arXiv:1903.00081v1 [math.CT] 28 Feb 2019 Abandoning Monomorphisms: Partial Maps, Fractions and Factorizations ✩ S.N. Hosseini a , A.R. Shir Ali Nasab a , W. Tholen b,∗ a Mahani Mathematical Research Center Shahid Bahonar University of Kerman, Kerman, Iran b Department of Mathematics and Statistics York University, Toronto, Canada Abstract For a composition-closed and pullback-stable class S of morphisms in a cat- egory 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 con- struction 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. Without confining S to be a class of monomorphisms of C , the second of these two quotient processes leads us to the category Par(C , S ) of S -partial maps in C . Under mild additional hypotheses on S , Par(C , S ) has a localization, which is a split restriction category, or even a split range category (in the sense of Cockett, Guo and Hofstra), but which is still large enough to admit C [S −1 ] as its quo- tient. The construction of the range category is part of a global adjunction between relatively stable factorization systems and split range categories. Keywords: span category, partial map category, category of fractions, localization, restriction category, range category. 2000 MSC: 18A99, 18B99, 18A32. ✩ Much of the work presented in this paper was done while the second author visited York University in 2018, with the partial financial assistance from the third author’s NSERC Discovery Grant (no. 050126), which is gratefully acknowledged. The first two authors acknowledge also the partial support by Mahani Mathematical Research Center. * Corresponding author Email addresses: nhoseini@uk.ac.ir (S.N. Hosseini), ashirali@math.uk.ac.ir (A.R. Shir Ali Nasab), tholen@mathstat.yorku.ca (W. Tholen) Preprint submitted to Elsevier March 4, 2019