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An Introduction To Category Theory

An Introduction To Category Theory. By: Eden Burton McMaster University November 18, 2008. History of Category Theory. introduced to mathematics world by Samuel Eilenberg and Sauders MacLane in 1944 found as part of their work in topology. Categories Defined.

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An Introduction To Category Theory

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  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 (con’t) • 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 Є ob(C) ] • a composition function ○ • Homc(b, c) X Homc(a, b) → Homc(a, c) [ a,b,c Є ob(C)] • an identity function • 1aЄHomc(a, a) [a Є ob(C)]

  5. Categories Defined (con’t) • the category must satisfy the following rules • associativity • (h ○ g) ○ f = h ○(g ○ f) [ a,b,c,d Є ob(C), f ЄHomc(a, b) g ЄHomc(b, c), h ЄHomc(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 Є 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)

  7. 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 @ Riverside (http://math.ucr.edu/home/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)

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