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I am trying to understand the meaning of a pushforward in the simplest context possible where the functions involved are defined on sets. From my readings in differential geometry, I have arrived at the following understanding that I have attempted to codify in a precise definition. Unfortunately, I have not been able to find a reference that defines the pushforward in this minimal context. My proposed definition is as follows:


Let $\phi:X \rightarrow Y$ be a bijection and let $f:X\rightarrow Z$ be any function from $X$ to the set $Z$. Then, the pushforward of $f$ by $\phi$ is a map $$ \phi_*:Z^X \rightarrow Z^Y $$ defined by $$ \phi_* f := f \circ \phi^{-1}. $$


So my question: Is this definition correct and is this is the right way to think of pushforward when only maps between sets are involved?

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    Pardon my ignorance, but isn't the terminology of "pushforward" (as well pullback/pushout/etc.) is a bit categorical? Shouldn't you therefore add a tag for [category-theory] or something similar?2012-02-23
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    @AsafKaragila Not really sure how it should be tagged. The only place I've encountered "pushforward" is within the context of manifolds but I'm trying to understand the operation at the most fundamental level. For all I know, the "pushforward" may not even technically be defined when only sets are involved...hence the question.2012-02-23
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    @Asaf: For what it is worth, Mac Lane's "Categories for the working mathematician" does not have "pushforward" in the index. I think you are confusing it with "pushout".2012-02-23
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    This is just a special case of the standard fact that a map $f\colon A\to B$ induces a homomorphism $f\colon\mathrm{Hom}(B,Z)\to\mathrm{Hom}(A,Z)$ by precomposition; the only difference is that you are "looking" at $\phi\colon X\to Y$ instead of looking at $\phi^{-1}\colon Y\to X$, which is what you are using for the construction.2012-02-23
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    I think that pushforward usually refers to a *covariant* alternative to the pullback. Hence, in some cases, you can get a covariant map between your Hom-sets (for example, in the context of homology or sheaf theory), but I don't think it's possible in general (for example, on manifolds, to pushforward a vector field, you usually need a diffeomorphism).2012-02-23
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    The point of my comment is that, in general, your only choice for a pushforward would be a pullback in disguise, as in the example you gave. BUT: sometimes, genuine pushforwards exist.2012-02-23

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