The Stacks project

Lemma 57.6.1. Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{D}' \subset \mathcal{D}$ be a full triangulated subcategory. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$. The category of arrows $E \to X$ with $E \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}')$ is filtered.

Proof. We check the conditions of Categories, Definition 4.19.1. The category is nonempty because it contains $0 \to X$. If $E_ i \to X$, $i = 1, 2$ are objects, then $E_1 \oplus E_2 \to X$ is an object and there are morphisms $(E_ i \to X) \to (E_1 \oplus E_2 \to X)$. Finally, suppose that $a, b : (E \to X) \to (E' \to X)$ are morphisms. Choose a distinguished triangle $E \xrightarrow {a - b} E' \to E''$ in $\mathcal{D}'$. By Axiom TR3 we obtain a morphism of triangles

\[ \xymatrix{ E \ar[r]_{a - b} \ar[d] & E' \ar[d] \ar[r] & E'' \ar[d] \\ 0 \ar[r] & X \ar[r] & X } \]

and we find that the resulting arrow $(E' \to X) \to (E'' \to X)$ equalizes $a$ and $b$. $\square$

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