Monomorphisms in spaces with Lindelöf filters

Richard N. Ball; Anthony W. Hager

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 1, page 281-317
  • ISSN: 0011-4642

Abstract

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𝐒𝐩𝐅𝐢 is the category of spaces with filters: an object is a pair ( X , ℱ ) , X a compact Hausdorff space and ℱ a filter of dense open subsets of X . A morphism f ( Y , 𝒢 ) → ( X , ℱ ) is a continuous function f Y → X for which f - 1 ( F ) ∈ 𝒢 whenever F ∈ ℱ . This category arises naturally from considerations in ordered algebra, e.g., Boolean algebra, lattice-ordered groups and rings, and from considerations in general topology, e.g., the theory of the absolute and other covers, locales, and frames, though we shall specifically address only one of these connections here in an appendix. Now we study the categorical monomorphisms in 𝐒𝐩𝐅𝐢 . Of course, these monomorphisms need not be one-to-one. For general 𝐒𝐩𝐅𝐢 we derive a criterion for monicity which is rather inconclusive, but still permits some applications. For the category 𝐋𝐒𝐩𝐅𝐢 of spaces with Lindelöf filters, meaning filters with a base of Lindelöf, or cozero, sets, the criterion becomes a real characterization with several foci ( C ( X ) , Baire sets, etc.), and yielding a full description of the monofine coreflection and a classification of all the subobjects of a given ( X , ℱ ) ∈ 𝐋𝐒𝐩𝐅𝐢 . Considerable attempt is made to keep the discussion “topological,” i.e., within 𝐒𝐩𝐅𝐢 , and to not get involved with, e.g., frames. On the other hand, we do not try to avoid Stone duality. An appendix discusses epimorphisms in archimedean ℓ -groups with unit, roughly dual to monics in 𝐋𝐒𝐩𝐅𝐢 .

How to cite

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Ball, Richard N., and Hager, Anthony W.. "Monomorphisms in spaces with Lindelöf filters." Czechoslovak Mathematical Journal 57.1 (2007): 281-317. <http://eudml.org/doc/31130>.

