16. Category Theory

16.1 Definition:

A category $\QTR{bs}{C}$ consists of:

  1. a class of objects, written $\QTR{bs}{Ob(C)}$

  2. For every two objects $\QTR{Large}{A}$ and $\QTR{Large}{B}$ in $\QTR{bs}{C}$, a set MATH of morphisms from $\QTR{Large}{A}$ to $\QTR{Large}{B}$ such that if MATHand/or MATHthen MATH. One writes MATH or MATHfor MATH

  3. For every three objects MATHand$\QTR{Large}{\ C}$, there is a binary operation MATH called composition (we write $\QTR{Large}{gf}$ or sometimes MATH )such that the following axioms hold:

         a. associativity:

        If MATH ,MATH, and MATH then MATH,

        b. identity:

        For every object $\QTR{Large}{X}$ there exists a morphism MATH MATHcalled the identity morphism for $X $, such that for         every morphism MATH we have MATH.

16.2 Examples:

  1. The Category of Sets -

  2. The Categories of posets, Ordered, or Well Ordered Sets -

  3. The Category of Metric Spaces -

  4. The Category or Groups, Rings, Fields etc -

16.3 Theorem:

In any Category $\QTR{bs}{C}$ the identity morphisms are unique.

Proof:

Suppose we have and object $\QTR{Large}{A}$ and two morphisms MATH and MATH that both

satify 16.1 b. Then MATH.

16.4 Definition:

Fixing a Category,

  1. A morphism MATHis called an epimorphism if for any object $\QTR{Large}{D}$ and morphisms MATH , MATH if and only if MATH

  2. A morphism MATHis called a monomorphism if for any object $\QTR{Large}{A}$ and morphisms MATH , MATH if and only if MATH.

16.5 Examples:

  1. The Category of Sets - onto is equivalent to epimorphism, and one to one is equivalent to monomorphism.

  2. The Category of Metric Spaces - Consider $\QTR{Large}{id:(}$0,1MATH0,1$\QTR{Large}{]}$ . This is an epimorphism.

    Exercise Prove this. ( Hint- Review the definition of continuity)

  3. One can find examples of monomorphisms that are not one to one as Set maps.