S. N. Hosseini | Searcharxiv

arXiv2019

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

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

math.CT↗

arXiv2019

On General Closure Operators and Quasi Factorization Structures

In this article the notions of (quasi weakly hereditary) general closure operator $\mb{C}$ on a category $\cx$ with respect to a class $\cm$ of morphisms, and quasi factorization structures in a category $\cx$ are introduced. It is shown that under certain conditions, if $(\ce, \cm)$ is a quasi factorization structure in $\cx$, then $\cx$ has quasi right $\cm$-factorization structure and quasi left $\ce$-factorization structure. It is also shown that for a quasi weakly hereditary and quasi idempotent QCD-closure operator with respect to a certain class $\cm$, every quasi factorization structure $(\ce, \cm)$ yields a quasi factorization structure relative to the given closure operator; and that for a closure operator with respect to a certain class $\cm$, if the pair of classes of quasi dense and quasi closed morphisms forms a quasi factorization structure, then the closure operator is both quasi weakly hereditary and quasi idempotent. Several illustrative examples are furnished.

math.CT↗