III.8 Categories
Eugenia Cheng
When we study GROUPS [I.3 §2.1] or VECTOR SPACES [I.3 §2.3], we pay particular attention to certain classes of maps between them: the important maps between groups are the group HOMOMORPHISMS [I.3 §4.1], and the important maps between vector spaces are the LINEAR MAPS [I.3 §4.2]. What makes these maps important is that they are the functions that “preserve structure”: for example, if φ is a homomorphism from a group G to a group H, then it “preserves multiplication,” in the sense that φ(g1g2) = φ(g1)φ(g2) for any pair of elements gl and g2 of G. Similarly, linear maps preserve addition and scalar multiplication.
The notion of a structure-preserving map applies far more generally than just to these two examples, and one of the purposes of category theory is to understand the general properties of such maps. For instance, if A, B, and C are mathematical structures of some given type, and f and g are structure-preserving maps from A to B and from B to C, respectively, then their composite g o f is a structure-preserving map from A to C. That is, structure-preserving maps can be composed (at least if the range of one equals the domain of the other). We also use structure-preserving maps to decide when to regard two structures as “essentially the same”: we call A and B isomorphic if there is a structure-preserving map from A to B with an inverse that also preserves structure.
A category is a mathematical structure that allows one to discuss properties such as these in the abstract. It consists of a collection of objects, together with morphisms between those objects. That is, if a and b are two objects in the category, then there is a collection of morphisms between a and b. There is also a notion of composition of morphisms: if f is a morphism from a to b and g is a morphism from b to c, then there is a composite of f and g, which is a morphism from a to c. This composition must be associative. In addition, for each object a there is an “identity morphism” which has the property that if you compose it with another morphism f then you get f.
As the earlier discussion suggests, an example of a category is the category of groups. The objects of this category are groups, the morphisms are group homomorphisms, and composition and the identity are defined in the way we are used to. However, it is by no means the case that all categories are like this, as the following examples show.
(i) We can form a category by taking the natural numbers as its objects, and letting the morphisms from n to m be all the n × m matrices with real entries. Composition of morphisms is the usual matrix multiplication. We would not normally think of an n × m matrix as a map from the number n to the number m, but the axioms for a category are nevertheless satisfied.
(ii) Any set can be turned into a category: the objects are the elements of the set, and a morphism from x to y is the assertion “x = y.” We can also make an ordered set into a category by letting a morphism from x to y be the assertion “x≤ y.” (The “composite” of “x≤ y” and “y≤ z” is “x≤ z.”)
(iii) Any group G can be made into a category as follows: you have just one object, and the morphisms from that object to itself are the elements of the group, with the group multiplication defining the composition of two morphisms.
(iv) There is an obvious category where the objects are TOPOLOGICAL SPACES [III.90] and the morphisms are continuous functions. A less obvious category with the same objects takes as its morphisms not continuous functions but HOMOTOPY CLASSES [IV.6 §2] of continuous functions.
Morphisms are also called maps. However, as the above examples illustrate, the maps in a category do not have to be remotely map-like. They are also called arrows, partly to emphasize the more abstract nature of a general category, and partly because arrows are often used to represent morphisms pictorially.
The general framework and language of “objects and morphisms” enable us to seek and study structural features that depend only on the “shape” of the category, that is, on its morphisms and the equations they satisfy. The idea is both to make general arguments that are then applicable to all categories possessing particular structural features, and also to be able to make arguments in specific environments without having to go into the details of the structures in question. The use of the former to achieve the latter is sometimes referred to, endearingly or otherwise, as “abstract nonsense.”
As we mentioned above, the morphisms in a category are generally depicted as arrows, so a morphism f from a to b is depicted as 
and composition is depicted by concatenating the arrows 
This notation greatly eases complex calculations and gives rise to the so-called commutative diagrams that are often associated with category theory; an equality between composites of morphisms such as g o f = t o s is expressed by asserting that the following diagram commutes, that is, that either of the two different paths from a to c yields the same composite:
Proving that one long string of compositions equals another then becomes a matter of “filling in” the space in between with smaller diagrams that are already known to commute. Furthermore, many important mathematical concepts can be described in terms of commutative diagrams: some examples are free groups, free rings, free algebras, quotients, products, disjoint unions, function spaces, direct and inverse limits, completion, compactification, and geometric realization.
