Two types of Galois correspondences over quantaloid-typed sets. (English) Zbl 1475.54005
There exist the so-called Lowen functors, which provide a correspondence between the categories of crisp and fuzzy topological spaces [R. Lowen, J. Math. Anal. Appl. 56, 621–633 (1976; Zbl 0342.54003)]. These functors have already been generalized in several ways (see, e.g., [U. Höhle, Many valued topology and its applications. Boston, MA: Kluwer Academic Publishers (2001; Zbl 0969.54002); U. Höhle and T. Kubiak, Semigroup Forum 75, No. 1, 1–17 (2007; Zbl 1125.06006); D. Zhang, Fuzzy Sets Syst. 140, No. 3, 479–487 (2003; Zbl 1086.54502)]), either replacing the category of topological spaces with the category of, e.g., limit spaces or employing a certain lattice-theoretic structure instead of the unit interval underlying fuzzy topological spaces.
The present paper follows suit and provides yet another extension of Lowen functors. The authors combine the above-mentioned generalizations by, first, replacing the underlying lattice \(L\) of lattice-valued topology by a quantaloid \(\mathcal{Q}\) (see, e.g., [K. I. Rosenthal, The theory of quantaloids. Harlow: Addison Wesley Longman (1996; Zbl 0845.18003)] for more details on quantaloids); and, second, replacing the category of topological spaces by the category of suitably defined limit spaces. Thus, instead of lattice-valued sets, the authors rely on quantaloid-typed sets. Moreover, Lowen functors are represented in the form of a Galois correspondence between concrete categories, employing the language of [J. Adámek et al., Repr. Theory Appl. Categ. 2006, No. 17, 1–507 (2006; Zbl 1113.18001)]. The authors additionally arrive at a quantaloid-enriched extension of the notion of \(\top\)-filter of U. Höhle [Manuscr. Math. 38, 289–323 (1982; Zbl 1004.54500)], which leads to the concept of \(\top\)-limit space as well as a Galois correspondence between the categories of quantaloid-enriched limit spaces and quantaloid-enriched \(\top\)-limit spaces.
The paper is well written, provides most of its required preliminaries, but is quite technical and thus will require a bit of patience from the reader (supposedly a categorical fuzzy topologist), who would like to dwell into all its subtleties.
The present paper follows suit and provides yet another extension of Lowen functors. The authors combine the above-mentioned generalizations by, first, replacing the underlying lattice \(L\) of lattice-valued topology by a quantaloid \(\mathcal{Q}\) (see, e.g., [K. I. Rosenthal, The theory of quantaloids. Harlow: Addison Wesley Longman (1996; Zbl 0845.18003)] for more details on quantaloids); and, second, replacing the category of topological spaces by the category of suitably defined limit spaces. Thus, instead of lattice-valued sets, the authors rely on quantaloid-typed sets. Moreover, Lowen functors are represented in the form of a Galois correspondence between concrete categories, employing the language of [J. Adámek et al., Repr. Theory Appl. Categ. 2006, No. 17, 1–507 (2006; Zbl 1113.18001)]. The authors additionally arrive at a quantaloid-enriched extension of the notion of \(\top\)-filter of U. Höhle [Manuscr. Math. 38, 289–323 (1982; Zbl 1004.54500)], which leads to the concept of \(\top\)-limit space as well as a Galois correspondence between the categories of quantaloid-enriched limit spaces and quantaloid-enriched \(\top\)-limit spaces.
The paper is well written, provides most of its required preliminaries, but is quite technical and thus will require a bit of patience from the reader (supposedly a categorical fuzzy topologist), who would like to dwell into all its subtleties.
