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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ and $\\mathcal{B}$ be [[abelian categories|Y9CZ3MTNF1]].\nA *cohomological $\\delta$-functor* from $\\mathcal{A}$ to $\\mathcal{B}$ consists of [[additive functors|KAN07YC9T2]] $T^n \\colon \\mathcal{A} \\to \\mathcal{B}$ for $n \\ge 0$ and of connecting morphisms.\nFor every [[short exact sequence|KJKNJMGKZZ]] $0 \\to A' \\to A \\to A'' \\to 0$ in $\\mathcal{A}$ and every $n \\ge 0$, there is a morphism $\\delta^n \\colon T^n(A'') \\to T^{n+1}(A')$.\nThese data must satisfy:\n\n1. for every short exact sequence, the sequence\n$$0 \\to T^0(A') \\to T^0(A) \\to T^0(A'') \\xrightarrow{\\delta^0} T^1(A') \\to T^1(A) \\to T^1(A'') \\xrightarrow{\\delta^1} T^2(A') \\to \\cdots$$\nis [[exact|JB7ZBFBA9C]];\n2. the maps $\\delta^n$ are natural. Let a morphism from $0 \\to A' \\to A \\to A'' \\to 0$ to $0 \\to B' \\to B \\to B'' \\to 0$ have components $f' \\colon A' \\to B'$ and $f'' \\colon A'' \\to B''$. Then $\\delta^n \\circ T^n(f'') = T^{n+1}(f') \\circ \\delta^n$.\n\nA $\\delta$-functor $(T^n)$ is *universal* if the following holds for every $\\delta$-functor $(S^n)$ and every [[natural transformation|KYCPCC57FT]] $T^0 \\to S^0$.\nThere is a unique family of natural transformations $T^n \\to S^n$ extending it and commuting with the maps $\\delta^n$.\n\n## Remarks\n\nExample: for an $R$-module $M$, the functors $T^n = \\operatorname{Ext}^n_R(M, -)$ with their connecting maps form a $\\delta$-functor, since Ext has the long exact sequence in the second variable, and it is natural.\nNon-example: on abelian groups, put $T^0 = \\Hom_\\Z(\\Z/2\\Z, -)$ and $T^n = 0$ for $n \\ge 1$.\nFor $0 \\to \\Z \\xrightarrow{\\cdot 2} \\Z \\to \\Z/2\\Z \\to 0$ the sequence in clause 1 starts $0 \\to 0 \\to 0 \\to \\Z/2\\Z \\to 0$, which is not exact, so clause 1 fails.\nA theorem of Grothendieck says that a $\\delta$-functor is universal when each $T^n$ with $n \\ge 1$ is effaceable, meaning that every object $A$ embeds by a monomorphism $u$ with $T^n(u) = 0$.\nHence the right derived functors of a left exact functor form a universal $\\delta$-functor.\nHomological $\\delta$-functors $(T_n)$ are defined in the same way with the arrows $\\delta_n \\colon T_n(A'') \\to T_{n-1}(A')$.\n",
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