Definition 13.27.4. Let $\mathcal{A}$ be an abelian category. Let $A, B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. A degree $i$ *Yoneda extension* of $B$ by $A$ is an exact sequence

\[ E : 0 \to A \to Z_{i - 1} \to Z_{i - 2} \to \ldots \to Z_0 \to B \to 0 \]

in $\mathcal{A}$. We say two Yoneda extensions $E$ and $E'$ of the same degree are *equivalent* if there exists a commutative diagram

\[ \xymatrix{ 0 \ar[r] & A \ar[r] & Z_{i - 1} \ar[r] & \ldots \ar[r] & Z_0 \ar[r] & B \ar[r] & 0 \\ 0 \ar[r] & A \ar[r] \ar[u]^{\text{id}} \ar[d]_{\text{id}} & Z''_{i - 1} \ar[r] \ar[u] \ar[d] & \ldots \ar[r] & Z''_0 \ar[r] \ar[u] \ar[d] & B \ar[r] \ar[u]_{\text{id}} \ar[d]^{\text{id}} & 0 \\ 0 \ar[r] & A \ar[r] & Z'_{i - 1} \ar[r] & \ldots \ar[r] & Z'_0 \ar[r] & B \ar[r] & 0 } \]

where the middle row is a Yoneda extension as well.

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