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Monotone-light factorisation systems and torsion theories. (English) Zbl 1284.18024

Summary: Given a torsion theory \((\mathbb{Y,X})\) in an abelian category \(\mathbb C\), the reflector \(I:\mathbb C\to\mathbb X\) to the torsion-free subcategory \(\mathbb X\) induces a reflective factorisation system \((\mathcal{E,M})\) on \(\mathbb C\). It was shown by A. Carboni et al. [Appl. Categ. Struct. 5, No. 1, 1–58 (1997; Zbl 0866.18003)] that \((\mathcal{E,M})\) induces a monotone-light factorisation system \((\mathcal E^\prime,\mathcal M^\ast)\) by simultaneously stabilising \(\mathcal E\) and localising \(\mathcal M\), whenever the torsion theory is hereditary and any object in \(\mathbb C\) is a quotient of an object in \(\mathbb X\). We extend this result to arbitrary normal categories, and improve it also in the abelian case, where the heredity assumption on the torsion theory turns out to be redundant. Several new examples of torsion theories where this result applies are then considered in the categories of abelian groups, groups, topological groups, commutative rings, and crossed modules.

MSC:

18E40 Torsion theories, radicals
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
16N80 General radicals and associative rings
20K15 Torsion-free groups, finite rank
54H11 Topological groups (topological aspects)

Citations:

Zbl 0866.18003
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References:

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