible".

Historical notes

Categories, functors and natural transformations were introduced by Samuel Eilenberg and Saunders Mac Lane 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/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 theoryan updated universal algebra with many new features allowing for semantic flexibility and higher-order logiccame 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).

Morphisms

Main article morphism 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 it is both monic and epic; equivalently, if there exists a morphism g : b → a with fg = 1_b and gf = 1_a.
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).

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(1_x) = 1_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" ("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. 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:

Diagram defining natural transformations

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

hom(x, hom(y, z)) → hom(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.

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 generalise this by considering "higher-dimensional processes". For example, (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. 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 John Baez: The Tale of n-categories.

References

Admek, Jiř, 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.