Injective locales over perfect embeddings and algebras of the upper powerlocale monad

Martín Escardó

doi:10.4995/agt.2003.2018

Authors

Martín Escardó University of Birmingham

DOI:

https://doi.org/10.4995/agt.2003.2018

Keywords:

Injective locale, Perfect embedding, Powerlocale, Free frame, Kock-Zöberlein monad, Stably supercontinuous lattice

Abstract

We show that the locales which are injective over perfect sublocale embeddings coincide with the underlying objects of the algebras of the upper powerlocale monad, and we characterize them as those whose frames of opens enjoy a property analogous to stable supercontinuity.

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Author Biography

Martín Escardó, University of Birmingham

School of Computer Science

References

B. Banaschewski, Another look at the localic Tychonoff theorem, Comment. Math. Univ. Carolin. 29 (4) (1988), 647-656.

B. Banaschewski and G. C. L. Brümmer, Stably continuous frames, Math. Proc. Cambridge Philos. Soc. 104 (1) (1988), 7-19. http://dx.doi.org/10.1017/S0305004100065208

B. Banaschewski and S. B. Niefield, Projective and supercoherent frames, J. Pure Appl. Algebra 70 (1-2) (1991), 45-51. http://dx.doi.org/10.1016/0022-4049(91)90005-M

A. Day, Filter monads, continuous lattices and closure systems, Canad. J. Maths. XXVII (1) (1975), 50-59. http://dx.doi.org/10.4153/CJM-1975-008-8

M. H. Escardó, Injective spaces via the filter monad, Topology Proceedings 22 (2) (1997), 97-110.

M. H. Escardó, Properly injective spaces and function spaces, Topology Appl. 89 (1-2) (1998), 75-120. http://dx.doi.org/10.1016/S0166-8641(97)00225-3

M. H. Escardó, The regular-locally-compact coreflection of stably locally compact locale, J. Pure Appl. Algebra 157 (1) (2001), 41-55. http://dx.doi.org/10.1016/S0022-4049(99)00172-3

M. H. Escardó and R.C. Flagg, Semantic domains, injective spaces and monads (Electronic Notes in Theoretical Computer Science 20, 1999).

G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, Continuous lattices and domains (Cambridge University Press, 2002).

R. Heckmann, Lower and upper power domain constructions commute on all cpos, Information Processing Letters 40 (1) (1991), 7-11. http://dx.doi.org/10.1016/S0020-0190(05)80003-1

K. H. Hofmann, Stably continuous frames, and their topological manifestations, In Categorical Topology, pages 282-307. (Heldermann, 1984).

K. H. Hofmann and J. D. Lawson, On the order theoretical foundation of a theory of quasicompactly generated spaces without separation axiom, volume 27 of Mathematik- Arbeitspapiere, pages 143-160. (University of Bremen, 1982).

P. T. Johnstone, The Gleason cover of a topos, II, J. Pure Appl. Algebra 22 (1981), 229-247. http://dx.doi.org/10.1016/0022-4049(81)90100-6

P. T. Johnstone, Stone spaces (Cambridge University Press, 1982).

P. T. Johnstone, Vietoris locales and localic semilattices, In Continuous lattices and their applications, pages 155-180. (Dekker, New York, 1985).

P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, Number 43-44 in Oxford Logic Guides (Oxford Science Publications, 2002).

Ho Weng Kin and Zhao Dongsheng, On the characterization of Scott-closed set lattices, Preprint. Mathematics Department, National Institute of Education University, Singapore, 2002.

A. Kock, Monads for which structures are adjoint to units (version 3), J. Pure Appl. Algebra 104 (1995), 41-59. http://dx.doi.org/10.1016/0022-4049(94)00111-U

S. Mac Lane, Categories for the Working Mathematician (Springer-Verlag, 1971). http://dx.doi.org/10.1007/978-1-4612-9839-7

L. Nachbin, Topologia e ordem (University of Chicago Press, 1950). In Portuguese. English translation published 1965 by Van Nostrand, Princeton, as Topology and order.

H. A. Priestley, Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc. (3) 24 (1972), 507-530. http://dx.doi.org/10.1112/plms/s3-24.3.507

A. Schalk, Algebras for Generalized Power Constructions, PhD thesis, Mathematics Department, Technische Hochschule Darmstadt, July 1993. Available at ftp://ftp.cl.cam.ac.uk/papers/as213/diss.dvi.gz.

A. Schalk, Domains arising as algebras for powerspace constructions, J. Pure Appl. Algebra 89 (3) (1993), 305-328. http://dx.doi.org/10.1016/0022-4049(93)90059-3

D. S. Scott, Continuous lattices, In F. W. Lawvere, editor, Toposes, Algebraic Geometry and Logic, volume 274 of Lecture Notes in Mathematics, pages 97-136, 1972.

H. Simmons, A couple of triples, Topology Appl. 13 (1982), 201-223. http://dx.doi.org/10.1016/0166-8641(82)90021-9

C. F. Townsend, Localic Priestley duality, J. Pure Appl. Algebra 116 (1-3) (1997), 323-335. http://dx.doi.org/10.1016/S0022-4049(96)00169-7

J. J. C. Vermeulen, Proper maps of locales, J. Pure Appl. Algebra 92 (1994), 79-107. http://dx.doi.org/10.1016/0022-4049(94)90047-7

S. J. Vickers, Locales are not pointless, In C. Hankin, I. Mackie, and R. Nagarajan, editors, Theory and Formal Methods 1994, (IC Press, 1995).

S. J. Vickers, Constructive points of powerlocales, Math. Proc. Cambridge Philos. Soc. 122 (2) (1997), 207-222. http://dx.doi.org/10.1017/S0305004196001636

O. Wyler, Algebraic theories for continuous semilattices, Archive for Rational Mechanics and Analysis 90 (2) (1985), 99-113. http://dx.doi.org/10.1007/BF00250716