User:Helohe/Categories

=Notes on Category Theory=

Note: These are my personal notes and may be incorrect.

Categories
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 * A Category C is a collection of Objects Ob and Morphisms (Arrows) Hom between the Objects.
 * There is a Identity Morphism Id for each object o with cod(ido) = dom(ido) such that ido = o

Functor
A functor is a morphism of categories consisting of a object function and a arrow function.

Example: Functor F:C→D, assigns each object of c of C an object Fc of D and each arrow f:c→c' an arrow Ff:Fc→Fc'. Such that a identity morphism is mapped to a identity morphism and a composite is mapped to a composite ( F(Idc) = IdFc and F(g ○ f) = Fg ○ Ff).


 * A functor is full if to every pair of objects c, c' of C and to every arrow g:Fc→Fc' of D there is a arrow f:c→c' of C with g=Ff.
 * A functor is faithful if to every parir of objects c, c' and to every pair of arrows f,g:c→c' of (parallel) arrows: Ff = Fg: Fc→Fc' implies f=g.

Natural Transformation
A natural transformation n:F→G (with functors F,G:C→D) assigns each object c of C an arrow nc:Fc→Gc of D. Such that for each arrow f:c→c' of C: Gf○nc = nc'○Ff

Initial and Terminal Objects

 * A object i of a category C is initial iff for every object a there is exactly one arrow i→a.
 * A object t of a category C is terminal iff for every object a there is exactly one arrow a→t.
 * A object which is both initial and terminal is called a null object.

Examples:


 * In the category Set, the empty set is an initial object and any one point set is a terminal object.

Sets of Morphisms
For any two objects c,d of a category C the set of (homo)morphisms (hom-set) consists of all arrows f of the category such that: c = dom(f) and f = cod(f) The set is denoted by homC(c,d) or hom(c,d) if the category is clear. (Sometimes the notations C(a,b), (a,b) and (a,b)C are also used).

Duality

 * To every category C, Copp denotes the category with the same objects as C but with all arrows reversed. eg: if f:c→c' is an arrow of c then there is an arrow fopp:c'→c in Copp.

Products
Given two categories C, D one may construct a new category C × D called the product of C and D. Defined as follows: An object of C×D is a (ordered) pair  of objects c of C and d of D; while an arrow → of C×D is a (ordered) pair  of arrows f:c→c' and g:d→d'. The composite is defined as ○ = .

Functors P:C×D→C and Q:C×D→D are called projections of the product. Defined by P = f, Q = g and P = c, Q = d. Sometimes the notation pr1 and pr2 are used to denote projections.

Comma category
The category of objects under a (a ↓ C) is defined with objects all pairs  with c an object of C and f:a→c an arrow of C and arrows h:→ those arrows h:c→c' of C for which h○f=f'.

Similarly the category of objects over a, (C ↓ a) has objects  (f:c→a) and arrows such that f'○h=f.

If instead of a category C a functor F:B→C is used, with a an object of C, we obtain (a ↓ F) defined with objects all pairs  with b an object of B and f:a→Fb and arrows h:<f,b>→<f',b'> all the arrows h:b→b' in B for which f'=Fh ○ f.

=Notations= I used most of the notations as they are in [1]. In other publications (a,b) instead of <a,b> may be used to denote a ordered set and {a,b} is used to denote a (not-ordered) set.

=References=
 * [1] Saunders Mac Lane - Categories for the Working Mathematician (Second Edition)
 * [2] http://math.ifi.unizh.ch/book/master.html