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

## TL;DR

This paper introduces new categorical constructions for fractions and restrictions of morphisms, generalizing existing concepts to broader classes of morphisms beyond monomorphisms, with applications to restriction and range categories.

## Contribution

It provides an alternative construction of quotient categories of fractions using S-spans and introduces new categories Par and RaPar that generalize restriction and range categories beyond monomorphisms.

## Key findings

- Constructed quotient categories of fractions via S-spans and universal methods.
- Established Par(C,S) as a split restriction category and RaPar(C,S) as a range category.
- Demonstrated adjunctions and 2-equivalences linking these categories to existing frameworks.

## Abstract

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