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Induced Homomorphs, Isomorphs.

  1. Feb 3, 2008 #1

    WWGD

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    Hi, everyone:
    I keep seeing, mostly in homological algebra, the use of "induced
    homomorphs" or "induced isomorphisms". I get the idea of what is
    going on, but I have not been able to find a formal result that
    rigorously explains this, i.e, under what conditions does a map
    induce an isomorphism or a homomorphism?. The only patterns
    there seems to be in all these induced maps is that they are all
    defined in some quotient space of the domain, i.e, if
    we have f:X->Y , then the induced maps f* are , or seem to be,
    defined in X/~ , for some equiv. relation ~ (e.g, in homotopy and
    homology). Also, maybe obviously, f is a continuous map.

    Basically: I would like to know a result that would allow me to
    give a yes/no answer to the question : does f:X->Y induce an
    isomorphism/homomorphism of some sort?

    Thanks.
     
  2. jcsd
  3. Mar 10, 2008 #2

    WWGD

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    Just in case someone else is interested, I think I have at least a partial answer,
    i.e, 2 cases in which maps are induced:

    In some cases, maps are induced when we consider categories associated with
    spaces, i.e, if we are given a map f (continuous) , f:X->Y , we consider (f_x,O_x)
    and (f_y,O_y) as categories , i.e, f_x,f_y are morphisms, and O_x,O_y are
    objects, respectfully (of course, we can only do this in some cases only).

    As an example, we would consider homology and homotopy as functors.
    Anyway, that is a sketch.
     
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