The left heart and exact hull of an additive regular category

• Ruben Henrard ^[1] ; Sondre Kvamme ^[2] ; Adam-Christiaan van Roosmalen ^[4] ; Sven-Ake Wegner ^[3]
  1. [1] University of Hasselt

     University of Hasselt

     Arrondissement Hasselt, Bélgica

     
  2. [2] Norwegian University of Science and Technology

     Norwegian University of Science and Technology

     Noruega

     
  3. [3] University of Hamburg

     University of Hamburg

     Hamburg, Freie und Hansestadt, Alemania

     
  4. [4] Xi’an Jiaotong-Liverpool University, Suzhou, China

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 2, 2023, págs. 439-494
• Idioma: inglés
• DOI: 10.4171/RMI/1388
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• Resumen

  • Quasi-abelian categories are abundant in functional analysis and representation theory. It is known that a quasi-abelian category E is a cotilting torsionfree class of an abelian category. In fact, this property characterizes quasi-abelian categories. This ambient abelian category is derived equivalent to the category E, and can be constructed as the heart LH(E) of a t-structure on the bounded derived category Db(E) or as the localization of the category of monomorphisms in E.

    However, there are natural examples of categories in functional analysis which are not quasi-abelian, but merely one-sided quasi-abelian or even weaker. Examples are the category of LB-spaces or the category of complete Hausdorff locally convex spaces. In this paper, we consider additive regular categories as a generalization of quasi-abelian categories that covers the aforementioned examples. Additive regular categories can be characterized as those subcategories of abelian categories which are closed under subobjects.

    As for quasi-abelian categories, we show that such an ambient abelian category of an additive regular category E can be found as the heart of a t-structure on the bounded derived category Db(E), or as the localization of the category of monomorphisms of E. In our proof of this last construction, we formulate and prove a version of Auslander's formula for additive regular categories.

    Whereas a quasi-abelian category is an exact category in a natural way, an additive regular category has a natural one-sided exact structure. Such a one-sided exact category can be 2-universally embedded into its exact hull. We show that the exact hull of an additive regular category is again an additive regular category.