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In mathematics, categories allow one to formalize notions involving abstract structure and processes that preserve structure. Categories appear in virtually every branch of modern mathematics and are a central unifying notion. more...
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The study of categories in their own right is known as category theory.
For more extensive motivational background and historical notes, see category theory and the list of category theory topics.
Definition
A category C consists of
a class ob(C) of objects:;
a class hom(C) of morphisms. Each morphism f has a unique source object a and target object b where a and b are in ob(C). We write f: a → b, and we say "f is a morphism from a to b". We write hom(a, b) (or homC(a, b)) to denote the hom-class of all morphisms from a to b. (Some authors write Mor(a, b).);
for every three objects a, b and c, a binary operation hom(a, b) × hom(b, c) → hom(a, c) called composition of morphisms; the composition of f : a → b and g : b → c is written as g o f or gf (Some authors write fg or f;g.);
such that the following axioms hold:
(associativity) if f : a → b, g : b → c and h : c → d then h o (g o f) = (h o g) o f, and;
(identity) for every object x, there exists a morphism 1x : x → x called the identity morphism for x, such that for every morphism f : a → b, we have 1b o f = f = f o 1a.;
From these axioms, one can prove that there is exactly one identity morphism for every object. Some authors use a slight variation of the definition in which each object is identified with the corresponding identity morphism.
A small category is a category in which both ob(C) and hom(C) are actually sets and not proper classes. A category that is not small is said to be large. A locally small category is a category such that for all objects a and b, the hom-class hom(a, b) is a set, called a homset. Many important categories in mathematics (such as the category of sets), although not small, are at least locally small.
The morphisms of a category are sometimes called arrows due to the influence of commutative diagrams.
Examples
Each category is presented in terms of its objects, its morphisms, and its composition of morphisms.
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