In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and have been a unifying notion. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942–1945, in connection with algebraic topology.
The term "abstract nonsense" has been used by some critics to refer to its high level of abstraction, compared to more classical branches of mathematics. Homological algebra is category theory in its aspect of organising and suggesting calculations in abstract algebra. Diagram chasing is a visual method of arguing with abstract 'arrows'. Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideas such as pointless topology.
In mathematics, an Abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative attempt to unify several cohomology theories by Alexander Grothendieck and has major applications in algebraic geometry, cohomology and pure category theory.
Alexander Grothendieck (March 28, 1928 Berlin, Germany - November 13, 2014) is considered to be one of the greatest mathematicians of the 20th century. He made major contributions to algebraic geometry, homological algebra, and functional analysis. He was awarded the Fields Medal in 1966, and was co-awarded the Crafoord Prize with Pierre Deligne in 1988. He declined the latter prize on ethical grounds in an open letter to the media. His work in algebraic geometry led to considerable developments in category theory, such as the concept of Abelian category and derived category.
Homological algebra • Additive categories
Duality theories • Sheaf theory
Higher category theory • Monoidal categories
Categorical logic • Topos theory
Categories in category theory
Objects • Functors
In homological algebra, the snake lemma, a statement valid in every abelian category, is the crucial tool used to construct the long exact sequences.
Homological algebra: Abelian category • Sheaf theory • K-theory
Topos theory • Enriched category theory • Higher category theory
Monoidal category • Closed category • Dagger category
More category theory topics
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