= Hereditary abelian category with enough projectives
{title2=$L_2F=0$}
An <abelian category> with enough projectives is hereditary when every <subobject> of a projective object is projective. For $0\to K\to P\to A\to0$ with $P$ projective, dimension shifting gives $L_2F(A)\cong L_1F(K)$. Hence hereditaryness is equivalent to vanishing of $L_2F$ for every additive <right exact functor> into any abelian <category>, by <projectivity detected by all first left derived functors>. It is also equivalent to left exactness of every $L_1F$: the derived long exact sequence gives injectivity of $L_1F(K)\to L_1F(P)$ when $L_2F$ vanishes, and applying left exactness to the displayed sequence proves the converse.
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