Definition 13.41.1. 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 an isomorphism $Y_0 \to X_0$ and 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

13.41.1.1
$$\label{derived-equation-map-complexes} \vcenter { \xymatrix{ X_ n \ar[r] \ar[d] & X_{n - 1} \ar[r] \ar[d] & \ldots \ar[r] & X_0 \ar[d] \\ X'_ n \ar[r] & X'_{n - 1} \ar[r] & \ldots \ar[r] & X'_0 } }$$

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

Comment #7802 by Nicolás on

It seems that (2) is missing an explicit mention of the inductive construction, as it is not specified which morphism $Y_0 \to X_0$ are we working with. Something like "If $n=1$, then it is a choice of a Postnikov system for $X_0$ and a choice of a distinguished [...]." (And probably, in that case (2) and (3) could be unified.)

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