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  • 1
    Electronic Resource
    Electronic Resource
    Closure Operators and Polarities (1993)
    Oxford, UK : Blackwell Publishing Ltd
    Annals of the New York Academy of Sciences 704 (1993), S. 0 
    Details
    ISSN: 1749-6632
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Natural Sciences in General
    Notes: . Basic results are obtained concerning Galois connections between collections of closure operators (of various types) and collections consisting of subclasses of (pairs of) morphisms in ?? for an (E, ??)-category ?? In effect, the “lattice” of closure operators on ?? is shown to be equivalent to the fixed-point lattice of the polarity induced by the orthogonality relation between composable pairs of morphisms in ??.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1749-6632.1993.tb52508.x
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  • 2
    Electronic Resource
    Electronic Resource
    A Primer on Galois Connections (1993)
    Oxford, UK : Blackwell Publishing Ltd
    Annals of the New York Academy of Sciences 704 (1993), S. 0 
    Details
    ISSN: 1749-6632
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Natural Sciences in General
    Notes: . The rudiments of the theory of Galois connections (or residuation theory, as it is sometimes called) are provided, together with many examples and applications. Galois connections occur in profusion and are well known to most mathematicians who deal with order theory; they seem to be less known to topologists. However, because of their ubiquity and simplicity, they (like equivalence relations) can be used as an effective research tool throughout mathematics and related areas. If one recognizes that a Galois connection is involved in a phenomenon that may be relatively complex, then many aspects of that phenomenon immediately become clear, and thus, the whole situation typically becomes much easier to understand.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1749-6632.1993.tb52513.x
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  • 3
    Electronic Resource
    Electronic Resource
    Regular closure operators (1994)
    Springer
    Applied categorical structures 2 (1994), S. 219-244 
    Details
    ISSN: 1572-9095
    Keywords: Primary: 18A32 ; Secondary: 06A15, 54B30 ; Galois connection ; polarity ; closure operator ; separated object ; dense morphism ; composable pair of morphisms ; factorization structure for sinks
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In an 〈E,M〉-categoryX for sinks, we identify necessary conditions for Galois connections from the power collection of the class of (composable pairs) of morphisms inM to factor through the “lattice” of all closure operators onM, and to factor through certain sublattices. This leads to the notion ofregular closure operator. As one byproduct of these results we not only arrive (in a novel way) at the Pumplün-Röhrl polarity between collections of morphisms and collections of objects in such a category, but obtain many factorizations of that polarity as well. (One of these factorizations constituted the main result of an earlier paper by the same authors). Another byproduct is the clarification of the Salbany construction (by means of relative dominions) of the largest idempotent closure operator that has a specified class ofX-objects as separated objects. The same relation that is used in Salbany's relative dominion construction induces classical regular closure operators as described above. Many other types of closure operators can be obtained by this technique; particular instances of this are the idempotent and modal closure operators that in a Grothendieck topos correspond to the Grothendieck topologies.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00878099
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  • 4
    Electronic Resource
    Electronic Resource
    Injectivity of topological categories (1989)
    Springer
    Algebra universalis 26 (1989), S. 284-306 
    Details
    ISSN: 1420-8911
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Injective objects in concrete categories frequently turn out to be objects with particularly pleasant properties. Often some form of “completeness” provides a characterization of injectivity in such a category, with injective hulls achieved through certain standard completion processes. Several results during the past decade have shown that certain specific ”topological” categories are precisely the injective objects in various natural quasicategories of concrete categories, with injective hulls obtained via certain sieve constructions. When the base category is trivial, some of these results specialize to classical results in certain categories of ordered structures; e.g., the injectives in posets characterized as complete lattices, with injective hulls the MacNeille completions, and the injectives in semilattices characterized as locales, with injective hulls the locale hulls. This paper contains two main results. The first provides a characterization of injective objects in a setting sufficiently general as to include the above mentioned characterizations as well as many others. The second theorem gives a characterization of those objects that have injective hulls, and provides a construction of the hulls as well. Corollaries of this theorem yield numerous known injective hull constructions. The second theorem uses a much stronger hypothesis than the first. That this hypothesis is indispensible follows from a result of E. Nelson on the non-existive of injective hulls of certain σ-semilattices.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01211836
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  • 5
    Electronic Resource
    Electronic Resource
    Realization of cartesian closed topological hulls (1985)
    Springer
    Manuscripta mathematica 53 (1985), S. 1-33 
    Details
    ISSN: 1432-1785
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The cartesian closed topological (CCT) hull of a concrete categoryK is the smallest full CCT extension ofK. A general method for describing the CCT hulls is presented and applied to render unified proofs for the basic examples ofK: topological spaces, uniform spaces, pretopological spaces, compact T2 spaces, metrizable spaces, and completely regular spaces.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01174009
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  • 6
    Electronic Resource
    Electronic Resource
    H-closed spaces and reflective subcategories (1968)
    Springer
    Mathematische Annalen 177 (1968), S. 302-309 
    Details
    ISSN: 1432-1807
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01350722
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    Paper (German National Licenses)
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