Closure operators in the category of quandles. (English) Zbl 1432.18003
Summary: We study a regular closure operator in the category of quandles. We show that the regular closure operator and the pullback closure operator corresponding to the reflector from the category of quandles to its full subcategory of trivial quandles coincide, we give a simple description of this closure operator, and analyze some of its properties. The category of algebraically connected quandles turns out to be a connectedness in the sense of Arhangel’skiǐ and Wiegandt corresponding to the full subcategory of trivial quandles, while the disconnectedness associated with it is shown to contain all quasi-trivial quandles. The separated objects for the pullback closure operator are precisely the trivial quandles. A simple formula describing the effective closure operator on congruences corresponding to the same reflector is also given.
MSC:
| 18A40 | Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) |
| 03C05 | Equational classes, universal algebra in model theory |
| 20N02 | Sets with a single binary operation (groupoids) |
| 57K18 | Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) |
References:
| [1] | Arhangel’skiı̌, A. V.; Wiegandt, R., Connectedness and disconnectedness in topology, Gen. Topol. Appl., 5, 9-33 (1975) · Zbl 0329.54008 |
| [2] | Barr, M.; Grillet, P. A.; van Osdol, D. H., Exact Categories and Categories of Sheaves, Lecture Notes in Math., vol. 236 (1971), Springer: Springer Berlin · Zbl 0223.18009 |
| [3] | Borceux, F., Handbook of Categorical Algebra 2. Categories and Structures, Encycl. Math. its Applications, vol. 51 (1994), Cambridge University Press · Zbl 0843.18001 |
| [4] | Borceux, F.; Gran, M.; Mantovani, S., On closure operators and reflections in Goursat categories, Rend. Ist. Mat. Univ. Trieste, 39, 87-104 (2007) · Zbl 1153.18003 |
| [5] | Bunch, E.; Lofgren, P.; Rapp, A.; Yetter, D. N., On quotients of quandles, J. Knot Theory Ramif., 19, 9, 1145-1156 (2010) · Zbl 1210.57013 |
| [6] | Burris, S.; Sankappannavar, H. P., A Course in Universal Algebra, Graduate Texts in Mathematics, vol. 78 (1981), Springer-Verlag · Zbl 0478.08001 |
| [7] | Carboni, A.; Lambek, J.; Pedicchio, M. C., Diagram chasing in Mal’cev categories, J. Pure Appl. Algebra, 69, 271-284 (1991) · Zbl 0722.18005 |
| [8] | Cassidy, C.; Hébert, M.; Kelly, G. M., Reflective subcategories, localizations, and factorization systems, J. Aust. Math. Soc. Ser. A, 38, 3, 287-329 (1985) · Zbl 0573.18002 |
| [9] | Clementino, M. M., Weakly hereditary regular closure operators, Topol. Appl., 49, 2, 129-139 (1993) · Zbl 0781.18001 |
| [10] | Clementino, M. M.; Tholen, W., Separated and connected maps, Appl. Categ. Struct., 6, 3, 373-401 (1998) · Zbl 0935.18003 |
| [11] | Dikranjan, D.; Giuli, E., Closure operators. I, Topol. Appl., 27, 2, 129-143 (1987) · Zbl 0634.54008 |
| [12] | Dikranjan, D.; Tholen, W., Categorical Structure of Closure Operators with Applications to Topology, Algebra and Discrete Mathematics (1995), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht · Zbl 0853.18002 |
| [13] | Dikranjan, D.; Tholen, W., Dual closure operators and their applications, J. Algebra, 439, 373-416 (2015) · Zbl 1320.18002 |
| [14] | Even, V.; Gran, M., On factorization systems for surjective quandle homomorphisms, J. Knot Theory Ramif., 23, 11, 1450060 (2014) · Zbl 1306.18001 |
| [15] | Holgate, D., The pullback closure operator and generalisations of perfectness, Appl. Categ. Struct., 4, 1, 107-120 (1996) · Zbl 0912.18002 |
| [16] | Inoue, A., Quasi-triviality of quandles for link-homotopy, J. Knot Theory Ramif., 22, 6, 1350026 (2013) · Zbl 1403.57010 |
| [17] | Joyce, D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra, 23, 1, 37-65 (1982) · Zbl 0474.57003 |
| [18] | Matveev, S. V., Distributive groupoids in knot theory, Mat. Sb. (N.S.), 119, 161, 78-88 (1982), 160 · Zbl 0523.57006 |
| [19] | Salbany, S., Reflective Subcategories and Closure Operators, Lecture Notes in Math., vol. 540, 548-565 (1976), Springer: Springer Berlin · Zbl 0335.54003 |
| [20] | Smith, J. D.H., Mal’cev Varieties, Lecture Notes in Math., vol. 554 (1976) · Zbl 0344.08002 |
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