Category theory is a general theory of mathematical structures and their relations, providing a unifying language for constructions that recur across different areas of mathematics. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionWhat is common to the structures and constructions that recur across different branches of mathematics, independent of the particular objects involved? 1 Key DebateThe subject was informally nicknamed general abstract nonsense at its own 1942 founding, in Saunders Mac Lane's recollection of introducing categories with Samuel Eilenberg, a self-deprecating name for its high level of abstraction that stuck as an affectionate, non-derogatory term among mathematicians rather than a real objection to the subject's usefulness. 2 Classification
Pure or Applied Connections
Includes
Source Alexander Grothendieck (Wikipedia)
Source Commutative Diagram (Wikipedia)
Source Embedding (Wikipedia)
Source Fundamental theorem of topos theory (Wikipedia)
Source Lawvere's fixed-point theorem, Wikipedia
Source Linton's theorem (equational theory), Wikipedia
Source Natural Transformation (Wikipedia)
Source Wikipedia: Samuel Eilenberg
Source Wikipedia: Saunders Mac Lane
Source Wikipedia: William Lawvere
Sources
1. Category Theory (Wikipedia)
Wikimedia Foundationlead paragraphQuote, lead paragraph
Category theory is a general theory of mathematical structures and their relations.
2. Abstract Nonsense (Wikipedia)
Wikimedia FoundationHistoryQuote, History
Referring to a joint paper with Samuel Eilenberg that introduced the notion of a "category" in 1942, Saunders Mac Lane wrote the subject was 'then called "general abstract nonsense"'.
Alexander Grothendieck (Wikipedia)
WikipediaIncludes: Alexander Grothendieck, Lead sectionQuote, Includes: Alexander Grothendieck, Lead section
His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics.
Commutative Diagram (Wikipedia)
Includes: Commutative Diagram, Lead sentenceQuote, Includes: Commutative Diagram, Lead sentence
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram
Natural Transformation (Wikipedia)
Includes: Natural Transformation, Lead sentenceQuote, Includes: Natural Transformation, Lead sentence
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another whil
Lawvere's fixed-point theorem, Wikipedia
Includes: Lawvere's Fixed-Point Theorem, Lead sentenceQuote, Includes: Lawvere's Fixed-Point Theorem, Lead sentence
e's fixed-point theorem is an important result in category theory.
Wikipedia: William Lawvere
Includes: William Lawvere, LeadQuote, Includes: William Lawvere, Lead
known for his foundational work in category theory, topos theory
Wikipedia: Samuel Eilenberg
Includes: Samuel Eilenberg, LeadQuote, Includes: Samuel Eilenberg, Lead
Samuel Eilenberg (September 30, 1913, January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra.
Wikipedia: Saunders Mac Lane
Includes: Saunders Mac Lane, LeadQuote, Includes: Saunders Mac Lane, Lead
Saunders Mac Lane (August 4, 1909, April 14, 2005), born Leslie Saunders MacLane, was an American mathematician who co-founded category theory with Samuel Eilenberg.
Linton's theorem (equational theory), Wikipedia
Includes: Linton's Theorem (Equational Theory), Lead paragraphQuote, Includes: Linton's Theorem (Equational Theory), Lead paragraph
These two notions are the two main category theoretic formulations of universal algebra.
Embedding (Wikipedia)
Includes: Embedding, Lead paragraphQuote, Includes: Embedding, Lead paragraph
In the terminology of category theory, a structure-preserving map is called a morphism.
Fundamental theorem of topos theory (Wikipedia)
Includes: Fundamental Theorem of Topos Theory, Lead sectionQuote, Includes: Fundamental Theorem of Topos Theory, Lead section
topos theory
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