# Portal:Category theory

## Category theory

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.

## Selected Article

In category theory, the **derived category** of an Abelian category is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors. The development of the theory, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s. Derived categories have since appeared outside of algebraic geometry, for example in D-modules theory and microlocal analysis.

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## Selected Biography

**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.

## Categories

## Selected Picture

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.

## Did you know?

- ... that in
**higher category theory**, there are two major notions of higher categories, the strict one and the weak one ? - ... that
**factorization systems**generalize the fact that every function is the composite of a surjection followed by an injection ? - ... that in a
**multicategory**, morphisms are allowed to have a multiple arity, and that multicategories with one object are operads ? - ... that it is possible to define the
**end**and the coend of certain functors ? - ... that in the
**category of rings**, the coproduct of two commutative rings is their tensor product ? - ... that the
**Yoneda lemma**proves that any small category can be embedded in a presheaf category ? - ... that it is possible to compose
**profunctors**so that they form a bicategory?

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