The Stacks project

Definition 13.40.1. Let $\mathcal{D}$ be a triangulated category. Let

\[ X_ n \to X_{n - 1} \to \ldots \to X_0 \]

be a complex in $\mathcal{D}$. A Postnikov system is defined inductively as follows.

  1. If $n = 0$, then it is an isomorphism $Y_0 \to X_0$.

  2. If $n = 1$, then it is a choice of a distinguished triangle

    \[ Y_1 \to X_1 \to Y_0 \to Y_1[1] \]

    where $X_1 \to Y_0$ composed with $Y_0 \to X_0$ is the given morphism $X_1 \to X_0$.

  3. If $n > 1$, then it is a choice of a Postnikov system for $X_{n - 1} \to \ldots \to X_0$ and a choice of a distinguished triangle

    \[ Y_ n \to X_ n \to Y_{n - 1} \to Y_ n[1] \]

    where the morphism $X_ n \to Y_{n - 1}$ composed with $Y_{n - 1} \to X_{n - 1}$ is the given morphism $X_ n \to X_{n - 1}$.

Given a morphism
\begin{equation} \label{derived-equation-map-complexes} \vcenter { \xymatrix{ X_ n \ar[r] \ar[d] & X_{n - 1} \ar[r] \ar[d] & \ldots \ar[r] & X_0 \ar[d] \\ X'_ n \ar[r] & X'_{n - 1} \ar[r] & \ldots \ar[r] & X'_0 } } \end{equation}

between complexes of the same length in $\mathcal{D}$ there is an obvious notion of a morphism of Postnikov systems.

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