What is the 'Push-Forward' Function f_{*}g?

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Discussion Overview

The discussion revolves around the concept of the "push-forward" function f_{*}g in the context of functions acting on other functions. Participants explore the definitions and implications of push-forward and pullback operations, particularly in relation to covariant and contravariant objects.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks a simple explanation of the push-forward function f_{*}g, noting their understanding of the pullback f^{*}g as g \circ f.
  • Another participant questions the context of the discussion and suggests that the operations involve functions acting on functions.
  • A participant argues that g cannot be pushed forward with f since g is defined on B and f maps A to B, indicating that push-forward operations are only applicable to objects defined on A and that functions are contravariant.
  • In response, a participant expresses confusion about the previous points and requests a simple example, while also asking for clarification on the terms covariant and contravariant.
  • One participant clarifies that functions are contravariant and explains that a function on B can be pulled back by f to give a function on A, reiterating that g \circ f = f * g.
  • Another participant elaborates on the nature of contravariance and covariance in the context of functions, suggesting that the set of functions from X to Y behaves as a functor that is contravariant in X and covariant in Y.
  • An example involving manifolds is provided, where a curve on a manifold is discussed in relation to push-forward operations, illustrating how curves can be pushed around covariantly through continuous functions.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of push-forward operations to functions, with some asserting that functions are contravariant and cannot be pushed forward, while others provide examples and clarify the definitions. The discussion remains unresolved regarding the precise nature and application of the push-forward function.

Contextual Notes

The discussion includes assumptions about the definitions of covariant and contravariant objects, as well as the context in which push-forward operations are applicable. There are unresolved aspects regarding the implications of these definitions in specific scenarios.

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"push-forward" ?

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

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

But what about the push forward f_{*}g? What is that?

Thanks for any help.
 
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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)
 


... said:
But what about the push forward f_{*}g? What is that?
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.
 


Thanks for your help guys...

Yes Hurkyl I'm just doing functions on functions.

dx I'm a little confused - if g_{*}f=g\circ f 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?
 


I'm not sure what Hurkyl meant, but g o f = f*g, not g∗f. 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.
 


As I said, functions are contravariant, so a function on B
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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