Semi-localizations of semi-abelian categories

Marino Gran*, Stephen Lack

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    3 Citations (Scopus)


    A semi-localization of a category is a full reflective subcategory with the property that the reflector is semi-left-exact. There are many interesting examples of semi-localizations, as for instance any torsion-free subcategory of a semi-abelian category. By specializing a result due to S. Mantovani, we first characterize the categories which are semi-localizations of exact Mal'tsev categories. We then prove a new characterization of protomodular categories in terms of binary relations, allowing us to obtain an abstract characterization of the semi-localizations of exact protomodular categories. This result is very useful to study the (hereditarily)-torsion-free subcategories of semi-abelian categories. Some examples are considered in detail in the categories of groups, crossed modules, commutative rings and topological groups. We finally explain how these results extend the corresponding ones obtained in the abelian context by W. Rump.

    Original languageEnglish
    Pages (from-to)206-232
    Number of pages27
    JournalJournal of Algebra
    Publication statusPublished - 15 May 2016


    • Semi-abelian category
    • Protomodular category
    • Mal'tsev category
    • Localization
    • Semi-left-exact reflector
    • Torsion theory
    • Exact completion


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