I've seen the word "pullback" used in a lot of situations. Examples are as follows.
(In category theory) Given morphisms $f:X\to M,\ g:Y\to M$. A pullback square consists of $f,g$ and two other morphisms $\eta:L\to X$ and $\xi:L\to Y$ such that
- $f\eta=g\xi$
- Any other pair of morphisms $\eta':L'\to X$ and $\xi':L'\to Y$ for which $f\eta'=g\xi'$ can be factored through $\eta$ and $\xi$.
Another usage is in differential geometry.
(In differential geometry) If $F:M\to N$ is a smooth map and $\omega$ is a differential form on $N$, the pullback $F^*\omega$ is a differential form on $M$, defined as...(omitted)
I am curious whether there is a link between these two concepts, as they are both called pullback. Tntuition tells me the pullback in category theory is supposed to be a boarder concept, and we should be able to describe the pullback of a differential form using the language of category theory. However I fail to see the connection.
Other examples are welcome.