@article{Ball2007,
abstract = {$\mathbf \{SpFi\}$ is the category of spaces with filters: an object is a pair $(X,\mathcal \{F\}) $, $X$ a compact Hausdorff space and $\mathcal \{F\}$ a filter of dense open subsets of $X$. A morphism $f\: (Y,\mathcal \{G\}) \rightarrow (X,\mathcal \{F\}) $ is a continuous function $f\: Y\rightarrow X$ for which $f^\{-1\}(F) \in \mathcal \{G\}$ whenever $F\in \mathcal \{F\}$. This category arises naturally from considerations in ordered algebra, e.g., Boolean algebra, lattice-ordered groups and rings, and from considerations in general topology, e.g., the theory of the absolute and other covers, locales, and frames, though we shall specifically address only one of these connections here in an appendix. Now we study the categorical monomorphisms in $\mathbf \{SpFi\}$. Of course, these monomorphisms need not be one-to-one. For general $\mathbf \{SpFi\}$ we derive a criterion for monicity which is rather inconclusive, but still permits some applications. For the category $\mathbf \{LSpFi\}$ of spaces with Lindelöf filters, meaning filters with a base of Lindelöf, or cozero, sets, the criterion becomes a real characterization with several foci ($C(X) $, Baire sets, etc.), and yielding a full description of the monofine coreflection and a classification of all the subobjects of a given $(X,\mathcal \{F\}) \in \mathbf \{LSpFi\}$. Considerable attempt is made to keep the discussion “topological,” i.e., within $\mathbf \{SpFi\}$, and to not get involved with, e.g., frames. On the other hand, we do not try to avoid Stone duality. An appendix discusses epimorphisms in archimedean $\ell $-groups with unit, roughly dual to monics in $\mathbf \{LSpFi\}$.},
author = {Ball, Richard N., Hager, Anthony W.},
journal = {Czechoslovak Mathematical Journal},
keywords = {compact Hausdorff space; Lindelöf set; monomorphism; compact Hausdorff space; Lindelöf set; monomorphism},
language = {eng},
number = {1},
pages = {281-317},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Monomorphisms in spaces with Lindelöf filters},
url = {http://eudml.org/doc/31130},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Ball, Richard N.
AU - Hager, Anthony W.
TI - Monomorphisms in spaces with Lindelöf filters
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 1
SP - 281
EP - 317
AB - $\mathbf {SpFi}$ is the category of spaces with filters: an object is a pair $(X,\mathcal {F}) $, $X$ a compact Hausdorff space and $\mathcal {F}$ a filter of dense open subsets of $X$. A morphism $f\: (Y,\mathcal {G}) \rightarrow (X,\mathcal {F}) $ is a continuous function $f\: Y\rightarrow X$ for which $f^{-1}(F) \in \mathcal {G}$ whenever $F\in \mathcal {F}$. This category arises naturally from considerations in ordered algebra, e.g., Boolean algebra, lattice-ordered groups and rings, and from considerations in general topology, e.g., the theory of the absolute and other covers, locales, and frames, though we shall specifically address only one of these connections here in an appendix. Now we study the categorical monomorphisms in $\mathbf {SpFi}$. Of course, these monomorphisms need not be one-to-one. For general $\mathbf {SpFi}$ we derive a criterion for monicity which is rather inconclusive, but still permits some applications. For the category $\mathbf {LSpFi}$ of spaces with Lindelöf filters, meaning filters with a base of Lindelöf, or cozero, sets, the criterion becomes a real characterization with several foci ($C(X) $, Baire sets, etc.), and yielding a full description of the monofine coreflection and a classification of all the subobjects of a given $(X,\mathcal {F}) \in \mathbf {LSpFi}$. Considerable attempt is made to keep the discussion “topological,” i.e., within $\mathbf {SpFi}$, and to not get involved with, e.g., frames. On the other hand, we do not try to avoid Stone duality. An appendix discusses epimorphisms in archimedean $\ell $-groups with unit, roughly dual to monics in $\mathbf {LSpFi}$.
LA - eng
KW - compact Hausdorff space; Lindelöf set; monomorphism; compact Hausdorff space; Lindelöf set; monomorphism
UR - http://eudml.org/doc/31130
ER -

