Portal: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. Template:/box-footer
Lua error in package.lua at line 80: module 'Module:Box-header/colours' not found.
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.
...Other articles | Read more... |
Lua error in package.lua at line 80: module 'Module:Box-header/colours' not found. 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.
Template:/box-header Template:/Categories Template:/box-footer
Lua error in package.lua at line 80: module 'Module:Box-header/colours' not found.
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.
- ... 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?
Homological algebra: Abelian category • Sheaf theory • K-theory
Topos theory • Enriched category theory • Higher category theory
|
|
|
|
|
|
|
Algebra | Analysis | Category theory |
Computer science |
Cryptography | Discrete mathematics |
Geometry |
|
|
|
|
|
|
|
|
Logic | Mathematics | Number theory |
Physics | Science | Set theory | Statistics | Topology |
- What are portals?
- List of portals
- Featured portals
- Improve the category theory articles, expand the category theory stubs
- Keep building this portal
- What are portals?
- List of portals
- Featured portals