Duality theory for the category of stable compactifications
Keywords:
Stably compact space, stable compactification, stably compact frame, proximity frame, Raney latticeAbstract
We introduce the category of stable compactifications of $T_0$-spaces and obtain a dual description of it in terms of what we call Raney extensions of proximity frames. These are proximity frame embeddings of a regular proximity frame into a Raney lattice, i.e. the lattice of upsets of a poset. This duality generalizes the duality between compactifications of completely regular spaces involving de Vries extensions given in [8]. It also specializes to give a duality between $T_0$-spaces and Raney extensions that are maximal in a certain sense. This duality is related to the duality for $T_0$-spaces given in [7] using the notion of a Raney algebra, i.e. a Raney lattice with a certain type of interior operator.
References
R. Balbes and Ph. Dwinger, Distributive lattices, University of Missouri Press, Columbia, Mo., 1974.
B. Banaschewski, Coherent frames, Lecture Notes in Math., vol. 871, SpringerVerlag (1981), 1-11.
B. Banaschewski, Compactification of frames, Math. Nachr. 149 (1990), 105-115.
B. Banaschewski and C. J. Mulvey, Stone-C̄ech compactification of locales. I, Houston J. Math. 6 (1980), no. 3, 301-312.
G. Bezhanishvili and J. Harding, Proximity frames and regularization, Appl. Categ. Structures 22 (2014), 43-78.
G. Bezhanishvili and J. Harding, Stable compactifications of frames, Cah. Topol. Géom. Différ. Catég. 55 (2014), no. 1, 37-65.
G. Bezhanishvili and J. Harding, Raney algebras and duality for $T_0$-spaces, Appl. Categ. Structures 28 (2020), no. 6, 963-973.
G. Bezhanishvili, P. J. Morandi, and B. Olberding, An extension of de Vries duality to completely regular spaces and compactifications, Topology Appl. 257 (2019), 85-105.
G. Bezhanishvili, P. J. Morandi, and B. Olberding, An extension of de Vries duality to normal spaces and locally compact Hausdorff spaces, J. Pure Appl. Algebra 224 (2020), no. 2, 703-724.
H. de Vries, Compact spaces and compactifications. An algebraic approach, Ph.D. thesis, University of Amsterdam, 1962.
R. Engelking, General topology, second ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989.
J. L. Frith, The category of uniform frames, Cah. Topol. Géom. Différ. Catég. 31 (1990), no. 4, 305-313.
G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, Continuous lattices and domains, Cambridge University Press, Cambridge, 2003.
P. T. Johnstone, Stone spaces, Cambridge University Press, Cambridge, 1982.
J. Picado and A. Pultr, Frames and locales, Frontiers in Mathematics, Birkhāuser/Springer Basel AG, Basel, 2012.
G. N. Raney, Completely distributive complete lattices, Proc. Amer. Math. Soc. 3 (1952), 677-680.
Y. M. Smirnov, On proximity spaces, Mat. Sbornik N.S. 31(73) (1952), 543-574, (Russian).
M. B. Smyth, Stable compactification. I, J. London Math. Soc. 45 (1992), no. 2, 321-340.
