Topological Functors via Closure Operators

Yıl 2013, Cilt: 10 Sayı: 1, - , 01.05.2013

Mina Jamshidi Seyed Naser Hosseini

Öz

In this article for a given category X , we fully embed certain categories of
closure operators on a given collection M ⊆ X1, in certain categories of preclass-valued
lax presheaves on X . We then fully embed the just mentioned categories of preclass-valued
lax presheaves on X , in certain categories of topological functors on X . Combining the full
embeddings obtained, we construct a topological functor from a given closure operator.

Anahtar Kelimeler

Closure operator, lax presheaf, lax natural transformation, (complete) preordered or partially ordered class, (weak) topological functor

Kaynakça

• [1] J. Ad´amek, H. Herrlich, J. Rosicky and W. Tholen, Weak factorization systems and topological functors, Applied Categorical Structures 10 (2002), 237–249.
• [2] J. Adam´ek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories, John Wiley and Sons, 1990. http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf
• [3] M. Baran, Compactness, perfectness, separation, minimality and closedness with respect to closure operators, Applied Categorical Structures 10 (2002), 403–415.
• [4] H. L. Bentley and H. Herrlich, Merotopological spaces, Applied Categorical Structures 12 (2004), 155–180.
• [5] H. L. Bentley and E. Lowen-Colebunders, Initial morphisms versus embeddings, Applied Categorical Structures 12 (2004), 361–367.
• [6] L. M. Brown, R. Ert¨urk and S¸. Dost, Ditopological texture spaces and fuzzy topology, II. Topological considerations, Fuzzy Sets and Systems 147 (2004), 201–231.
• [7] G. Castellini, Categorical Closure Operators, Birkh¨auser, Boston 2003.
• [8] G. Castellini, Connectedness with respect to a closure operator, Applied Categorical Structures 9 (2001), 285–302.
• [9] M. M. Clementino, On categorical notions of compact objects, Applied Categorical Structures 4 (1996), 15–29.
• [10] M. M. Clementino and D. Hofmann, Topological features of lax algebras, Applied Categorical Structures 11 (2003), 267–286.
• [11] M. M. Clementino and W. Tholen, Tychonoff’s theorem in a category, Proceedings of the American Mathematical Society 124 (1996), 3311–3314.
• [12] D. Dikranjan and W. Tholen, Categorical Structure of Closure Operators, Kluwer Academic Publishers, Netherlands 1995.
• [13] D. Dikranjan, E. Giuli and A. Tozzi, Topological categories and closure operators, Quaestiones Mathematicae 11 (1988), 323–337.
• [14] T. H. Fay, Weakly hereditary initial closure operators, Applied Categorical Structures 8 (2000), 415–431.
• [15] T. H. Fay and S. V. Joubert, Isolated submodules and skew fields, Applied Categorical Structures 8 (2000), 317–326.
• [16] J. Fillmore, D. Pumpl¨un and H. R¨ohrl, On N-summations, I, Applied Categorical Structures 10 (2002), 291–315.
• [17] W. G¨ahler, A. S. Abd-Allah and A. Kandil, On extended fuzzy topologies, Fuzzy Sets and Systems 109 (2000), 149–172.
• [18] E. Giuli and W. Tholen, Openness with respect to a closure operator, Applied Categorical Structures 8 (2000), 487–502.
• [19] S. N. Hosseini and S. Sh. Mousavi, A relation between closure operators on a small category and its category of presheaves, Applied Categorical Structures 14 (2006), 99–110.
• [20] S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Logic, A First Introduction to Topos Theory, Springer-Verlag New York Inc. 1992.
• [21] M. V. Mielke, Final lift actions associated with topological functors, Applied Categorical Structures 10 (2002), 495–504.