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Category theory

  1. Feb 5, 2005 #1
    One question for now:
    1. If C is a full subcategory of D and D is a full subcategory of C, what can be said, if anything, about C and D being either equal or equivalent in some way?
     
  2. jcsd
  3. Feb 5, 2005 #2

    Hurkyl

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    They'd have to be equal.

    Every object of C is an object of D, and vice cersa. Full subcategories are determined by their objects.
     
  4. Feb 6, 2005 #3
    2. Could you explain slice categories?

    I have a quote that I don't get:
    In the "above diagram," f:B-->A, g:C-->A, and h:B-->C s.t. gh=f.

    Thanks in advance...
     
  5. Feb 6, 2005 #4

    Hurkyl

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    Have you worked much with fiber bundles? I think they're a good model to understand what's going on.

    You can imagine the objects of the slice category are the projections from objects of the original category to the base object.

    For a given projection, each "point" X of the base object corresponds to a "fiber", those "points" of the source object that project onto X. (Thus, we have a B-indexed family of fibers)

    The morphisms of the slice category, then, are the morphisms that act "fiber-wise". That is, if P is in a fiber of X, then f(P) is also in a fiber of X.

    Of course, in general you won't have points and fibers to manipulate, but I think this is the spirit behind it.
     
    Last edited: Feb 6, 2005
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