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Push-forward ?

  1. Jun 6, 2009 #1
    "push-forward" ?

    Can someone help me understand what this is, in as simple terms as possible?

    If I have a function [tex]f: A\rightarrow B[/tex] and another one [tex]g:B\rightarrow C[/tex] I know the "pullback" [tex]f^{*}g: A\rightarrow C[/tex] is [tex]f^{*}g = g\circ f[/tex] (correct?)

    But what about the push forward [tex]f_{*}g[/tex]? What is that?

    Thanks for any help.
     
  2. jcsd
  3. Jun 6, 2009 #2

    Hurkyl

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    Re: "push-forward" ?

    What is the context?



    Anyways, I imagine you're just talking about functions acting on functions, in which case

    f*(g) = g o f = g*(f)
     
  4. Jun 6, 2009 #3

    dx

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    Re: "push-forward" ?


    Firstly, g is defined on B, so you can't push it forward with f because f: A → B. You can only push things forward with f when they are defined on A, and then too only if they are covariant. Functions are contraviarant, so even if G was a function defined on A, you cannot push it forward with f.
     
  5. Jun 6, 2009 #4
    Re: "push-forward" ?

    Thanks for your help guys...

    Yes Hurkyl I'm just doing functions on functions.

    dx I'm a little confused - if [tex]g_{*}f=g\circ f[/tex] as Hurkly says then we'd have f mapping A to B, then g mapping B to C ... wouldn't that be OK? Perhaps I'm missing something serious here :S ... could you give me a simple example?

    Also what do you mean by covariant and contravariant?
     
  6. Jun 7, 2009 #5

    dx

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    Re: "push-forward" ?

    I'm not sure what Hurkyl meant, but g o f = f*g, not gf. Contravariant objects are things that can be pulled back, and covariant objects are things that can be pushed forward. As I said, functions are contravariant, so a function on B can be pulled back by f : A → B to give you a function on A: f*g = g o f.
     
  7. Jun 7, 2009 #6

    Hurkyl

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    Re: "push-forward" ?

    Functions are contravariant on their domain, but covariant on their codomain.

    Well, that's somewhat of an abuse of language. More accurately, "the set of functions from X to Y" is a functor contravariant in the variable X and covariant in the variable Y.

    If I denote it by Hom(X,Y), then:

    For any function f:Y->Z, I have f*:Hom(X,Y)->Hom(X,Z) given by f*(g) = f o g

    For any function f:W->X, I have f*:Hom(X,Y)->Hom(W,Y) given by f*(g) = g o f


    For an example in the setting of manifolds, recall that for a manifold M, we define a "curve on M" to be a continuous function [0,1]->M. This is a case where we fix the domain and vary the codomain, so curves get pushed around covariantly: given any continuous function f:M->N and curve c on M, we have a pushforward curve f*(c) on N, given by f*(c) = f o c.
     
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