The limit object of Hausdorff spectrum in the category TLC

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.