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For a composition-closed and pullback-stable class S of morphisms in acategory C containing all isomorphisms, we form the category Span(C,S) ofS-spans (s,f) in C with first "leg" s lying in S, and give an alternativeconstruction of its quotient category C[S^{-1}] of S-fractions. Instead oftrying to turn S-morphisms "directly" into isomorphisms, we turn themseparately into retractions and into sections in a universal manner, thusobtaining the quotient categories Retr( C,S) and Sect(C,S). The fractioncategory C[S^{-1}] is their largest joint quotient category.  Without confining S to be a class of monomorphisms of C, we show thatSect(C,S) admits a quotient category, Par(C,S), whose name is justified by twofacts. On one hand, for S a class of monomorphisms in C, it returns thecategory of S-spans in C, also called S-partial maps in this case; on the otherhand, we prove that Par(C,S) is a split restriction category (in the sense ofCockett and Lack). A further quotient construction produces even a rangecategory (in the sense of Cockett, Guo and Hofstra), RaPar(C,S), which is stilllarge enough to admit C[S^{-1}] as its quotient.  Both, Par and RaPar, are the left adjoints of global 2-adjunctions. Whenrestricting these to their "fixed objects", one obtains precisely the2-equivalences by which their name givers characterized restriction and rangecategories. Hence, both Par(C,S)$ and RaPar(C,S may be naturally presented asPar(D,T)$ and RaPa(D,T), respectively, where now T is a class of monomorphismsin D. In summary, while there is no {\em a priori} need for the exclusiveconsideration of classes of monomorphisms, one may resort to them naturally

Fraction, Restriction and Range Categories from Non-Monic Classes of Morphisms