Two varieties are birationally equivalent if and only if their function fields are isomorphic. (But note that non-isomorphic varieties may have the same function field!) Assigning to each variety its function field yields a duality (contravariant equivalence) between the category of varieties over "k" (with dominant rational maps as morphisms) and the category of algebraic function fields over "k". (The varieties considered here are to be taken in the scheme sense; they need not have any "k"-rational points, like the curve defined over the reals, that is with .)