Definition 13.3.6. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{D}$ be a triangulated category. A $\delta$-functor from $\mathcal{A}$ to $\mathcal{D}$ is given by a functor $G : \mathcal{A} \to \mathcal{D}$ and a rule which assigns to every short exact sequence

$0 \to A \xrightarrow {a} B \xrightarrow {b} C \to 0$

a morphism $\delta = \delta _{A \to B \to C} : G(C) \to G(A)[1]$ such that

1. the triangle $(G(A), G(B), G(C), G(a), G(b), \delta _{A \to B \to C})$ is a distinguished triangle of $\mathcal{D}$ for any short exact sequence as above, and

2. for every morphism $(A \to B \to C) \to (A' \to B' \to C')$ of short exact sequences the diagram

$\xymatrix{ G(C) \ar[d] \ar[rr]_{\delta _{A \to B \to C}} & & G(A)[1] \ar[d] \\ G(C') \ar[rr]^{\delta _{A' \to B' \to C'}} & & G(A')[1] }$

is commutative.

In this situation we call $(G(A), G(B), G(C), G(a), G(b), \delta _{A \to B \to C})$ the image of the short exact sequence under the given $\delta$-functor.

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