The Stacks project

Lemma 13.5.2. Let $\mathcal{D}$ be a pre-triangulated category. Let $S$ be a set of morphisms of $\mathcal{D}$ and assume that axioms MS1, MS5, MS6 hold (see Categories, Definition 4.27.1 and Definition 13.5.1). Then MS2 holds.

Proof. Suppose that $f : X \to Y$ is a morphism of $\mathcal{D}$ and $t : X \to X'$ an element of $S$. Choose a distinguished triangle $(X, Y, Z, f, g, h)$. Next, choose a distinguished triangle $(X', Y', Z, f', g', t[1] \circ h)$ (here we use TR1 and TR2). By MS5, MS6 (and TR2 to rotate) we can find the dotted arrow in the commutative diagram

\[ \xymatrix{ X \ar[r] \ar[d]^ t & Y \ar[r] \ar@{..>}[d]^{s'} & Z \ar[r] \ar[d]^1 & X[1] \ar[d]^{t[1]} \\ X' \ar[r] & Y' \ar[r] & Z \ar[r] & X'[1] } \]

with moreover $s' \in S$. This proves LMS2. The proof of RMS2 is dual. $\square$


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