A categorical account of the localic closed subgroup theorem

Christopher Townsend

Commentationes Mathematicae Universitatis Carolinae (2007)

  • Volume: 48, Issue: 3, page 541-553
  • ISSN: 0010-2628

Abstract

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Given an axiomatic account of the category of locales the closed subgroup theorem is proved. The theorem is seen as a consequence of a categorical account of the Hofmann-Mislove theorem. The categorical account has an order dual providing a new result for locale theory: every compact subgroup is necessarily fitted.

How to cite

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Townsend, Christopher. "A categorical account of the localic closed subgroup theorem." Commentationes Mathematicae Universitatis Carolinae 48.3 (2007): 541-553. <http://eudml.org/doc/250198>.

@article{Townsend2007,
abstract = {Given an axiomatic account of the category of locales the closed subgroup theorem is proved. The theorem is seen as a consequence of a categorical account of the Hofmann-Mislove theorem. The categorical account has an order dual providing a new result for locale theory: every compact subgroup is necessarily fitted.},
author = {Townsend, Christopher},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {locale; power locale; Hofmann-Mislove theorem; closed subgroup; compact locale; fitted sublocale; categorical logic; category of locales; group; subspace; open object; closed subset},
language = {eng},
number = {3},
pages = {541-553},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A categorical account of the localic closed subgroup theorem},
url = {http://eudml.org/doc/250198},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Townsend, Christopher
TI - A categorical account of the localic closed subgroup theorem
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 3
SP - 541
EP - 553
AB - Given an axiomatic account of the category of locales the closed subgroup theorem is proved. The theorem is seen as a consequence of a categorical account of the Hofmann-Mislove theorem. The categorical account has an order dual providing a new result for locale theory: every compact subgroup is necessarily fitted.
LA - eng
KW - locale; power locale; Hofmann-Mislove theorem; closed subgroup; compact locale; fitted sublocale; categorical logic; category of locales; group; subspace; open object; closed subset
UR - http://eudml.org/doc/250198
ER -

References

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  2. Johnstone P.T., Stone Spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, Cambridge, 1982. Zbl0586.54001MR0698074
  3. Johnstone P.T., Vickers S.J., Preframe presentations present, in: Carboni, Pedicchio and Rosolini (Eds.), Category Theory (Como, 1990); Springer Lecture Notes in Mathematics 1488, Springer, Berlin, 1991, pp.193-212. Zbl0764.18004MR1173013
  4. Johnstone P.T., Sketches of an Elephant: A Topos Theory Compendium, Vols 1, 2, Oxford Logic Guides 43, 44, Oxford Science Publications, Oxford, 2002. Zbl1071.18002MR1953060
  5. Joyal A., Tierney M., An extension of the Galois theory of Grothendieck, Memoirs of the American Mathematical Society 309, 1984. Zbl0541.18002MR0756176
  6. Isbell J.R., Kříž I., Pultr A., Rosický J., Remarks on localic groups, in: Categorical Algebra and its Applications, (ed. F. Borceux), Lecture Notes in Mathematics 1349, Springer, Berlin, 1988, pp.154-172. MR0975968
  7. MacLane S., Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer, New York-Berlin, 1971. Zbl0705.18001MR1712872
  8. Townsend C.F., Vickers S.J., A universal characterization of the double power locale, Theoret. Comput. Sci. 316 (2004), 297-321. (2004) MR2074935
  9. Townsend C.F., An axiomatic account of weak localic triquotient assignments, J. Pure Appl. Algebra, to appear. 
  10. Townsend C.F., A categorical account of the Hofmann-Mislove theorem, Math. Proc. Cambridge Philos. Soc. 139 (2005), 441-456. (2005) Zbl1104.06008MR2177170
  11. Vickers S.J., Topology via Logic, Cambridge Tracts in Theoretical Computer Science 5, Cambridge University Press, Cambridge, 1989. Zbl0922.54002MR1002193

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