$L$-enriched topological systems---a common framework of $L$-topology and $L$-frames

Document Type: Research Paper


School of Sciences, Chang'an University, Xi'an, China


Employing the notions of the strong $L$-topology introduced by Zhang
and the $L$-frame introduced by Yao  and the concept of $L$-enriched
topological system defined in the present paper, we construct
adjunctions among the categories {\bf St$L$-Top} of strong
$L$-topological spaces, {\bf S$L$-Loc} of strict $L$-locales and
{\bf $L$-EnTopSys} of $L$-enriched topological systems. All of these
concepts are essentially based on the theory of $L$-enriched
categories, thus we obtain a unified enriched-categorical version of
the classical adjunctions among the categories {\bf Top} of
topological spaces, {\bf Loc} of locales and {\bf TopSys} of
topological systems, as well as a unified enriched-categorical
approach to treating these concepts.


J. Ad'{a}mek, H. Herrlich and G. E. Strecker, {it Abstract and
Concrete Categories}, Wiley, New York, 1990.

R. Bv{e}lohl'{a}vek, {it Concept lattices and order in fuzzy
logic}, Annals of Pure and Applied Logic, {bf 128(1-3)} (2004),

R. Bv{e}lohl'{a}vek, {it Fuzzy relational systems: foundations
and principles}, Kluwer Academic/Plenum Publishers, New York, 2002.

C. L. Chang, {it Fuzzy topological spaces}, J. Math. Anal. Appl.,
{bf 24(1)} (1968), 182-190.

J. T. Denniston, A. Melton and S. E. Rodabaugh, {it Enriched
topological systems and variable-basis enriched functors}, In: U.
H"{o}hle, L. N. Stout, E. P. Klement,  Abstracts of the 33th Linz
Seminar on Fuzzy Set Theory, Universit"{a}tsdirektion Johannes
Kepler Universit"{a}t (Linz, Austria), 14-18 February (2012),

J. T. Denniston, A. Melton and S. E. Rodabaugh, {it Interweaving
algebra and topology: Lattice-valued topological systems}, Fuzzy
Sets and Systems, {bf 192} (2012), 58-103.

J. T. Denniston, A. Melton and S. E. Rodabaugh, {it Lattice-valued
topological systems}, In: U. Bodenhofer, B. De Baets, E. P. Klement,
Abstracts of the 30th Linz Seminar, Universit"{a}tsdirektion
Johannes Kepler Universit"{a}t (Linz, Austria), 3-7 February
(2009), 24-31.

bibitem{Fan:ANAT}L. Fan, {it A new approach to quantitative domain theory},
Electronic Notes in Theoretical Computer Science, {bf 45} (2001),

J. A. Goguen, {it $L$-fuzzy sets}, Journal of Mathematical Analysis
and Applications, {bf 18(1)} (1967), 145-174.

J. A. Goguen, {it The fuzzy tychonoff theorem}, J. Math. Anal. Appl.,
{bf 43(3)} (1973), 734-742.

D. Hofmann and P. L. Waszkiewicz, {it Approximation in
quantale-enriched categories}, Topology and its Applications, {bf
158(8)} (2011), 963-977.

bibitem{Johnstone:SS} P. T. Johnstone, {it Stone spaces}, Cambridge University Press, Cambridge, 1982.

G. M. Kelly, {it Basic concepts of enriched category theory},
London Mathematical Society Lecture Notes Series 64, Cambridge
University Press, 1982.

H. Lai and D. Zhang, {it Complete and directed complete
$Omega$-categories}, Theoret. Comput. Sci., {bf 388(1-3)} (2007),

H. Lai and D. Zhang, {it Many-valued complete distributivity},
arXiv:math/0603590v2 [math.CT], 12 May 2006.

 F. W. Lawvere, {it Metric spaces, generalized logic, and closed
categories}, Rend. Sem. Mat. Fis. Milano, {bf 43(1)} (1973),

M. Liu, {it On some related problems in $Omega$-categories and
fuzzy domains}, Ph.D. Thesis, Shannxi normal university, Xi'an,
China, 2013.

R. Lowen, {it Fuzzy topological spaces and fuzzy compactness}, J.
Math. Anal. Appl., {bf 56(3)} (1976), 623-633.

S. E. Rodabaugh, {it Powerset operator foundations for poslat fuzzy
set theories and topologies}, in: U. H"{o}hle, S. E. Rodabaugh
(Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure
Theory, The Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic
Publishers, Boston, Dordrecht, London, (1999), 91-116.

K. I. Rosenthal, {it Quantales and Their Applications}, Pitman
Research Notes in Mathematics Series, Longman, {bf 234} (1990).

I. Stubbe, {it Categorical structures enriched in a quantaloid:
Categories, distributors and functors}, Theory Appl. Categ., {bf
14(1)} (2005), 1-45.

I. Stubbe, {it Categorical structures enriched in a quantaloid:
Tensored and cotensored categories}, Theory Appl. Categ., {bf
16(14)} (2006), 283-306.

S. J. Vickers, {it Topology via logic}, Cambridge University Press,
Cambridge, 1989.

K. R. Wagner, {it Solving recursive domain equations with enriched
categories}, Ph.D. Thesis, Carnegie Mellon University, Tech. Report
CMU-CS-94-159, July, 1994.

bibitem{Yao:AATF}  W. Yao, {it An approach to fuzzy frames via fuzzy
posets}, Fuzzy Sets and Systems, {bf 166(1)} (2011), 75-89.

bibitem{Yao:QDVF} W. Yao, {it Quantitative domains via fuzzy sets: Part I: continuity of
fuzzy directed-complete poset}, Fuzzy Sets and Systems, {bf 161(7)}
(2010), 983-987.

D. Zhang, {it An enriched category approach to many valued
topology}, Fuzzy Sets and Systems, {bf 158(4)} (2007), 349-366.

Q. Y. Zhang, W. X. Xie and L. Fan, {it Fuzzy complete lattices}, Fuzzy
Sets and Systems, {bf 160(16)} (2009), 2275-2291.