Notes on Category Theory[edit]

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

Categories[edit]

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(id_o) = dom(id_o) such that id_o = o

Examples[edit]

Name of category	Objects	Arrows	Description
Set	All small sets	All functions between sets	Category of sets
Cat	All small categories	All functors	Category of small categories
Mon	All small monoids	All morphisms of monoids
Grp	All small groups	All morphisms of groups	Category of groups
Rng	All small rings	All morphisms between rings
Top	All small topological spaces	All continuous maps	Category of topological spaces

Functor[edit]

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(Id_c) = Id_Fc 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[edit]

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[edit]

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[edit]

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 hom_C(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[edit]

To every category C, C^opp 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 f^opp:c'→c in C^opp.

Products[edit]

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 <c,d> of objects c of C and d of D; while an arrow <c,d>→<c',d'> of C×D is a (ordered) pair <f,g> of arrows f:c→c' and g:d→d'. The composite is defined as

<f',g'>○<f,g> = <f'○f,g'○g>.

Functors P:C×D→C and Q:C×D→D are called projections of the product. Defined by

P<f,g> = f, Q<f,g> = g and  P<c,d> = c, Q<c,d> = d.

Sometimes the notation pr₁ and pr₂ are used to denote projections.

Comma category[edit]

The category of objects under a (a ↓ C) is defined with objects all pairs <f,c> with c an object of C and f:a→c an arrow of C and arrows h:<f,c>→<f',c'> 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> (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 <f,b> 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[edit]

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[edit]

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