Reviewer: Sergejs Solovjovs (Praha)
MSC:
| 54A40 | Fuzzy topology |
| 54A20 | Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) |
| 06F07 | Quantales |
| 18D20 | Enriched categories (over closed or monoidal categories) |
Keywords:
convergence structure; frame; fuzzy set; Galois correspondence; limit space; Lowen functors; presheaf; quantale; quantaloid; quantaloid-enriched category; quantaloid of diagonals; quantaloid-typed set; Scott continuous presheaf; set filter; slice category; stratified topological space; topological category; \(\top\)-filterCitations:
Zbl 0342.54003; Zbl 0969.54002; Zbl 1125.06006; Zbl 1086.54502; Zbl 0845.18003; Zbl 1113.18001; Zbl 1004.54500References:
| [1] | Adámek, J.; Herrlich, H.; Strecker, G. E., Abstract and Concrete Categories (1990), Wiley: Wiley, New York · Zbl 0695.18001 |
| [2] | Boustique, H.; Richardson, G., Regularity: lattice-valued Cauchy spaces, Fuzzy Sets and Systems, 190, 94-104 (2012) · Zbl 1263.54009 · doi:10.1016/j.fss.2011.08.006 |
| [3] | Cook, C. H.; Fischer, H. R., Regular convergence spaces, Math. Annalen, 174, 1-7 (1967) · Zbl 0152.39603 · doi:10.1007/BF01363119 |
| [4] | Craig, C. H.; Jäger, G., A common framework for lattice-valued uniform limit spaces and probabilistic uniform limit spaces, Fuzzy Sets and Systems, 160, 1177-1203 (2009) · Zbl 1184.54006 · doi:10.1016/j.fss.2008.11.002 |
| [5] | Fang, J.-M., Stratified L-ordered convergence structures, Fuzzy Sets and Systems, 161, 2130-2149 (2010) · Zbl 1197.54015 · doi:10.1016/j.fss.2010.04.001 |
| [6] | Fang, J.-M., Relationships between L-ordered convergence structures and strong L-topologies, Fuzzy Sets and Systems, 161, 2923-2944 (2010) · Zbl 1271.54030 · doi:10.1016/j.fss.2010.07.010 |
| [7] | Fang, J.-M., Residuated Lattices and Fuzzy Sets (2012), China Science Publishing: China Science Publishing, Beijing |
| [8] | Fang, J.-M.; Yue, Y., ⊤-Diagonal conditions and continuous extension theorem, Fuzzy Sets and Systems, 321, 73-89 (2017) · Zbl 1379.54007 · doi:10.1016/j.fss.2016.09.003 |
| [9] | Fang, J.-M. and Yue, Y., ⊤-Q-filters and their applications, Abstracts of the 37th Linz Seminar on Fuzzy Set Theory, pp. 31-34, Linz, Austria, 2017. |
| [10] | Flores, P. V.; Mohapatra, R. N.; Richardson, G., Lattice-valued spaces: fuzzy convergence, Fuzzy Sets and Systems, 157, 2706-2704 (2006) · Zbl 1123.54002 · doi:10.1016/j.fss.2006.03.023 |
| [11] | Gutiérrez García, J., On stratified L-filters induced by ⊤-filters, Fuzzy Sets and Systems, 157, 813-819 (2006) · Zbl 1101.54006 · doi:10.1016/j.fss.2005.09.003 |
| [12] | Gutiérrez García, J.; de Prada Vicente, M. A., Characteristic values of ⊤- filters, Fuzzy Sets and Systems, 156, 55-67 (2005) · Zbl 1087.54003 · doi:10.1016/j.fss.2005.04.012 |
| [13] | Heymans, H., Q-∗-Categories, Applied Categorical Structures, 17, 1-28 (2009) · Zbl 1205.18007 · doi:10.1007/s10485-008-9149-0 |
| [14] | Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S., Continuous Lattices and Domains, 93 (2003), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1088.06001 |
| [15] | Höhle, U., Probabilistic topologies induced by L-fuzzy uniformities, Manuscripta Math, 38, 289-323 (1982) · Zbl 1004.54500 · doi:10.1007/BF01170928 |
| [16] | Höhle, U., Many-valued topology and its applications (2001), Kluwer Academic Publishers: Kluwer Academic Publishers, Dordrecht/Boston/London · Zbl 0969.54002 |
| [17] | Höhle, U.; Kubiak, T., Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems, 256, 166-210 (2014) · Zbl 1339.54009 · doi:10.1016/j.fss.2013.03.010 |
| [18] | Höhle, U., Many valued topologies and lower semicontinuity, Semigroup Forum, 75, 1-17 (2007) · Zbl 1125.06006 · doi:10.1007/s00233-006-0652-z |
| [19] | Höhle, U., A non-commutative and non-idempotent theory of quantale sets, Fuzzy Sets and Systems, 166, 1-43 (2011) · Zbl 1226.06011 · doi:10.1016/j.fss.2010.12.001 |
| [20] | Hofmann, D.; Stubbe, I., Topology from enrichment: the curious case of partial metrics, Cahiers De Topologie Et Geometrie Differentielle Categoriques, 59, 4, 307-353 (2018) · Zbl 1407.18008 |
| [21] | Jäger, G., A category of L-fuzzy convergence spaces, Quaestiones Mathematicae, 24, 501-507 (2001) · Zbl 0991.54004 · doi:10.1080/16073606.2001.9639237 |
