Nov 26, 2012
CATEGORY THEORY_Documented by Marsigit
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.