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Definition 13.3.4. Let $(\mathcal{D}, [\ ], \mathcal{T})$ be a pre-triangulated category. A pre-triangulated subcategory1 is a pair $(\mathcal{D}', \mathcal{T}')$ such that

  1. $\mathcal{D}'$ is an additive subcategory of $\mathcal{D}$ which is preserved under $[1]$ and $[-1]$,

  2. $\mathcal{T}' \subset \mathcal{T}$ is a subset such that for every $(X, Y, Z, f, g, h) \in \mathcal{T}'$ we have $X, Y, Z \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}')$ and $f, g, h \in \text{Arrows}(\mathcal{D}')$, and

  3. $(\mathcal{D}', [\ ], \mathcal{T}')$ is a pre-triangulated category.

If $\mathcal{D}$ is a triangulated category, then we say $(\mathcal{D}', \mathcal{T}')$ is a triangulated subcategory if it is a pre-triangulated subcategory and $(\mathcal{D}', [\ ], \mathcal{T}')$ is a triangulated category.

[1] This definition may be nonstandard. If $\mathcal{D}'$ is a full subcategory then $\mathcal{T}'$ is the intersection of the set of triangles in $\mathcal{D}'$ with $\mathcal{T}$, see Lemma 13.4.15. In this case we drop $\mathcal{T}'$ from the notation.

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