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The concept of "pullback" has several definitions depending on the context in which it is applied, e.g., smooth functions on manifolds, differential forms, multilinear forms and so forth. See, for example, the Wikipedia Page for an enumeration of these definitions. Is there a "universal definition" of pullback from which these various specialized definitions can be derived or should these definitions be viewed as intrinsic and independent? I am aware of the categorical representation as discussed here, but I don't believe (perhaps I'm mistaken) that these various specialized definitions can be derived from the categorical one.

Also, is it a coincidence that the adjoint of an operator and the pullback operation both share the exponentiated $*$ as an indicator or is it reflective of a deeper relationship?

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    The pullback in category theory is just an equation f(x)=g(y), with any x and y. So many different things will match this pattern, including everything in that wikipedia page. The equations in the wikipedia page seem to be special cases where pullback solution is needed, so they're probably listing cases which are difficult to solve because of requirement of a pullback equation.2011-09-29
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    Not an expert on category theory, but in differential geometry, you really don't have any liberty to define a different pullback, i.e. they're defined in the only reasonable way possible.2011-09-29
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    possible duplicate of [Question on pullbacks and compositions](http://math.stackexchange.com/questions/81385/question-on-pullbacks-and-compositions)2012-01-24

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