The Stacks project

Definition 13.5.1. Let $\mathcal{D}$ be a pre-triangulated category. We say a multiplicative system $S$ is compatible with the triangulated structure if the following two conditions hold:

1. For a morphism $f$ of $\mathcal{D}$ we have $f \in S \Leftrightarrow f[1] \in S$¹.

2. Given a solid commutative square

   $$\xymatrix{ X \ar[r] \ar[d]^ s & Y \ar[r] \ar[d]^{s'} & Z \ar[r] \ar@{-->}[d] & X[1] \ar[d]^{s[1]} \\ X' \ar[r] & Y' \ar[r] & Z' \ar[r] & X'[1] }$$

   whose rows are distinguished triangles with $s, s' \in S$ there exists a morphism $s'' : Z \to Z'$ in $S$ such that $(s, s', s'')$ is a morphism of triangles.