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Main Notions of the Category Theory

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Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

A category C consists of the following data:
  1. a)

    A class ObC whose elements are called objects of C.

     
  2. b)

    A collection of sets Hom(X, Y), one for each ordered pair of objects X, Y ∈ ObC, whose elements are called morphisms (from X to Y); they are denoted by ϕ: X → Y.

     
  3. c)
    A collection of mappings
    $$ Hom(X,Y) \times Hom(Y,Z) \to Hom(X,Z), $$
    one for each ordered triple of objects X,Y,Z ∈ ObC. Any mapping in this collection associates with a pair ϕ: X → Y, ψ: Y → Z a morphism from X to Z, denoted by ψ ◦ ϕ or ψϕ: X → Z, and called the composition or product of ϕ and ψ.
     

Keywords

Abelian Group Exact Sequence Category Theory Full Subcategory Abelian Category 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.American Mathematical SocietyProvidenceUSA
  2. 2.Max Planck Institute for MathematicsBonnGermany

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