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Note on multisolid categories. (English) Zbl 0939.18003
Let \(U:{\mathcal A}\to X\) be a faithful functor, with \({\mathcal A}\) cowellpowered and \({\mathcal X}\) cocomplete; then \(U\) is solid (= semi-topological) if and only if \(U\) is right adjoint and \({\mathcal A}\) is cocomplete [cf. W. Tholen, J. Pure Appl. Algebra 15, 53-73 (1979; Zbl 0413.18001)]. The note under review gives a direct proof of the “multi”-version of this theorem in the sense of Diers: the faithful functor \(U\) with \({\mathcal A}\) cowellpowered and \({\mathcal X}\) multicocomplete is multisolid if and only if \(U\) is right multiadjoint and \({\mathcal A}\) is multicocomplete. With the use of the formal product completion of a category, this latter fact may also be derived from the former theorem, as was observed by J. Adámek, L. Sousa, and W. Tholen in a recent paper [“Totality of product completions”, Comment. Math. Univ. Carolinae, to appear].
MSC:
18A22 Special properties of functors (faithful, full, etc.)
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18A05 Definitions and generalizations in theory of categories
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
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References:
[1] Adámek, J.; Herrlich, H.; Strecker, G.E., Abstract and concrete categories, (1990), Wiley New York · Zbl 0695.18001
[2] Diers, Y., Catégories localisables, ()
[3] Tholen, W., Semitopological functors I, J. pure appl. algebra, 15, 53-73, (1979) · Zbl 0413.18001
[4] Tholen, W., Macneille completion of concrete categories with local properties, Comment. math. univ. st. Pauli, XXVIII, 2, 179-202, (1979) · Zbl 0435.18005
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