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  • 1
    Book
    Book
    Categorical structure of closure operators : with applications to topology, algebra and discrete mathematics (1995)
    Dordrecht [u.a.] : Kluwer
    Details
    UID:
    b3kat_BV010597296
    Format: XVII, 356 S. , graph. Darst.
    ISBN: 0792337727
    Series Statement: Mathematics and its applications 346
    Language: English
    Subjects: Mathematics
    RVK:
    SK 320
    Keywords: Abgeschlossener linearer Operator ; Kategorientheorie ; Lehrbuch
    URL: Inhaltsverzeichnis
    URL: Inhaltsverzeichnis
    URL: Cover
    URL: Inhaltstext
    URL: http://www.loc.gov/catdir/enhancements/fy0822/95039768-t.html
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  • 2
    Online Resource
    Online Resource
    Categorical Structure of Closure Operators : With Applications to Topology, Algebra and Discrete Mathematics (1995)
    Dordrecht : Springer Netherlands
    Details
    UID:
    b3kat_BV042424054
    Format: 1 Online-Ressource (XVIII, 358 p)
    ISBN: 9789401584005 , 9789048146314
    Series Statement: Mathematics and Its Applications 346
    Note: Our motivation for gathering the material for this book over aperiod of seven years has been to unify and simplify ideas wh ich appeared in a sizable number of re­ search articles during the past two decades. More specifically, it has been our aim to provide the categorical foundations for extensive work that was published on the epimorphism- and cowellpoweredness problem, predominantly for categories of topological spaces. In doing so we found the categorical not ion of closure operators interesting enough to be studied for its own sake, as it unifies and describes other significant mathematical notions and since it leads to a never-ending stream of ex­ amples and applications in all areas of mathematics. These are somewhat arbitrarily restricted to topology, algebra and (a small part of) discrete mathematics in this book, although other areas, such as functional analysis, would provide an equally rich and interesting supply of examples. We also had to restrict the themes in our theoretical exposition. In spite of the fact that closure operators generalize the uni­ versal closure operations of abelian category theory and of topos- and sheaf theory, we chose to mention these aspects only en passant, in favour of the presentation of new results more closely related to our original intentions. We also needed to refrain from studying topological concepts, such as compactness, in the setting of an arbitrary closure-equipped category, although this topic appears prominently in the published literature involving closure operators
    Language: English
    Keywords: Abgeschlossener linearer Operator ; Kategorientheorie
    DOI: 10.1007/978-94-015-8400-5
    URL: Volltext
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  • 3
    Online Resource
    Online Resource
    Categorical foundations : special topics in order, topology, algebra, and Sheaf theory (2004)
    Cambridge : Cambridge University Press
    Details
    UID:
    gbv_875082718
    Format: 1 Online-Ressource (xii, 417 pages) , digital, PDF file(s).
    ISBN: 9781107340985 , 9780521834148
    Series Statement: Encyclopedia of mathematics and its applications volume 97
    Content: The book offers categorical introductions to order, topology, algebra and sheaf theory, suitable for graduate students, teachers and researchers of pure mathematics. Readers familiar with the very basic notions of category theory will learn about the main tools that are used in modern categorical mathematics but are not readily available in the literature. Hence, in eight rather independent chapters the reader will encounter various ways of how to study 'spaces': order-theoretically via their open-set lattices, as objects of a fairly abstract category merely via their interaction with other objects, or via their topoi of set-valued sheaves. Likewise, 'algebras' are treated both as models for Lawvere's algebraic theories and as Eilenberg-Moore algebras for monads, but they appear also as the objects of an abstract category with various levels of 'exactness' conditions. The abstract methods are illustrated by applications which, in many cases, lead to results not yet found in more traditional presentations of the various subjects, for instance on the exponentiability of spaces and embeddability of algebras.
    Note: Title from publisher's bibliographic system (viewed on 05 Oct 2015) , Ordered sets via adjunction , Locales , Functional approach to general topology , Regular, protomodular, and Abelian categories , Aspects of monads , Algebraic categories , Sheaf theory , Beyond Barr exactness
    Additional Edition: ISBN 9780521834148
    Additional Edition: ISBN 9780521834148
    Additional Edition: Erscheint auch als Categorical foundations Cambridge [u.a.] : Cambridge University Press, 2004 ISBN 0521834147
    Language: English
    Subjects: Mathematics
    RVK:
    SK 320
    Keywords: Kategorie
    DOI: 10.1017/CBO9781107340985
    URL: Volltext  (URL des Erstveröffentlichers)
    URL: Volltext
    URL: Inhaltstext
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    UB Potsdam
  • 4
    Online Resource
    Online Resource
    Monoidal topology : a categorical approach to order, metric and topology (2014)
    Cambridge : Cambridge University Press
    Details
    UID:
    gbv_883350122
    Format: 1 Online-Ressource (xvii, 503 pages) , digital, PDF file(s).
    ISBN: 9781107517288
    Series Statement: Encyclopedia of mathematics and its applications volume 153
    Content: Monoidal Topology describes an active research area that, after various past proposals on how to axiomatize 'spaces' in terms of convergence, began to emerge at the beginning of the millennium. It combines Barr's relational presentation of topological spaces in terms of ultrafilter convergence with Lawvere's interpretation of metric spaces as small categories enriched over the extended real half-line. Hence, equipped with a quantale V (replacing the reals) and a monad T (replacing the ultrafilter monad) laxly extended from set maps to V-valued relations, the book develops a categorical theory of (T,V)-algebras that is inspired simultaneously by its metric and topological roots. The book highlights in particular the distinguished role of equationally defined structures within the given lax-algebraic context and presents numerous new results ranging from topology and approach theory to domain theory. All the necessary pre-requisites in order and category theory are presented in the book.
    Content: Introduction / Robert Lowen and Walter Tholen -- Monoidal structures / Gavin J. Seal and Walter Tholen -- Lax algebras / Dirk Hofmann, Gavin J. Seal, and Walter Tholen -- Kleisli monoids / Dirk Hofmann, Robert Lowen, Rory Lucyshyn-Wright, and Gavin J. Seal -- Lax algebras as spaces / Maria Manuel Clementino, Eva Colebunders, and Walter Tholen
    Note: Title from publisher's bibliographic system (viewed on 05 Oct 2015)
    Additional Edition: ISBN 9781107063945
    Additional Edition: ISBN 9781107063945
    Additional Edition: Erscheint auch als Hofmann, Dirk, 1978 - Monoidal topology Cambridge, United Kingdom : Cambridge University Press, 2014 ISBN 9781107063945
    Additional Edition: Print version ISBN 9781107063945
    Language: English
    Subjects: Mathematics
    RVK:
    SK 340
    Keywords: Monoid
    DOI: 10.1017/CBO9781107517288
    URL: Volltext
    URL: Inhaltstext
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    UB Potsdam
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