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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? – Asaf Karagila Feb 23 '12 at 15:45
@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. – ItsNotObvious Feb 23 '12 at 15:48
@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". – Arturo Magidin Feb 23 '12 at 15:50
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. – Arturo Magidin Feb 23 '12 at 15:51
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). – M Turgeon Feb 23 '12 at 16:01

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