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  • Some categorical aspects of coarse spaces and balleans
    Dikranjan, Dikran N., 1950- ; Zava, Nicolò
    Coarse spaces [J. Roe, Lectures on Coarse Geometry, Univ. Lecture Ser., vol. 31, American Mathematical Society, Providence, RI, 2003] and balleans [I. Protasov, T. Banakh, Ball Structures and ... Colorings of Groups and Graphs, Mat. Stud. Monogr. Ser., vol. 11, VNTL, Lviv, 2003] are known to be equivalent constructions ([I. Protasov, M. Zarichnyi, General Asymptology, VNTL Publishers, Lviv, Ukraine, 2007]). The main subject of this paper is the category, ▫$\mathbf{Coarse}$▫, having as objects these structures, and its quotient category ▫$\mathbf{Coarse} /_\sim$▫. We prove that the category ▫$\mathbf{Coarse}$▫ is topological and hence ▫$\mathbf{Coarse}$▫ is complete and co-complete and one has a complete description of its epimorphisms and monomorphisms. In particular, ▫$\mathbf{Coarse}$▫ has products and coproducts, quotients, etc., and ▫$\mathbf{Coarse}$▫ is not balanced. A special attention is paid to investigate quotients in ▫$\mathbf{Coarse}$▫ by introducing some particular classes of maps, i.e. (weakly) soft maps which allow one to explicitly describe when the quotient ball structure of a ballean is a ballean. A particular type of quotients, namely the adjunction spaces, is considered in detail in order to obtain a description of the epimorphisms in ▫$\mathbf{Coarse} /_\sim$▫, shown to be the bornologous maps with large image. The monomorphisms in ▫$\mathbf{Coarse} /_\sim$▫ are the coarse embeddings; consequently, the bimorphisms in ▫$\mathbf{Coarse} /_\sim$▫ are precisely the isomorphisms, i.e., ▫$\mathbf{Coarse} /_\sim$▫ is a balanced category.
    Vir: Topology and its Applications. - ISSN 0166-8641 (Vol. 225, 2017, str. 164-194)
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2017
    Jezik - angleški
    COBISS.SI-ID - 18024537

    Povezava(-e):

    http://dx.doi.org/10.1016/j.topol.2017.04.011
    DOI

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