A function is normally defined to be an object *f : X →* Y which assigns to each point *x* in a set X, known as the *domain,* a point *f (x)* in another set Y, known as the *range* (see THE LANGUAGE AND GRAMMAR OF MATHEMATICS [I.2 §2.2]). Thus, the definition of functions is set-theoretic and the fundamental operation that one can perform on a function is *evaluation:* given an element *x* of X, one evaluates *f* at *x* to obtain the element *f(x)* of Y.

However, there are some fields of mathematics where this may not be the best way of describing functions. In geometry, for instance, the fundamental property of a function is not necessarily how it acts on points, but rather how it *pushes forward* or *pulls back* objects that are ...

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