Fraction, restriction, and range categories from stable systems of morphisms
| Autor: | Walter Tholen, A.R. Shir Ali Nasab, S. N. Hosseini |
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| Rok vydání: | 2020 |
| Předmět: |
Class (set theory)
Algebra and Number Theory Quotient category Span (category theory) 010102 general mathematics 0102 computer and information sciences 01 natural sciences Combinatorics Morphism 010201 computation theory & mathematics Mathematics::Category Theory Fraction (mathematics) 0101 mathematics Quotient Mathematics Range (computer programming) |
| Zdroj: | Journal of Pure and Applied Algebra. 224:106361 |
| ISSN: | 0022-4049 |
| DOI: | 10.1016/j.jpaa.2020.106361 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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