A category is a mathematical structure consisting of objects and morphisms that relate them.Formally, a category is defined by a class of objects and, for each pair of objects, a set of morphisms (arrows representing admissible transformations) with a composition operation. Composition is associative: if you apply arrows f, then g, then h, it does not matter whether you group them as h after (g after f) or as (h after g) after f. For each object, there is an identity arrow that leaves it unchanged. The essence of the theory is to ignore the internal structure of objects and focus on the connections between them, providing a universal language for constructing such things as the product of objects (a way to combine two objects) or the colimit (a way to merge information from several objects).
Categories emerged in the work of Samuel Eilenberg and Saunders Mac Lane (1942–1945) on the axiomatization of natural transformations in algebraic topology. In the 1950s–60s, Alexander Grothendieck rebuilt algebraic geometry in categorical language and introduced the notion of a topos. William Lawvere developed categorical logic. Since the 1980s, categories have been actively applied in computer science (e.g., monads in functional programming). In the 21st century, applications appeared in quantum physics (categorical quantum mechanics) and type theory.
The work is based on the axioms of composition and universal properties. For example, the product of objects A and B in a category is defined not by construction but by a property: it is an object P with a pair of projections p₁: P→A, p₂: P→B, such that for any object X with morphisms f: X→A and g: X→B, there exists a unique morphism h: X→P making the diagram commute. This allows proving theorems without referring to the elements of objects. The principle of duality (reversing all arrows) doubles the number of provable statements.
This approach facilitates the transfer of ideas between different branches of mathematics. In computer science, categories are used for strict type checking of programs, and in physics, for modeling quantum computations.
Category theory is sometimes jokingly called 'abstract nonsense', emphasizing its extreme generality, which nonetheless has provided a rigorous foundation for many areas of mathematics.
| w_1 | variable | weight for a — weight |
| a | variable | first value |
| w_2 | variable | weight for b — weight |
| b | variable | second value |
| X | variable | first random variable — random variable |
| Y | variable | second random variable — random variable |
| Var(X) | variable | variance of X — variance |
| Var(Y) | variable | variance of Y — variance |
| Var(X+Y) | variable | variance of the sum X+Y — variance |
| U | variable | open set (topological space) — topological space |
| r | variable | number of independent Z/2Z factors — rank |
| X | variable | a set — set |
| κ | variable | a cardinal number — cardinal number |
| |X| | operator | cardinality of the set X — cardinality |
| cf(κ) | operator | cofinality of the cardinal κ — cofinality |
