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 RaPar ( D , T ) , respectively, where now T is a class of monomorphisms in D . In summary, while there is no a priori need for the exclusive consideration of classes of monomorphisms, one may resort to them naturally.

中文翻译：

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来自稳定态射系统的分数、限制和范围类别

摘要 对于包含所有同构的类别 C 中的态射的复合封闭和回拉稳定类 S，我们形成 C 中 S 跨度 ( s , f ) 的范畴跨度 ( C , S )，其中第一条“腿”位于在 S 中，并给出 S 分数的商类别 C [ S − 1 ] 的替代构造。我们没有尝试将 S 态射“直接”转化为同构，而是以通用方式将它们分别转化为缩回和节，从而获得商类别 Retr (C, S) 和 Sect (C, S)。分数类别 C [ S − 1 ] 是它们最大的联合商类别。不将 S 限制为 C 的一类单态，我们证明 Sect ( C , S ) 承认商范畴 Par ( C , S ) ，其名称由两个事实证明。一方面，对于 S 中的一类单态，它返回 C 中 S -spans 的类别，在这种情况下也称为 S -部分映射；另一方面，我们证明 Par ( C , S ) 是一个分裂限制范畴（在 Cockett 和 Lack 的意义上）。进一步的商构造甚至产生了一个范围范畴（在 Cockett、Guo 和 Hofstra 的意义上），RaPar ( C , S ) ，它仍然足够大以允许 C [ S - 1 ] 作为其商。Par 和 RaPar 都是全局 2-adjunctions 的左伴随。当将这些限制到它们的“固定对象”时，人们精确地获得了它们的命名者用来表征限制和范围类别的 2-等价。因此， Par ( C , S ) 和 RaPar ( C , S ) 可以自然地分别表示为 Par ( D , T ) 和 RaPar ( D , T ) ，其中现在 T 是 D 中的一类单态。总之，