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Fraction, restriction, and range categories from stable systems of morphisms. (English) Zbl 1454.18001

The formation of the category \(\mathcal{C}\left[ \mathcal{S}^{-1}\right]\) of fractions with respect to a sufficiently well-behaved class \(\mathcal{S}\) of morphisms in \(\mathcal{C}\), which is a fundamental device in homotopy theory, was first given in [P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory. Berlin-Heidelberg-New York: Springer-Verlag (1967; Zbl 0186.56802)] . The construction is characterized by its localizing functor \[ \mathcal{C}\rightarrow\mathcal{C}\left[ \mathcal{S}^{-1}\right] \] which is universal with respect to the property of turning morphisms in \(\mathcal{S}\) into isomorphisms. The question of the size of the “homs” of \(\mathcal{C}\left[ \mathcal{S}^{-1}\right]\) is highly delicate.
This paper aims, assuming that \(\mathcal{S}\) contain all isomorphisms, is closed under composition, and is stable under pullbacks in \(\mathcal{C}\), to take a stepwise approach to the formation of \(\mathcal{C}\left[ \mathcal{S}^{-1}\right]\), considering separately the two processes of transforming every morphism in \(\mathcal{S}\) into a retraction and into a section before amalgamating them to obtain the category of fractions.
A synopsis of the paper consisting of ten sections goes as follows. §2 is concerned with span categories \(\mathsf{Span}(\mathcal{C},\mathcal{S})\) and their quotients. §3 forms the \(S\)-retractable span category \(\mathsf{Retr}(\mathcal{C},\mathcal{S})\) of \(\mathcal{C}\), while §4 forms the \(S\)-sectional span category \(\mathsf{Sect}(\mathcal{C},\mathcal{S})\) of \(\mathcal{C}\). §5 shows how to amalgamate the two constructions to obtain the category \(\mathcal{C}\left[ \mathcal{S}^{-1}\right]\), performing and characterizing these constructions strictly at the ordinary category level. The \(2\)-categorical structure of \(\mathsf{Span}(\mathcal{C},\mathcal{S})\) [J. Bénabou, Lect. Notes Math. 47, 1–77 (1967; Zbl 1375.18001); C. Hermida, Adv. Math. 151, No. 2, 164–225 (2000; Zbl 0960.18004)] is alluded to in §10, where it is indicated how the constructions of \(\mathsf{Retr}(\mathcal{C},\mathcal{S})\) and \(\mathsf{Sect}(\mathcal{C},\mathcal{S})\) are naturally motivated.
§6 elaborates on how to obtain the \(S\)-partial map category \(\mathsf{Par}(\mathcal{C},\mathcal{S})\) as a quotient category of \(\mathsf{Sect}(\mathcal{C},\mathcal{S})\), which is a restriction category. Under a fairly mild additional hypothesis on \(S\) holding in particular under the weak left cancellation condition (\(s,s\cdot t\in\mathcal{S}\Longrightarrow t\in\mathcal{S}\)), \(\mathsf{Par}(\mathcal{C},\mathcal{S})\) is a localization of \(\mathsf{Sect}(\mathcal{C},\mathcal{S})\) making \(\mathsf{Retr}(\mathcal{C},\mathcal{S})=\mathcal{C}\left[ \mathcal{S}^{-1}\right]\) its quotient category.§8 presents the construction of the \(\mathcal{S}\)-partial map range category \(\mathsf{RaPar}(\mathcal{C},\mathcal{S})\), completing the quotient construction in the paper and yielding the commutative diagram \[ \begin{matrix} \mathcal{C} & \rightarrow & \mathsf{Span}(\mathcal{C},\mathcal{S}) & \rightarrow & \mathsf{Sect}(\mathcal{C},\mathcal{S})& \rightarrow & \mathsf{Par}(\mathcal{C},\mathcal{S})\\ & & \downarrow & & \downarrow & & \\ & & \mathsf{Retr}(\mathcal{C},\mathcal{S}) & \rightarrow & \mathcal{C}\left[ \mathcal{S}^{-1}\right] & & \end{matrix} \]
Extending a key result in [J. R. B. Cockett and S. Lack, Theor. Comput. Sci. 270, No. 1–2, 223–259 (2002; Zbl 0988.18003)] , §7 provides a setting which presents \[ (\mathcal{C},\mathcal{S})\longmapsto\mathsf{Par}(\mathcal{C},\mathcal{S}) \] as the left adjoint to the formation of the category \(\mathsf{Total}\left( \mathcal{X}\right)\) for every split restriction category \(\mathcal{X}\). Extending one of the principal results in [J. R. B. Cockett et al., Theory Appl. Categ. 26, 412–452 (2012; Zbl 1252.18003)] , §9 provides a setting which presents \[ (\mathcal{C},\mathcal{S})\longmapsto\mathsf{RaPar}(\mathcal{C},\mathcal{S}) \] as the left adjoint to the formation of the category \(\mathsf{Total}\left( \mathcal{X}\right)\) for every split range category \(\mathcal{X}\).
To conclude this review, let me remark that:
The authors give their earlier version of the paper [arXiv:1903.00081] as the eighth item in the References, note that the initial title in v1 (identical to the paper) has changed in v2 \[ \text{Abandoning monomorphisms:partial maps, fractions, factorizations} \Longrightarrow\text{Fraction, restriction, and range categories from non-monic classes of morphisms} \]
In §1, \[ \text{See Sections 3 and 2, respectively}\Longrightarrow\text{See Sections 4 and 3, respectively} \] \[ \text{In Section 4}\Longrightarrow\text{ In Section 5} \] \[ \text{That is why, in Section 5}\Longrightarrow\text{That is why, in Section 6} \]

MSC:

18A99 General theory of categories and functors
18B99 Special categories
18A32 Factorization systems, substructures, quotient structures, congruences, amalgams
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References:

[1] Bénabou, J., Introduction to Bicategories, Lecture Notes in Mathematics, vol. 47, 1-77 (1967), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York · Zbl 1375.18001
[2] Borceux, F., Handbook of Categorical Algebra 1, Basic Category Theory (1994), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0803.18001
[3] Cockett, J. R.B.; Guo, X.; Hofstra, P., Range categories I: general theory, Theory Appl. Categ., 26, 17, 412-452 (2012) · Zbl 1252.18003
[4] Cockett, J. R.B.; Lack, S., Restriction categories I, Theor. Comput. Sci., 270, 223-259 (2002) · Zbl 0988.18003
[5] Gabriel, P.; Zisman, M., Calculus of Fractions and Homotopy Theory (1967), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-NewYork · Zbl 0186.56802
[6] Hermida, C., Representable multicategories, Adv. Math., 151, 164-225 (2000) · Zbl 0960.18004
[7] Hosseini, S. N.; Mielke, M. V., Universal monos in partial morphism categories, Appl. Categ. Struct., 17, 435-444 (2009) · Zbl 1192.18001
[8] Hosseini, S. N.; Shir Ali Nasab, A. R.; Tholen, W., Abandoning monomorphisms: partial maps, fractions, factorizations
[9] Schubert, H., Categories (1972), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York · Zbl 0253.18002
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