The Stacks project

Example 13.41.2. Let $\mathcal{A}$ be an abelian category. Let $\ldots \to A_2 \to A_1 \to A_0$ be a chain complex in $\mathcal{A}$. Then we can consider the objects

\[ X_ n = A_ n \quad \text{and}\quad Y_ n = (A_ n \to A_{n - 1} \to \ldots \to A_0)[-n] \]

of $D(\mathcal{A})$. With the evident canonical maps $Y_ n \to X_ n$ and $Y_0 \to Y_1[1] \to Y_2[2] \to \ldots $ the distinguished triangles $Y_ n \to X_ n \to Y_{n - 1} \to Y_ n[1]$ define a Postnikov system as in Definition 13.41.1 for $\ldots \to X_2 \to X_1 \to X_0$. Here we are using the obvious extension of Postnikov systems for an infinite complex of $D(\mathcal{A})$. Finally, if colimits over $\mathbf{N}$ exist and are exact in $\mathcal{A}$ then

\[ \text{hocolim} Y_ n[n] = (\ldots \to A_2 \to A_1 \to A_0 \to 0 \to \ldots ) \]

in $D(\mathcal{A})$. This follows immediately from Lemma 13.33.7.

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