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:

1. a collection $F^ n : \mathcal{A} \to \mathcal{B}$, $n \geq 0$ of additive functors, and

2. 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

1. 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

2. 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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