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Category and Subcategory

  1. Oct 5, 2012 #1

    SVD

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    Using the definition given below, I wonder whether we can deduce that for each object A in C',
    the identity for A in C' coincides with the identity for A in C.

    Let C' and C be two categories which satisfies that
    (i)each objects in C' belongs to C
    (ii)each hom-set in C' is contained in the corresponding hom-set in C.
    (iii)each composition in C' is the restriction of that in C .
     
  2. jcsd
  3. Oct 5, 2012 #2
    No. Let C and C' consist out of exactly one object A. Let

    [tex]Hom_{C^\prime}(A,A)=\{a\}~\text{and}~Hom_C(A,A)=\{a,b\}[/tex]

    with

    [tex]a\circ a = a\circ b=b\circ a=a~\text{and}~b\circ b=b[/tex]
     
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