Let A be an abelian category and let $A \in $ A. Denote by InjA the category of injective objects of A. We denote by $A\langle0\rangle$ the complex concentrated in degree zero. I define an injective resolution to be a complex $B$ in InjA quasi-isomorphic to $A\langle0\rangle$.
Question. Is this equivalent to the regular definition or do I need to specify that $B^i = 0$ for $i<0$? If this is obvious, I apologise. I am tired right now.