# 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 mathematics, a **category** is a fundamental and abstract way to describe mathematical entities and their relationships. A category is composed of a collection of abstract "objects" of any kind, linked together by a collection of abstract "arrows" of any kind that have a few basic properties (the ability to compose the arrows associatively and the existence of an identity arrow for each object). Many well-known categories are conventionally identified by a short capitalized word or abbreviation in bold or italics such as **Set** (category of sets) or **Ring** (category of rings).

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## Selected Biography

**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.

## 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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