An Introduction To Category Theory

1
An Introduction To Category Theory

• By Eden Burton
• McMaster University
• November 18, 2008

2
History of Category Theory

• introduced to mathematics world by Samuel
  Eilenberg and Sauders MacLane in 1944
• found as part of their work in topology

3
Categories Defined

• theories of functions with only composition as
  an operation
• collection of abstract objects and the
  relationships between them
• objects typically belong to a class of
  mathematical structures
• relationships are defined as morphisms mapping
  between one object and another
• rules that must be satisfied preserve the
  structure of the objects

4
Categories Defined (cont)

• a category C is .
• a collection of objects ob(C) .. a, b, c .
• a collection of morphisms f, g .
• A set of morphisms from object a into b is
  denoted by Homc(a, b) or a?b. a,b ? ob(C)
• a composition function ?
• Homc(b, c) X Homc(a, b) ? Homc(a, c) a,b,c ?
  ob(C)
• an identity function
• 1a ? Homc(a, a) a ? ob(C)

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Categories Defined (cont)

• the category must satisfy the following rules
• associativity
• (h ? g) ? f h ?(g ? f) a,b,c,d ? ob(C), f ?
  Homc(a, b) g ? Homc(b, c), h ? Homc(c, d)
• unit laws
• f ? 1a f 1b ? f

6
Functors

• a category of categories
• objects are categories, morphisms are mappings
  between categories
• preserves identity and composition properties
• definition - functor F from category C1 and C2
• map of all objects from a ? C1 to an object F(a)
  in C2
• map of all morphisms from Homc1(a, b) to
  Homc1(F(a), F(b) )
• F(1a) 1F(a) (preserves identity)
• F(f ? g) F(f) ? F(g) (preserves composition)

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Category Examples

• deductive systems
• objects are propositions
• morphisms are proofs of a b
• sets
• objects are sets, morphisms are functions
• pre-orders
• objects are elements of partial order, morphism
  is the relation
• specifications
• objects are specifications, morphisms translate
  vocabulary of one specification to another

8
Resources

• online
• John Baez - UC _at_ Riverside (http//math.ucr.edu/ho
  me/baez/categories.html)
• Steve Easterbrook UT (http//www.cs.toronto.edu/
  sme/presentations/cat101.pdf)
• Tom Leinster University of Glasgow
  (http//www.maths.gla.ac.uk/tl/ct/)
• books
• An Introduction to Category Theory (Asperti,
  Longo)
• Introduction To The Theory Of Categories (Bucur)
• Category Theory (Herrlich, Strecker)