# Joyal's CatLab Introduction

**Category theory** ##Contents * Contributors * References * Introduction * Basic category theory * Weak factorisation systems * Factorisation systems * Distributors and barrels * Model structures on Cat * Homotopy factorisation systems in Cat * Accessible categories * Locally presentable categories * Algebraic theories and varieties of algebras **Categorical mathematics** ##Contents * Set theory * Category theory * Homotopical algebra * Higher category theory * Theory of quasi-categories * Higher quasi-categories * Geometry * Elementary geometry * Differential geometry * Lie theory * Algebraic geometry * Homotopical algebraic geometry * Number theory * Elementary number theory * Algebraic number theory * Algebra * Universal algebra * Group theory * Rings and modules * Commutative algebras * Lie algebras * Representation theory * Operads * Homological algebra * Logic * Boolean algebra * First order theory * Model theory * Categorical logic * Topology * General topology * Algebraic topology * Homotopy theory * Theory of locales * Topos theory * Higher topos theory * Combinatorics * Combinatorial geometry * Enumerative combinatorics * Algebraic combinatorics

# What is category theory?

###### Prehistory

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 $f:X\to Y$, instead of $f(X)\subseteq Y$, for a function $f$ with domain $X$ and codomain $Y$, and even to write

$\xymatrix{X\ar[r]^f & Y.}$

He used commutatives squares of spaces and maps, or of groups and homomorphisms,

$\xymatrix{A\ar[r]^{f}\ar[d]_{u}& B \ar[d]^{v}\\ C\ar[r]_{g}& D.}$

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,

$H_n:Man \to Ab.$

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

$\pi_n:Man_* \to Ab.$

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

$\pi_n\to H_n.$

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.

###### History

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 objects to the study of maps or relations between objects: of (homo)morphisms between objects.

The classical examples of categories are concrete categories whose objects are sets with extra structure and whose morphisms 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 constructions in categories, such as limits, 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 January 30, 2010 at 19:24:02 by joyal