Analysis of a problem of Raikov with applications to barreled and bornological spaces |
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Authors: | Wolfgang Rump |
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Affiliation: | Institute for Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany |
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Abstract: | Raikov’s conjecture states that semi-abelian categories are quasi-abelian. A first counterexample is contained in a paper of Bonet and Dierolf who considered the category of bornological locally convex spaces. We prove that every semi-abelian category I admits a left essential embedding into a quasi-abelian category Kl(I) such that I can be recovered from Kl(I) by localization. Conversely, it is shown that left essential full subcategories I of a quasi-abelian category are semi-abelian, and a criterion for I to be quasi-abelian is given. Applied to categories of locally convex spaces, the criterion shows that barreled or bornological spaces are natural counterexamples to Raikov’s conjecture. Using a dual argument, the criterion leads to a simplification of Bonet and Dierolf’s example. |
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Keywords: | Primary, 18E10, 18E35, 46M15, 46A08, 46A17 Secondary, 18G25, 18A40, 46A03 |
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