Nov 26, 2012

CATEGORY THEORY_Documented by Marsigit




CATEGORY THEORY

A category E consists of two classes, the members of the first of which -- denoted by letters X, Y, ...--are called objects (structures) and the members of the second of which -- denoted by the letters f,g,... --are called arrows (morphisms). 

Each arrow f is assigned an object X as domain and an object Y as   codomain, indicated by writing f: X -> Y. If g is any arrow g: Y -> Z with domain Y, the codomain of f, there is an arrow fg: X -> Z called the composition of f and g. 

For each object Y there is an arrow idY:Y -> Y called the identity arrow of Y. These notions are assumed to satisfy the following identity and associativity axioms:

f C idY = f,  idY C g = g, f(gh) = (fg)h for any arrows f: X -> Y, g: Y -> Z, h: Z -> W .

Given two categories D and E, a functor F from D to E     consists of a pair of functions(both denoted by F), one from the class of objects of D to that of E, and the           other from the class of arrows of D to that of E, such that if f: X -> Y in D, then F(f): F(X) -> F(Y) in E; F(idX) = idF(X)  and      F(fg) = F(f)F(g) for composable arrows f,g of D. A functor can be thought of as a morphism of categories. Categories and functors are found in many seemingly diverse branches of mathematics. Some categories are:

 Set: objects - the sets; arrows - the (set) functions
 Grp: objects - the groups; arrows - group homomorphisms
 Top: objects - topological spaces; arrows - continuous functions
         
An example of a functor is the "forgetful" functor from Grp to Top of Set which assigns to each group or topological space its underlying set. This functor has the effect of "forgetting" the structure and just maintaining the elements.

1 comment:

  1. Syahrial
    16701251015
    S2 PEP kelas B 2016
    Teori kategori berhubungan dengan struktur matematika dan hubungan antar struktur tersebut secara abstrak. Saat ini kategori digunakan dalam matematika, informatika teori, dan fisika matematis. Kategori diperkenalkan pertama kali oleh Samuel Eilenberg dan Saunders Mac Lane pada tahun 1942-1945, dalam hubungannya dengan topologi aljabar.
    Misalkan kita mempunyai himpunan (yang lalu kita sebut dengan object beserta fungsi total di antar himpunan tersebut (yang lalu kita sebut morphism, maka properti kategori adalah sebagai berikut.
    Tipe Fungsi. f: A -> B berarti fungsi f memetakan dari himpunan A ke himpunan B.
    Komposisi. Kita bisa menggabungkan dua fungsi f dan g, jika himpunan target dari fungsi pertama sama dengan himpunan sumber dari fungsi kedua, misal f: A -> B dan g: B -> C untuk beberapa himpunan A,B, dan C. Komposisi biasanya dilambangkan dengan g o f.
    Fungsi Identitas. Untuk setiap himpunan A, terdapat fungsi identitas id A : A -> A

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