Definition 12.12.1. Let \mathcal{A}, \mathcal{B} be abelian categories. A cohomological \delta -functor or simply a \delta -functor from \mathcal{A} to \mathcal{B} is given by the following data:
a collection F^ n : \mathcal{A} \to \mathcal{B}, n \geq 0 of additive functors, and
for every short exact sequence 0 \to A \to B \to C \to 0 of \mathcal{A} a collection \delta _{A \to B \to C} : F^ n(C) \to F^{n + 1}(A), n \geq 0 of morphisms of \mathcal{B}.
These data are assumed to satisfy the following axioms
for every short exact sequence as above the sequence
\xymatrix{ 0 \ar[r] & F^0(A) \ar[r] & F^0(B) \ar[r] & F^0(C) \ar[lld]^{\delta _{A \to B \to C}} \\ & F^1(A) \ar[r] & F^1(B) \ar[r] & F^1(C) \ar[lld]^{\delta _{A \to B \to C}} \\ & F^2(A) \ar[r] & F^2(B) \ar[r] & \ldots }is exact, and
for every morphism (A \to B \to C) \to (A' \to B' \to C') of short exact sequences of \mathcal{A} the diagrams
\xymatrix{ F^ n(C) \ar[d] \ar[rr]_{\delta _{A \to B \to C}} & & F^{n + 1}(A) \ar[d] \\ F^ n(C') \ar[rr]^{\delta _{A' \to B' \to C'}} & & F^{n + 1}(A') }are commutative.
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