The limit object of Hausdorff spectrum in the category TLC

Eugeny Ivanovich Smirnov, Sergey Alexandrovich Tikhomirov

Abstract


The category ${\mathcal H}$ of Hausdorff spectra ${\mathcal X}=\{X_s,{\gF},h_{s's}\}$ is introduced by E.I.Smirnov into the discussion by means of an appropriate factorization of the category of Hausdorff spectra ${\rm Spect}\,{\mathcal G}$ over the category ${\mathcal G}$ [4]. If ${\mathcal G}$ is a semiabelian complete subcategory of the category $TG$, then ${\mathcal H}$ is a semiabelian category in the sense of V.~P.~Palamodov. The direct and inverse spectra of a family of objects are particular cases of Hausdorff spectra -- it suffices to put ${\gF}=|{\gF}|$, $h_{s's}=q_{F'F}$ in the direct case and ${\gF}=\{|{\gF}|\}$, $h_{s's}:X_s\rightsquigarrow X_{s'}\ (s'\rightarrow s)$, $q_{F'F}=i_{|F|}=i_{|{\sgF}|}$ in the inverse case. In this case for each Hausdorff spectrum ${\mathcal X}=\{X_s,{\gF},h_{s's}\}$ over ${\mathcal G}$ there exists a unique (up to isomorphism) object of the category ${\mathcal G}$, the $H$-limit of the Hausdorff spectrum ${\mathcal X}$, which we denote by $\displaystyle\hl\,h_{s's}X_s\,$. Thus the additive and covariant functor of the $H$-limit of a Hausdorff spectrum ${\rm Haus}:{\mathcal H} \rightarrow{\mathcal G}$ is defined and we remark that it is natural in the categorical sense.

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How to Cite this Article:

Eugeny Ivanovich Smirnov, Sergey Alexandrovich Tikhomirov, The limit object of Hausdorff spectrum in the category TLC, J. Math. Comput. Sci., 5 (2015), 222-236

Copyright © 2015 Eugeny Ivanovich Smirnov, Sergey Alexandrovich Tikhomirov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

 

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