Under a splitting assumption, the paper asserts that a chain complex's homology object is the categorified eigenvalue of that complex with the ground ring as eigenobject; the converse half of the assertion is false over Z.
Schapira : Categories and homological algebra https://webusers.imj-prg.fr/ pierre.schapira/lectnotes/HomAl.pdf
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Homology Groups and Categorical Diagonalization
Under a splitting assumption, the paper asserts that a chain complex's homology object is the categorified eigenvalue of that complex with the ground ring as eigenobject; the converse half of the assertion is false over Z.