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

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

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

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 $X_1 \to X_0$.

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

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

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