Abstract
We study the notions of coproduct and subdirect product of preadditive categories and prove a Birkhoff type theorem showing that every skeletally small preadditive category is a subdirect product of subdirectly irreducible, skeletally small, preadditive categories. Moreover, we show that every direct-sum decomposition of the monoid \(V({\cal A})\) of the isomorphism classes of objects of \(\cal A\) is weakly induced by a coproduct decomposition of the preadditive category \(\cal A\).
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Partially supported by Ministero dell’Università e della Ricerca (PRIN 2005 “Perspectives in the theory of rings, Hopf algebras and categories of modules”), by Gruppo Nazionale Strutture Algebriche e Geometriche e loro Applicazioni of Istituto Nazionale di Alta Matematica, and by Università di Padova (Progetto di Ateneo CDPA048343 “Decomposition and tilting theory in modules, derived and cluster categories”).
Partially supported by Gruppo Nazionale Strutture Algebriche e Geometriche e loro Applicazioni of Istituto Nazionale di Alta Matematica, Italy. This paper was written during a visit of the second author at the Dipartimento di Matematica Pura e Applicata (Università di Padova, Italy). He acknowledges the kind hospitality received.
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Facchini, A., Fernández-Alonso, R. Subdirect Products of Preadditive Categories and Weak Equivalences. Appl Categor Struct 16, 103–122 (2008). https://doi.org/10.1007/s10485-007-9093-4
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DOI: https://doi.org/10.1007/s10485-007-9093-4