Ideals and continuity for quantaloid-enriched categories. (English) Zbl 1518.18005
Whereas a quantale is a monoid in the symmetric monoidal closed category \(\boldsymbol{Sup}\)of complete lattices and supremum-preserving morphisms, a quantaloid \(\mathcal{Q}\)is a category enriched in \(\boldsymbol{Sup}\).
Domain theory [S. Abramsky (ed.) et al., Handbook of logic in computer science. Vol. 3: Semantic structures. Oxford: Clarendon Press (1994; Zbl 0829.68111); G. Gierz et al., Continuous lattices and domains. Cambridge: Cambridge University Press (2003; Zbl 1088.06001)] originated from the work of D. Scott [Lect. Notes Math. 274, 97–136 (1972; Zbl 0239.54006); Courant Computer Sci. Sympos. 2(1970), 65–106 (1972; Zbl 0279.68042); D. Scott and C. Strachey, Microwave Res. Inst. Sympos. Ser. 21, 19–46 (1971; Zbl 0268.68004); D. Scott, Lect. Notes Math. 188, 311–366 (1971; Zbl 0228.68016)] as a formal basis for the semantics of programming languages, developing into so-called quantitative domain theory, where one uses quantale-enriched categories, for which general concepts of approxiamtion and convergence are addressed.
I. Stubbe [Theor. Comput. Sci. 373, No. 1–2, 142–160 (2007; Zbl 1111.68073)] studied \(\mathcal{Q}\)-categories as crucial mathematical structure for a so-called dynamic logic as common mathematical foundation for dynamic phenomena in both computer science and physics.This paper aims to further develop [loc. cit.] by defining \(\mathcal{Q}\)-categories relative to any saturated class of presheaves. The authors propose several saturated classes of presheaves that are natural generalizations of the concept of ideals in posetal domain theory. They establish quantitative domain theory based on \(\mathcal{Q}\)-categories, which admits of unification of quantale-enriched categories and the approach of partial metric spaces into one framework.
The synopsis of the paper goes as follows.
Domain theory [S. Abramsky (ed.) et al., Handbook of logic in computer science. Vol. 3: Semantic structures. Oxford: Clarendon Press (1994; Zbl 0829.68111); G. Gierz et al., Continuous lattices and domains. Cambridge: Cambridge University Press (2003; Zbl 1088.06001)] originated from the work of D. Scott [Lect. Notes Math. 274, 97–136 (1972; Zbl 0239.54006); Courant Computer Sci. Sympos. 2(1970), 65–106 (1972; Zbl 0279.68042); D. Scott and C. Strachey, Microwave Res. Inst. Sympos. Ser. 21, 19–46 (1971; Zbl 0268.68004); D. Scott, Lect. Notes Math. 188, 311–366 (1971; Zbl 0228.68016)] as a formal basis for the semantics of programming languages, developing into so-called quantitative domain theory, where one uses quantale-enriched categories, for which general concepts of approxiamtion and convergence are addressed.
I. Stubbe [Theor. Comput. Sci. 373, No. 1–2, 142–160 (2007; Zbl 1111.68073)] studied \(\mathcal{Q}\)-categories as crucial mathematical structure for a so-called dynamic logic as common mathematical foundation for dynamic phenomena in both computer science and physics.This paper aims to further develop [loc. cit.] by defining \(\mathcal{Q}\)-categories relative to any saturated class of presheaves. The authors propose several saturated classes of presheaves that are natural generalizations of the concept of ideals in posetal domain theory. They establish quantitative domain theory based on \(\mathcal{Q}\)-categories, which admits of unification of quantale-enriched categories and the approach of partial metric spaces into one framework.
The synopsis of the paper goes as follows.
- § 2
- is concerned with preliminaries on quantaloid-enriched categories.
- § 3
- studies, based on a saturated class \(\Phi\)of presheaves, \(\Phi \)-cocompleteness of \(\mathcal{Q}\)-categories.
- § 4
- gives three saturated classes of presheves, namely, that of all irreducible ideals, that of all flat ideals and that of all conical ideals.
- § 5
- defines \(\Phi\)-continuity in \(\mathcal{Q}\)-categories, studying it as a generalization of continuity in domain theory and computing several examples to show the non-triviality of the generalization.
- § 6
- defines \(\Phi\)-algebraic \(\mathcal{Q}\)-categories, studying them.
- § 7
- ends the paper with a conclusion.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18B35 | Preorders, orders, domains and lattices (viewed as categories) |
| 18D20 | Enriched categories (over closed or monoidal categories) |
| 06F07 | Quantales |
Citations:
Zbl 0829.68111; Zbl 1088.06001; Zbl 0239.54006; Zbl 0279.68042; Zbl 0268.68004; Zbl 0228.68016; Zbl 1111.68073References:
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