(delta functor)
Let \(\mca, \mcb\) be abelian categories. A covariant \(\delta\)-functor from \(\mca\) to \(\mcb\) is a collection of functors \(T = (T^i)_{i\geq 0}\) together with a boundary morphism \(\delta^i: T^i(A'') \to T^{i+1}(A')\) for every short exact sequence \(0 \to A' \to A \to A'' \to 0\), such that
each short exact sequence gives a long exact sequence \[ 0 \to T^0(A') \to T^0(A) \to T^0(A'') \mapsvia{\delta^0} T^1(A') \to T^1(A) \to \cdots , \]
for each morphism of short exact sequences the induced maps commute with the \(\delta^i\).
(effaceable)
An additive functor \(F: \mca \to \mcb\) is effaceable if for each object \(A\) there is a monomorphism \(u: A \injects M\) with \(F(u) = 0\), and coeffaceable if for each object \(A\) there is an epimorphism \(u: P \surjects A\) with \(F(u) = 0\).
(universal delta functor)
A covariant \(\delta\)-functor \(T=(T^i)\) from \(\mca\) to \(\mcb\) is universal if for every \(\delta\)-functor \(T'=(T'^i)\) from \(\mca\) to \(\mcb\), every natural transformation \(f^0\colon T^0\to T'^0\) extends uniquely to a sequence of natural transformations \(f^i\colon T^i\to T'^i\) commuting with the \(\delta^i\).
(Grothendieck)
A covariant \(\delta\)-functor \(T\) with \(T^i\) effaceable for every \(i>0\) is universal [@Har10a, Theorem III.1.3A]. If \(\mca\) has enough injectives and \(F\) is left exact, then \((R^iF)_{i\ge0}\) is a universal \(\delta\)-functor with \(R^0F\cong F\), since every \(A\) embeds in an injective object \(I\) and \(R^iF(I)=0\) for \(i>0\).
Two universal \(\delta\)-functors with isomorphic degree-\(0\) functors are isomorphic. For \(A\)-modules \(M,N\), the groups \(\Ext^i_A(M,N)\) computed from a projective resolution of \(M\) form, as functors of \(N\), a \(\delta\)-functor with \(\Ext^0_A(M,\wait)=\Hom_A(M,\wait)\) that vanishes on injective \(N\) in positive degrees; so they are the right derived functors of \(\Hom_A(M,\wait)\).
Dependencies
D-DEFDERIVDerived functorsuses