Appendix IV Basic Notions in Category Theory

Appendix IV

BASIC NOTIONS IN CATEGORY THEORY

We are using in this book standard notions of categorical algebra [41]. For reference purposes, definitions of the key concepts used are given below. 1. Let X be a class of objects, X, I: together with two functions as follows : (i) A function assigning to each pair(X, Y)of objects ofX a set Mor (X,Y). An element f E Mor (X,Y) in this set is called a morphism f :X + Y of X, with domain X and codomain Y. (ii) A function assigning to each triple (X, Y, 2)of objects of X a function

Mor (Y, Z) x Mor (X,Y) + Mor (X, 2) For morphisms g : Y + 2 and f:X+ Y,this function is written as ( g , f ) + g - f, and the morphism g - f:X + Z is called the composite of g with f. The class X with these two functions is called a category when the following two axioms hold. Associativity. If h:Z + W,g : Y -,Z, and f:X--* Y are morphisms of with the indicated domains and codomains, then

X

h . ( g * f )= ( h * g ) * f

Identity. For each object Y of that Z,-f=f

X,there exists a morphism I,: Y + Y such for f : X + Y 262

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and g.I,=g 2. The class of objects of

for g : Y + Z

X

will be denoted as Ob X.

3. A morphism f:X Y in a category X is called invertible in X if there exists a morphism g : Y + X in X with both g -f= I, and f.g = I,. g will be denoted by f - since if g exists, it is unique. -+

4. If X and X’ are two categories, a functor F : X + x’ is a pair of functions, an object function and a mapping function. The object function assigns to each object X of the first category X an object F ( X ) of 1’; the mapping function assigns to each morphism f :X + Y of the first category a morphism F(f):F(X) + F ( Y ) of the second category 8’. These functions must satisfy two requirements :

FU,) = I F ( X ) F(g .f)= F ( g ) . F(f)

for each identity I, of X for each composite g . f defined in X

5. To each category X we can construct another category as follows : Objects of Pare all the objects of X ;morphisms of Pare f O P : Y + X , one for each morphism f :X + Y in 8 ; composites fOP.gop = ( g .f ) O P in P are defined whenever g .f is defined. Pwill be denoted by FP.

6. If F, G : X -+ x’ are functors, a natural transformation T : F -+ G from F to G is a function which assigns to each object X of X a morphism z ( X ) : F ( X )-+ G ( X ) of x’ in such a way that every morphism f:X+ Y of X yields a commutative diagram

7. To each pair of categories X and X’,we can construct another category X x x’called product category as follows: An object of 8 x X’ is an ordered pair (X, X ) of objects of 8 and 8’,respectively. A morphism ( X , X’) + ( Y , Y’) with the indieted domain and codomain is an ordered pair (f,f’) of morphismsf :X + Y, f’:X‘ + Y’.The composite‘ofmorphisms is defined termwise.

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Appendix ZV 8. If F :X

X' is a functor, we can construct another functor Mor(F-, -):J?"' x X'+ Set

Mor (F -, -)(X, X') 3 Mor (FX, X') is the set of all morphisms FX + X'; Mor(F-, - ) ( f o p , g) E Mor(F'g) of a morphism ( f o p , g):(Y, X') + (X, Y') is a mapping Mor (Ff, 8):Mor(FX, X') + Mor (FY, Y ' ) such that for every h E Mor (FY, X')

-

Mor (Ff,g)(h) = g h . Ff 9. A natural transformation z:F --+ G from a functor F:X -,X' to a functor G :X + x'is termed a natural isomorphism if and only if for each X, z(X) is invertible in x'.If z is a natural isomorphism, we shall write F z G. 10. Each pair of functors F :X-, x'and G : X ' -, X is called a pair of Mor(-, G - ) : X o P x x'; F is called adjoint functors iff Mor (F-, -) left adjoint to G, and G is called right adjoint to F.