References

top
  1. Archimedean kernel-distinguishing extensions of archimedean ℓ -groups with weak unit, Indian J. Math. 29 (1987), 351–368. (1987) MR0971646
  2. Applications of spaces with filters to archimedean ℓ -groups with weak unit, Ordered Algebraic Structures, J. Martinez (ed.), Kluwer, Dordrecht, 1989, pp. 99–112. (1989) MR1094829
  3. Characterization of epimorphisms in archimedean lattice-ordered groups and vector lattices, Lattice-Ordered Groups, Advances and Techniques, A. Glass and C. Holland (eds.), Kluwer, Dordrecht, 1989, pp. 175–205. (1989) MR1036078
  4. Epicomplete archimedean ℓ -groups, Trans. Amer. Math. Soc. 322 (1990), 459–478. (1990) MR0943603
  5. Epicompletion of archimedean ℓ -groups and vector lattices with weak unit, J. Austral. Math. Soc. (1990), 25–56. (1990) MR1026835
  6. An α -disconnected space has no proper monic preimage, Top. Appl. 37 (1990), 141–151. (1990) MR1080348
  7. Spaces with filters, Proc. Symp. Cat. Top. Univ. Cape Town 1994, C. Gilmour, B. Banaschewski and H. Herrlich (eds.), Dept. Math. and Appl. Math., Univ. Cape Town, 1999, pp. 21–36. (1999) MR1722564
  8. The quasi- F κ cover of a compact Hausdorff space and the κ -ideal completion of an archimedean ℓ -group, General Topology and its Applications, R. M. Shortt (ed.), Dekker Notes 123, Marcel Dekker, 1989, pp. 1–40. (1989) 
  9. C - and C * -quotients in pointfree topology, Dissertationes Mathematicae 412, Warszawa, 2002. (2002) MR1952051
  10. Estimates for the number of real-valued continuous functions, Trans. Amer. Math. Soc. 150 (1970), 618–631. (1970) MR0263016
  11. General Topology, Revised and completed ed., Sigma Series in Pure Mathematics; Vol. 6, Heldermann, Berlin, 1989. (1989) Zbl0684.54001MR1039321
  12. On bianalytic spaces, Czech. Math. J. 88 (1963), 561–573. (1963) 
  13. Rings of Continuous Functions, Van Nostrand, Princeton, 1960, reprinted as Springer-Verlag Graduate Texts 43, Berlin-Heidelberg-New York, 1976. (1976) MR0116199
  14. Monomorphisms in Spaces with Filters, Lecture at Curacao Math. Foundation Conference on Locales and Topological Groups, 1989. (1989) 
  15. Representing and ringifying a Riesz space, Sympos. Math. XXI (1977), 411–431. (1977) MR0482728
  16. Lectures on Boolean Algebras, Van Nostrand, 1963. (1963) Zbl0114.01603MR0167440
  17. The structure of a class of archimedean lattice-ordered algebras, Fund. Math. 50 (1961/1962), 73–94. (1961/1962) MR0133698
  18. Category Theory, Allyn and Bacon, Inc., 1973. (1973) MR0349791
  19. Stone Spaces, Cambridge University Press, Cambridge, 1982. (1982) Zbl0499.54001MR0698074
  20. Handbook of Boolean Algebras, I, J. Monk, with R. Bonnet (ed.), North Holland, Amsterdam, 1989. (1989) MR0991565
  21. Archimedean vector lattices versus topological spaces with filters, Ph.D. thesis, Wesleyan University, 1989. (1989) 
  22. Monic sometimes means α -irreducible, General Topology and its Applications, S. J. Andima et al. (eds.), Dekker Notes 134, Marcel Dekker, 1991, pp. 239–260. (1991) Zbl0774.54009MR1142807
  23. 10.1016/0022-4049(90)90147-A, J. Pure Appl. Algebra 68 (1990), 243–252. (1990) MR1082794DOI10.1016/0022-4049(90)90147-A
  24. 10.1016/0022-4049(91)90011-P, J. Pure Appl. Algebra 70 (1991), 107–127. (1991) Zbl0721.06006MR1100510DOI10.1016/0022-4049(91)90011-P
  25. 10.1016/0022-4049(91)90012-Q, J. Pure Appl. Algebra 70 (1991), 129–132. (1991) MR1100511DOI10.1016/0022-4049(91)90012-Q
  26. 10.1016/S0001-8708(74)80011-3, Adv. Math. 12 (1974), 418–450. (1974) Zbl0278.26005MR0367911DOI10.1016/S0001-8708(74)80011-3
  27. H -closed and extremally disconnected Hausdorff spaces, Diss. Math. 66 (1969). (1969) MR0256353
  28. Covers of compact Hausdorff spaces via localic methods, Ph.D. thesis, Wesleyan University, 1992. (1992) 
  29. A localic construction of some covers of compact Hausdorff spaces, General Topology and Applications, R. M. Shortt (ed.), Dekker Notes 123, Marcel Dekker, 1990, pp. 219–226. (1990) Zbl0708.54030MR1057640
  30. Banach Spaces of Continuous Functions, PWN Publishers, 1971. (1971) Zbl0225.46030MR0296671
  31. Boolean Algebras, third ed, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, 1969. (1969) 
  32. 10.4153/CJM-1949-016-5, Canad. J. Math. 1 (1949), 176–186. (1949) Zbl0032.16901MR0029091DOI10.4153/CJM-1949-016-5
  33. Extended real-valued functions and the projective resolution of a compact Hausdorff space, Ph.D. thesis, Wesleyan Univ., 1970. (1970) 
  34. 10.1016/0166-8641(84)90043-9, Topology Appl. 17 (1984), 217–232. (1984) Zbl0593.54036MR0752272DOI10.1016/0166-8641(84)90043-9
  35. Covering properties and coreflective subcategories, Proc. CCNY Conference on Limits 1987, Ann. NY Acad. Sc. 552, 173–184. Zbl0946.54010MR1020785
  36. On the representation of the vector lattice, Proc. Imp. Acad. Tokyo 18 (1942), 339–342. (1942) Zbl0063.09070MR0015378

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