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Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense".See list of category theory topics for a breakdown of relevant articles.
1 Background
A category attempts to capture the essence of a class of related mathematical objects, for instance the class of groups. Instead of focusing on the individual objects (groups) as has been done traditionally, the morphisms—i.e. the structure-preserving maps between these objects—are emphasized. In the example of groups, these are the group homomorphisms. Then it becomes possible to relate different categories by functors, generalizations of functions which associate to every object of one category an object of another category and to every morphism in the first category a morphism in the second. Very commonly, certain "natural constructions", such as the fundamental group of a topological space, can be expressed as functors. Furthermore, different such constructions are often "naturally related" which leads to the concept of natural transformation, a way to "map" one functor to another. Throughout mathematics, one encounters "natural isomorphisms", things that are (essentially) the same in a "canonical way". This is made precise by special natural transformations, the natural isomorphisms.
2 Historical notes
Categories, functors and natural transformations were introduced by Samuel Eilenberg and Saunders Mac Lane in 1945. Initially, the notions were applied in topology, especially algebraic topologyTopology Algebraic topology Abstract algebra Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. The method of algebraic invariants The goal is to take topological spaces, and further ca, as an important part of the transition from homologyIn mathematics (especially algebraic topology and abstract algebra), homology is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). See homology theory (an intuitive and geometric concept) to homology theoryHomology theory Algebraic topology In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces. At th, an axiomFor the algebra software named Axiom, see Axiom (algebra software). For the 1970s Australian rock music group, see Axiom (band). In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built upatic approach. It has been claimed, for example by or on behalf of Ulam, that comparable ideas were current in the later 1930sCenturies: 19th century 20th century 21st century Decades: 1880s 1890s 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s Years: 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 Events and trends Technology Jet engine invented Link Trainer invented Sc in the Polish school.
Eilenberg/Mac Lane have said that their goal was to understand natural transformations; in order to do that, functors had to be defined; and to define functors one needed categories.
The subsequent development of the theory was powered first by the computational needs of homological algebraHomological algebra Abstract algebra Algebraic topology Algebraic geometry Homological algebra is that branch of mathematics which studies the methods of homology and cohomology in a general setting. These concepts originated in algebraic topology. Cohomo; and then by the axiomatic needs of algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations . When there is more than o, the field most resistant to the Russell-Whitehead view of united foundations. General category theory—an updated universal algebra with many new features allowing for semantic flexibility and higher-order logic—came later; it is now applied throughout mathematics.
Special categories called topoi can even serve as an alternative to axiomatic set theory as the foundation of mathematics. These broadly-based foundational applications of category theory are contentious; but they have been worked out in quite some detail, as a commentary on or basis for constructive mathematics. One can say, in particular, that axiomatic set theory still hasn't been replaced by the category-theoretic commentary on it, in the everyday usage of mathematicians. The idea of bringing category theory into earlier, undergraduate teaching (signified by the difference between the Birkhoff-Mac Lane and later Mac Lane-Birkhoff abstract algebra texts) has hit noticeable opposition.
Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with application to the theory of functional programming and domain theory, all in a setting of a cartesian closed category as non-syntactic description of a lambda calculus. At the very least, the use of category theory language allows one to clarify what exactly these related areas have in common (in an abstract sense).
