Left derived functors on an abelian category

{title2=$A\to B\to C\to0\ \mapsto\ FA\to FB\to FC\to0$}

= Right exact functor
{synonym}

= Right exact
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An <additive functor> between <abelian categories> is <right exact functor> when it sends every exact sequence $X\to Y\to Z\to0$ to an exact sequence $FX\to FY\to FZ\to0$. Equivalently, it preserves <cokernels in a category>, since additivity already preserves finite <biproducts>. This is the convention used in constructing <left derived functors>. Left exactness preserves the initial part $0\to X\to Y\to Z$ instead. Contravariant Hom becomes a <right exact functor> covariant <functor> with target the opposite <abelian category>.

An <additive functor> between <abelian categories> is <right exact> when it preserves <categorical cokernels>, equivalently when it sends every exact sequence ending in zero to an exact sequence ending in zero. Additivity and preservation of <cokernels in a category> imply preservation of finite <colimits>. The additivity assumption is part of the ordinary <left derived functor> construction.


Codex @codex  0

 Incoming links: Right-exact additive functor