| [22] | Jäger, G., Subcategories of lattice-valued convergence spaces, Fuzzy Sets and Systems, 156, 1-24 (2005) · Zbl 1086.54006 · doi:10.1016/j.fss.2005.04.013 |
| [23] | Jäger, G., Lattice-valued convergence spaces and regularity, Fuzzy Sets and Systems, 159, 2488-2502 (2008) · Zbl 1177.54003 · doi:10.1016/j.fss.2008.05.014 |
| [24] | Jäger, G., Convergence approach spaces and approach spaces as lattice-valued convergence spaces, Iranian Journal of Fuzzy Systems, 9, 4, 1-16 (2012) · Zbl 1260.54015 |
| [25] | Kowalski, H.-J., Limesräume und Komplettierung, Math. Nachrichten, 12, 301-340 (1954) · Zbl 0056.41403 · doi:10.1002/mana.19540120504 |
| [26] | Kubiak, T.; Rodabaugh, S. E.; Klement, E. P.; Höhle, U., The topological modification of the L-fuzzy unit interval, Applications of Category Theory to Fuzzy Subsets, 275-305 (1992), Kluwer Academic Publishers: Kluwer Academic Publishers, Boston/Dordrecht/London |
| [27] | Li, L.; Jin, Q., On adjunctions between Lim, SL-Top, and SL-Lim, Fuzzy Sets and Systems, 182, 66-78 (2011) · Zbl 1244.54018 · doi:10.1016/j.fss.2010.10.002 |
| [28] | Li, L.; Jin, Q.; Hu, K., On stratified L-convergence spaces: Fischer’s diagonal axiom, Fuzzy Sets and Systems, 267, 31-40 (2015) · Zbl 1392.54010 · doi:10.1016/j.fss.2014.09.001 |
| [29] | Lowen, R., Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl, 56, 623-633 (1976) · Zbl 0342.54003 · doi:10.1016/0022-247X(76)90029-9 |
| [30] | Matthews, Steve G., Partial metric topology, Papers on general topology and applications, 728, 183-197 (1994), New York Acad. Sci.: New York Acad. Sci., New York · Zbl 0911.54025 |
| [31] | Pang, B., On (L, M )-fuzzy convergence spaces, Fuzzy Sets and Systems, 238, 46-70 (2014) · Zbl 1315.54008 · doi:10.1016/j.fss.2013.07.007 |
| [32] | Pang, B., The category of stratified L-filters spaces, Fuzzy Sets and Systems, 182, 53-65 (2011) · Zbl 1244.54019 · doi:10.1016/j.fss.2011.04.003 |
| [33] | Pang, B.; Fang, J.-M., L-fuzzy Q-convergence structures, Fuzzy Sets and Systems, 182, 53-65 (2011) · Zbl 1244.54019 · doi:10.1016/j.fss.2011.04.003 |
| [34] | Preuss, G., Foundations of Topology (2002), Kluwer Academic Publishers: Kluwer Academic Publishers, Dordrecht/Boston/London · Zbl 1058.54001 |
| [35] | Pu, Q.; Zhang, D., Preordered sets valued in a GL-monoid, Fuzzy Sets and Systems, 187, 1-32 (2012) · Zbl 1262.18008 · doi:10.1016/j.fss.2011.06.012 |
| [36] | Reid, L.; Richardson, G., Connecting ⊤ and lattice-valued convergences. (4) (2018), Iranian Journal of Fuzzy Systems, 15, 151-169 · Zbl 1400.54005 |
| [37] | Rosenthal, K. I., Quantales and Their Applications, 234 (1990), Longman: Longman, Harlow · Zbl 0703.06007 |
| [38] | Rosenthal, K. I., The Theory of Quantaloids, 348 (1996), Longman: Longman, Harlow · Zbl 0845.18003 |
| [39] | Stubbe, I., Categorical structures enriched in a quantaloid: categories, distributors and functors, Theory Appl. Categ, 14, 1, 1-45 (2005) · Zbl 1079.18005 |
| [40] | Stubbe, I., Categorical structures enriched in a quantaloid: tensored and cotensored categories, Theory Appl. Categ, 16, 14, 283-306 (2006) · Zbl 1119.18005 |
| [41] | Stubbe, I., An introduction to quantaloid-enriched categories, Fuzzy Sets and Systems, 256, 95-116 (2014) · Zbl 1335.18002 · doi:10.1016/j.fss.2013.08.009 |
| [42] | Walters, Robert F. C., Sheaves and Cauchy-complete categories, Cahiers Topologie Géométrie Différentielle Catégoriques, 22, 3, 283-286 (1981) · Zbl 0495.18009 |
| [43] | Warner, M. W., Fuzzy topology with respect to continuous lattices, Fuzzy Sets and Systems, 35, 85-91 (1990) · Zbl 0707.54004 · doi:10.1016/0165-0114(90)90020-7 |
| [44] | Yao, W., On many-valued stratified L-fuzzy convergence spaces, Fuzzy Sets and Systems, 159, 2503-2519 (2008) · Zbl 1206.54012 · doi:10.1016/j.fss.2008.03.003 |
| [45] | Yu, Q.; Fang, J.-M., The category of ⊤-convergence spaces and its catesianclosedness, Iranian Journal of Fuzzy Systems, 14, 3, 121-138 (2017) · Zbl 1398.54007 |
| [46] | Yue, Y.L. and Fang, J.-M., Characterizations of topological quantaloid-enriched convergence spaces, Abstracts of the 37th Linz Seminar on Fuzzy Set Theory, pp. 85-89, Linz, Austria, 2017. |
| [47] | Zhang, D., Meet continuous lattices, limit spaces, and L-topological spaces, Fuzzy Sets and Systems, 140, 479-487 (2003) · Zbl 1086.54502 · doi:10.1016/S0165-0114(03)00134-9 |
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