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Note on separation in categories. (English) Zbl 0811.18001

Let \({\mathcal V} : {\mathcal A} \to {\mathcal X}\) be a covariant faithful functor between categories \(\mathcal A\) and \(\mathcal X\), and \(\mathcal D\) a class of \(\mathcal A\)-objects. \(\text{Sep }{\mathcal D}\) denotes the class of those objects \(A\) such that every morphism from an object in \(\mathcal D\) into \(A\) is constant, \(\text{Sep }{\mathcal V}\) the class of those \(\mathcal A\)-objects \(A\) such that every \(\mathcal V\)-initial source starting at \(A\) is a monosource and \(\text{Noncog }{\mathcal A}\) the class of \(\mathcal A\)-objects \(A\) that are not cogenerators of \(\mathcal A\). The authors investigate properties of and relations between these classes. Most of the papers dealing with transferring topological separation axioms into general categories uses functors \(\mathcal V\) with the range \({\mathcal X} = \text{Set}\). In this paper, they show that most of the relations known for the Set-based case are valid also in the general situation and that some new results are obtained. As a summary of their results, the following is stated. Suppose that \(\mathcal V\) preserves and lifts constants, \(\mathcal A\) contains a proper \(\mathcal V\)-initially ext-epireflective subcategory \(\mathcal R\), and \(C\) is a cogenerator of \(\mathcal A\) satisfying certain mild conditions. Then such \(\mathcal R\) is unique and coincides with \(\text{Sep}{\mathcal V}\), \(\text{Noncog }{\mathcal A}\) and \(\text{Sep }C\). Many examples related to discrete algebraic categories, relational structures, topological categories and topological algebraic structures are given.

MSC:

18A25 Functor categories, comma categories
18B30 Categories of topological spaces and continuous mappings (MSC2010)
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
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References:

[1] Herrlich H., Lect. Notes in Math. 78 (1978)
[2] Herrlich H., Category Theory (1979)
[3] Hoffmann R.-E., (E, M)-universally topological functors (1974)
[4] Hušek M., Proc. Berlin Top. Symp. (1967)
[5] Hušek M., Proc. Conf. Categ. Top. Mannheim 1975, Lect. Notes in Math. 540 pp 404– (1976)
[6] DOI: 10.1080/16073606.1990.9631972 · Zbl 0756.54010 · doi:10.1080/16073606.1990.9631972
[7] DOI: 10.1007/BF01299608 · Zbl 0466.18001 · doi:10.1007/BF01299608
[8] Marny T., Rechts-Bikategorienstrukturen in topologischen Kategorien (1973)
[9] Semadeni Z., Banach spaces of continuous functions (1971) · Zbl 0225.46030
[10] DOI: 10.1080/16073606.1980.9631584 · Zbl 0431.18001 · doi:10.1080/16073606.1980.9631584
[11] Weck-Schwarz, S. 1988.Cartesisch abgeschlossene monotopologische Hüllen und die Rolle der T0-Reflexion131Hannover PhD-Thesis, p
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