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In the theory of triangulated categories, we propose to replace hearts of$t$-structures by proper abelian subcategories, which may be plentiful evenwhen hearts are not. For instance, this happens in negative cluster categories.In support of our proposal, we show that proper abelian subcategories with afew vanishing negative self extensions permit a tilting theory which is adirect generalisation of Happel-Reiten-Smal{\o} tilting of hearts.

Proper abelian subcategories of triangulated categories and their tilting theory