Joyal's CatLab Introduction


Category theory

Category theory

Categorical mathematics

Categorical mathematics


What is category theory?


The algebraic topology of the 1930’s was a fertile ground for the future emergence of category theory. The idea of a category may have germinated from a change of notation introduced by Witold Hurewicz? during that time. He began to write f:XYf:X\to Y, instead of f(X)Yf(X)\subseteq Y, for a function ff with domain XX and codomain YY, and even to write He used commutatives squares of spaces and maps, or of groups and homomorphisms,

The simplicial approximation theorem? of L.E.J. Brouwer was showing that the homology groups, introduced by Emmy Noether?, were behaving functorially with respect to continuous maps. Thus providing a sequence of abelian group valued functors on the category of manifolds,

H n:ManAb.H_n:Man \to Ab.

The higher homotopy groups? π n(X)\pi_n(X) (n2)(n\geq 2) of a space, introduced by Eduard Cech and Hurewicz, were providing another sequence of abelian group valued functors on the category of pointed manifolds,

π n:Man *Ab.\pi_n:Man_* \to Ab.

The Hurewicz map? π n(X)H n(X)\pi_n(X)\to H_n(X) extends to higher dimensions the canonical map π 1(X)H 1(X)\pi_1(X)\to H_1(X) defined by Henri Poincaré?. It is a stricking example of a natural transformation,

π nH n.\pi_n\to H_n.

This account of the prehistory of category theory is based on a conversation I had with Eilenberg around 1983. My assertions should be checked since my memory is not perfect.


Category theory was introduced by Samuel Eilenberg and Saunders Mac Lane in the 1945 paper General theory of natural equivalences. The reason for introducing categories was to introduce [functor]s, and the reason for introducing functors was to introduce [natural transformation]s (more specifically natural equivalences) in order to define what natural means in mathematics.

The paper was a clash of ideas from abstract algebra (Mac Lane) and topology?/homotopy theory? (Eilenberg). It was first rejected on the ground that it had no content but was later published. Unexpectedly category theory has flourished into almost all areas of mathematics, has found many applications outside mathematics like computer science and applications to logic, even attempts to build a foundations? of mathematics.

Not all basic notions of category theory were introduced by Eilenberg and MacLane in 1945. For example, product and coproduct were introduced by Mac Lane in 195?. The notion of adjoint functors, which is truly central in category theory, was introduced by Daniel Kan in 1956?. The notion of equivalence of categories was introduced by Alexander Grothendieck in 1957?.


A typical example of a category is the category Set of (small) sets and maps between sets. The basic idea of category theory is to shift attention from the study of object?s to the study of maps or relations between objects: of (homo)morphism?s between objects.

The classical examples of categories are concrete categories? whose object?s are sets with extra structure? and whose morphism?s are structure preserving functions of sets, such as Top, Grp, Vect?. These are the examples from which the term category derives: a category is collection of structures of the same type (same category) together with a notion of mappings between them. A typical examples of non-concrete categories is the fundamental groupoid? of a topological space?.

But by far not all categories are of this type and categories are much more versatile than these classical examples suggest. After all, a category is just a directed graph? with a notion of composition of its edges. As such it generalizes the concepts of monoid? and poset?. If the category is a groupoid it generalizes the concept of group?. Thinking of a category as a generalized poset is particularly useful when studying limits? and adjunctions?.

Conceptual unification

One major driving force behind the development of category theory is its ability to abstract and unify concepts. General statements about categories apply to each specific concrete category? of mathematical structures. The general notion of universal construction?s in categories, such as limit?s, turns out to prevail throughout mathematics and manifest itself in myriads of special examples.

Duality principles

The principle of abstract duality? states that every theorem of pure category theory has a dual theorem, obtained by reversing all of the arrows involved. See opposite category?. Abelian duality states that the opposite of an abelian category? is abelian.

Category theory vs. order theory

A category may be thought of as a categorification? of a poset? rather than of a set; much (but by no means all) of category theory also appears in order theory?.

Revised on November 20, 2020 at 21:37:39 by Dmitri Pavlov