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.
Samuel Eilenberg (born in Warsaw, September 30, 1913 and died in New York City, January 30, 1998) was a Polish and American mathematician. He spent much of his career in USA as a professor at Columbia University. His main interest was algebraic topology and foundational grounds to homology theory. He cofounded category theory with Saunders Mac Lane and wrote in 1965, Homological Algebra with Henri Cartan. Later, he worked in automata theory and pure category theory.
Homological algebra • Additive categories
Duality theories • Sheaf theory
Higher category theory • Monoidal categories
Categorical logic • Topos theory
Categories in category theory
Objects • Functors
A natural transformation, as a morphism between functors, respects the "functor structure". This idea, as usual in category theory, is expressed in a commutative diagram, pictured on the right.
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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