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

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

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

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

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