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Graph and Free Graph in Category Theory

  1. Oct 16, 2008 #1
    Cat(FG, B) [tex]\cong[/tex]Grph(G, UB)

    Cat denotes the category of all small categories and Grph denotes the category of all small graphs.
    G is a small graph which consists of small set O of objects and small set A of arrows f (CWM, PP48-51)
    UB is a forgetful functor applied to category B which is an underlying graph of a category B.
    Morphism of graphs D:G->UB corresponds to D[tex]\acute{}[/tex]:FG->B

    I can't figure out how D[tex]\acute{}[/tex] looks like and how the mapping behaves.
    For D:G->UB, CWM (p50) says it sends each arrow f:a1->a2 of the given graph G to the string <a1,f,a2> of length 2 in UB.

    Now, if G is freely generated to make a category FG, how is it generated and how does it look like?
    A category itself can be described in a graph form. What would be the difference between B and UB if a forgetful functor is applied to B?

    Any advice will be appreciated.
    Last edited: Oct 16, 2008
  2. jcsd
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