Different Views of Category, Type & Set

Different views of objects 对象 by:

1. Category 范畴 (morphism* between Objects, Functors ‘F‘ between Cats);

2. Set 集合 (a “smaller Cat”, only objects);

3. Type 类型 (deal only with same kind of objects: Int, String, Boulean…).

Note : Category can be a Set (SET) , Group, Ring, Vector Space (Vect) , “Topo” (Topology) … any algebraic structure with Associative Morphism (Map or ‘Arrow’ ) between them.

Note (*) : A morphism 态 in layman’s term is best illustrated by geometry:

2 triangle objects are similar 相似 = homomorphism 同态

2 triangle objects are congruent = isomorphism 同构


Note: Analogy –
Category : School
Type : Boy Class, Girl Class
Set : Students mixed of Boys, Girls

An “Introduction of Introduction” to Category Theory

Category : 范畴 has 3 things: (hence richer than a Set 集合 which is only a collection of objects)

  1. Objects 对象
  2. Arrow (Morphism 态射) between Objects, includes identity morphism.
  3. Associativity 结合性

Functor (函子) between 2 Categories (preserve structure)

Natural Transformation 自然变换

  • Example :
    Matrices -> Determinants