Let us see how it is done in the case of disjoint unions. We say that a disjoint union of sets A and B is a set U equipped with morphisms 
and 
such that, given any set X and morphisms 
and 
there is a unique morphism 
that makes the following diagram commute:
Here p and q tell us how A and B inject into the disjoint union. The “such that” part of the definition above is a universal property. It expresses the fact that giving a function from the disjoint union to another set is precisely the same as giving a function from each of the individual sets; this completely characterizes a disjoint union (which we regard as defined up to isomorphism). Another viewpoint is that the universal property expresses the fact that a disjoint union is the “most free” way of having two sets map into another set, neither adding any information nor collapsing any information. Universal properties are central to the way category theory describes structures that are somehow “canonical.” (See also the discussion of free groups in GEOMETRIC AND COMBINATORIAL GROUP THEORY [IV.10].)
Another key concept in a category is that of an isomorphism. As one might expect, this is defined to be a morphism with a two-sided inverse. Isomorphic objects in a given category are thought of as “the same, as far as this particular category is concerned.” Thus, categories provide a framework in which the most natural way of classifying objects is “up to isomorphism.”
Categories are mathematical structures of a certain kind, and as such they themselves form a category (subject to size restrictions so as to avoid a Russell-type paradox). The morphisms, which are the structure-preserving maps for categories, are called functors. In other words, a functor F from a category X to a category Y takes the objects of X to the objects of Y and the morphisms of X to the morphisms of Y in such a way that the identity of a is taken to the identity of Fa and the composite of f and g is taken to the composite of Ff and Fg. An important example of a functor is the one that takes a topological space S with a “marked point” s to its fundamental group π(S, s): it is one of the basic theorems of algebraic topology that a continuous map between two topological spaces (that takes marked point to marked point) gives rise to a homomorphism between their fundamental groups.
Furthermore, there is a notion of morphism between functors called a natural transformation, which is analogous to the notion of homotopy between maps of topological spaces. Given continuous maps F,G : X →. Y, a homotopy from F to G gives us, for every point x in X, a path in Y from Fx to Gx; analogously, given functors F, G : X → Y, a natural transformation from F to G gives us, for every point x in X, a morphism in Y from Fx to Gx. There is also a commuting condition that is analogous to the fact that, in the case of homotopy, a path in X must have its image under F continuously transformed to its image under G without passing over any “holes” in the space Y. This avoidance of holes is expressed in the category case by the commutativity of certain squares in the target category Y, which is known as the “naturality condition.”
One example of a natural transformation encodes the fact that every vector space is canonically isomorphic to its double dual; there is a functor from the category of vector spaces to itself that takes each vector space to its double dual, and there is an invertible natural transformation from this functor to the identity functor via the canonical isomorphisms. By contrast, every finite-dimensional vector space is isomorphic to its dual, but not canonically so because the isomorphism involves an arbitrary choice of basis; if we attempt to construct a natural transformation in this case, we find that the naturality condition fails. In the presence of natural transformations, categories actually form a 2-category, which is a two-dimensional generalization of a category, with objects, morphisms, and morphisms between morphisms. These last are thought of as two-dimensional morphisms; more generally an n-category has morphisms for each dimension up to n.
Categories and the language of categories are used in a wide variety of other branches of mathematics. Historically, the subject is closely associated with algebraic topology; the notions were first introduced in 1945 by Eilenberg and Mac Lane. Applications followed in algebraic geometry, theoretical computer science, theoretical physics, and logic. Category theory, with its abstract nature and lack of dependency on other fields of mathematics, can be thought of as “foundational.” In fact, it has been proposed as an alternative candidate for the foundations of mathematics, with the notion of morphism as the basic one from which everything else is built up, instead of the relation of set membership that is used in SET-THEORETIC FOUNDATIONS [IV.22 §4].