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

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

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 ${\displaystyle F}$.¹