For a composition-closed and pullback-stable class S of morphisms in a
category C containing all isomorphisms, we form the category Span(C,S) of
S-spans (s,f) in C with first "leg" s lying in S, and give an alternative
construction of its quotient category C[S^{-1}] of S-fractions. Instead of
trying to turn S-morphisms "directly" into isomorphisms, we turn them
separately into retractions and into sections in a universal manner, thus
obtaining the quotient categories Retr( C,S) and Sect(C,S)...
The fraction
category C[S^{-1}] is their largest joint quotient category. Without confining S to be a class of monomorphisms of C, we show that
Sect(C,S) admits a quotient category, Par(C,S), whose name is justified by two
facts. On one hand, for S a class of monomorphisms in C, it returns the
category of S-spans in C, also called S-partial maps in this case; on the other
hand, we prove that Par(C,S) is a split restriction category (in the sense of
Cockett and Lack). A further quotient construction produces even a range
category (in the sense of Cockett, Guo and Hofstra), RaPar(C,S), which is still
large enough to admit C[S^{-1}] as its quotient. Both, Par and RaPar, are the left adjoints of global 2-adjunctions. When
restricting these to their "fixed objects", one obtains precisely the
2-equivalences by which their name givers characterized restriction and range
categories. Hence, both Par(C,S)$ and RaPar(C,S may be naturally presented as
Par(D,T)$ and RaPa(D,T), respectively, where now T is a class of monomorphisms
in D. In summary, while there is no {\em a priori} need for the exclusive
consideration of classes of monomorphisms, one may resort to them naturally
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Abstract