Concept

Category theory

Overview

Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the mid-20th century in their foundational work on algebraic topology. Category theory can be used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality.

Many areas of computer science also rely on category theory, such as functional programming and semantics.

A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the source and the target of the morphism. A morphism is often represented by an arrow from its source to its target (see the figure). Morphisms can be composed if the target of the first morphism equals the source of the second one. Morphism composition has similar properties as function composition (associativity and existence of an identity morphism for each object). Morphisms are often some sort of functions, but this is not always the case. For example, a monoid may be viewed as a category with a single object, whose morphisms are the elements of the monoid.

The second fundamental concept of category theory is the concept of a functor, which plays the role of a morphism between two categories

C

1

{\displaystyle {\mathcal {C}}_{1}}

and

C

2

{\displaystyle {\mathcal {C}}_{2}}

: it maps objects of

C

1

{\displaystyle {\mathcal {C}}_{1}}

to objects of

C

2

{\displaystyle {\mathcal {C}}_{2}}

and morphisms of

C

1

{\displaystyle {\mathcal {C}}_{1}}

to morphisms of

C

2

{\displaystyle {\mathcal {C}}_{2}}

in such a way that sources are mapped to sources, and targets are mapped to targets (or, in the case of a contravariant functor, sources are mapped to targets and vice-versa). A third fundamental concept is a natural transformation that may be viewed as a morphism of functors.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

Possible passages

Found by search and not yet checked by a person; they may only mention the idea in passing.

1. We have suffered much from critics who consider it sufficient to criticise our procedure on the slender basis of a knowledge of the dictionary meanings of such terms. For example, a one-to-one correlation depends on the notion of a class with only one member, and this notion is defined without appeal to the concept of the number one. The notion of diversity is all that is wanted.

   Alfred North Whitehead, The organisation of thought, educational and scientific

   States what Category theory is.

   Read the chapter · Open in the reader

2. All types of series are capable of formal definition, and their properties can be deduced from the principles of symbolic logic by means of the Algebra of Relatives. The notion of a limit, which is fundamental in the greater part of higher mathematics, used to be defined by means of quantity, as a term to which the terms of some series approximate as nearly as we please. But nowadays the limit is defined quite…

   Bertrand Russell, Mysticism and Logic and Other Essays

   States what Category theory is.

   Read the chapter · Open in the reader

3. The relations between these things then receive "genaue und vollstaendige Beschreibung" through the axioms of the geometry. And the fact that these "things" are called points, lines, and planes is not to give to them any of the connotations ordinarily associated with these words further than are determined by the axiom groups that follow. Indeed, other geometers are even more explicit on this point. Thus for Peano…

   John Dewey, Creative Intelligence

   States what Category theory is.

   Read the chapter · Open in the reader

Philosophers who discussed Category theory

• Per Martin-Löf
• Solomon Feferman
• Alain Badiou
• Grigore Moisil

Related ideas

• Intuitionistic logic
• Function (mathematics)
• Mathematical object
• Mathematical logic
• Constructivism (philosophy of mathematics)
• Natural number
• Physics
• Set (mathematics)
• Stanford University
• Morphism
• Identity (mathematics)
• Topology
• Saunders Mac Lane
• Stanford Encyclopedia of Philosophy
• Type theory
• Map (mathematics)
• Computer science
• Empty set
• Duality (mathematics)
• Samuel Eilenberg
• Categorical logic

Opposing perspectives

• Continuous function

Place in the tradition

Part of Philosophy of mathematics.

Explore Category theory in the interactive graph →