範疇 (數學)

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在數學裡，範疇允許公式化一包含抽象結構與保有結構之程序的概念。實際上，範疇在現代數學的每一分支內都會出現，且是一核心統一概念。有關範疇自身的研究稱做範疇論。

[编辑] 定義

範疇 C包含有

物件的類 ob(C)
態射的類 hom(C)。且每一態射 f 都有唯一的源物件a和目標物件b，且a和b都於ob(C)內。此寫成f: a → b，且稱f為由a至b的態射。所有由a至b的態射所組成的類，其標記為hom(a, b) (或 hom_C(a, b))。
對任三個物件a、b和c，二元運算hom(a, b) × hom(b, c) → hom(a, c)，稱之為態射複合；f : a → b and g : b → c的複合寫成g o f或gf。

使得下列公理成立：

(結合律) 若 f : a → b、g : b → c且h : c → d，則h o (g o f) = (h o g) o f；
(單位元) 對任一物件x，存在一態射 1_x : x → x，使得每一態射f : a → b，會有 1_b o f = f = f o 1_a。此一態射稱為x的單位態射。

由上述公理，可證明對每一個物件均存在一且唯一一個單位態射。一些作者會將每一個物件等同於其相對應的單位態射。

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.

[编辑] 例子

每一範疇都可由其物件、態射和態射複合來表示。

The category Set of all sets together with functions between sets, where composition is the usual function composition (The following are subcategories of Set, obtained by adding some type of structure onto a set, by requiring that morphisms are functions that respect this added structure, and where morphism composition is simply ordinary function composition.)
- The category Rel of all sets with relations
- The category Ord of all preordered sets with monotonic functions
- The category Mag consisting of all magmas with their homomorphisms
- The category Med consisting of all medial magmas with their homomorphisms
- The category Grp consisting of all groups with their group homomorphisms
- The category Ab consisting of all abelian groups with their group homomorphisms
- The category Vect_K of all vector spaces over the field K (which is held fixed) with their K-linear maps
- The category Top of all topological spaces with continuous functions
- The category Met of all metric spaces with short maps
- The category Uni of all uniform spaces with uniformly continuous functions
The category Cat of all small categories with functors
Any preordered set (P, ≤) forms a small category, where the objects are the members of P, the morphisms are arrows pointing from x to y when x ≤ y (The composition law is forced, because there is at most one morphism from any object to another.)
Any monoid forms a small category with a single object x. (Here, x is any fixed set.) The morphisms from x to x are precisely the elements of the monoid, and the categorical composition of morphisms is given by the monoid operation. In fact, one may view categories as generalizations of monoids; several definitions and theorems about monoids may be generalized for categories.
Any directed graph generates a small category: the objects are the vertices of the graph and the morphisms are the paths in the graph. Composition of morphisms is concatenation of paths. This is called the free category generated by the graph.
If I is a set, the discrete category on I is the small category that has the elements of I as objects and only the identity morphisms as morphisms. Again, the composition law is forced.)
Any category C can itself be considered as a new category in a different way: the objects are the same as those in the original category but the arrows are those of the original category reversed. This is called the dual or opposite category and is denoted C^op.
If C and D are categories, one can form the product category C × D: the objects are pairs consisting of one object from C and one from D, and the morphisms are also pairs, consisting of one morphism in C and one in D. Such pairs can be composed componentwise.

[编辑] 態射類型

A morphism f : a → b is called

a monomorphism (or monic) if fg₁ = fg₂ implies g₁ = g₂ for all morphisms g₁, g₂ : x → a.
an epimorphism (or epic) if g₁f = g₂f implies g₁ = g₂ for all morphisms g₁, g₂ : b → x.
a bimorphism if it is both a monomorphism and an epimorphism.
a retraction if it has a right inverse, i.e. if there exists a morphism g : b → a with fg = 1_b.
a section if it has a left inverse, i.e. if there exists a morphism g : b → a with gf = 1_a.
an isomorphism if it has an inverse, i.e. if there exists a morphism g : b → a with fg = 1_b and gf = 1_a.
an endomorphism if a = b. The class of endomorphisms of a is denoted end(a).
an automorphism if f is both an endomorphism and an isomorphism. The class of automorphisms of a is denoted aut(a).

Every retraction is an epimorphism. Every section is a monomorphism. The following three statements are equivalent:

f is a monomorphism and a retraction;
f is an epimorphism and a section;
f is an isomorphism.

Relations among morphisms (such as fg = h) can most conveniently be represented with commutative diagrams, where the objects are represented as points and the morphisms as arrows.

[编辑] 範疇類型

In many categories, the hom-sets hom(a, b) are not just sets but actually abelian groups, and the composition of morphisms is compatible with these group structures; i.e. is bilinear. Such a category is called preadditive. If, furthermore, the category has all finite products and coproducts, it is called an additive category. If all morphisms have a kernel and a cokernel, and all epimorphism are cokernels and all monomorphisms are kernels, then we speak of an abelian category. A typical example of an abelian category is the category of abelian groups.
A category is called complete if all limits exist in it. The categories of sets, abelian groups and topological spaces are complete.
A category is called cartesian closed if it has finite direct products and a morphism defined on a finite product can always be represented by a morphism defined on just one of the factors.
A topos is a certain type of cartesian closed category in which all of mathematics can be formulated (just like classically all of mathematics is formulated in the category of sets). A topos can also be used to represent a logical theory.
A groupoid is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups, group actions and equivalence relations.

[编辑] 參考文獻

Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990). Abstract and Concrete Categories. Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition)
Asperti, Andrea, & Longo, Giuseppe (1991). Categories, Types and Structures. Originally publ. M.I.T. Press.
Barr, Michael, & Wells, Charles (2002). Toposes, Triples and Theories. (revised and corrected free online version of Grundlehren der mathematischen Wissenschaften (278). Springer-Verlag,1983)
Borceux, Francis (1994). Handbook of Categorical Algebra.. Vols. 50-52 of Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press.
Lawvere, William, & Schanuel, Steve. (1997). Conceptual Mathematics: A First Introduction to Categories. Cambridge: Cambridge University Press.
Mac Lane, Saunders (1998). Categories for the Working Mathematician (2nd ed.). Graduate Texts in Mathematics 5. Springer. ISBN 0-387-98403-8.