Acyclic object

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

$F:{\mathcal {C}}\to {\mathcal {D}}$,

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

${\rm {R}}^{i}F(A)=0\,\!$ for all $i>0\,\!$,

where ${\rm {R}}^{i}F$ are the right derived functors of $F$.