Acyclic object

In mathematics, in the field of homological algebra, given an abelian category ${\displaystyle {\mathcal {C}}}$ having enough injectives and an additive (covariant) functor

an acyclic object with respect to ${\displaystyle F}$, or simply an ${\displaystyle F}$-acyclic object, is an object ${\displaystyle A}$ in ${\displaystyle {\mathcal {C}}}$ such that

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