The Stacks project

Lemma 13.40.8. Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{B} \subset \mathcal{D}$ be a full triangulated subcategory. The following are equivalent

  1. the inclusion functor $\mathcal{B} \to \mathcal{D}$ has a left adjoint, and

  2. for every $X$ in $\mathcal{D}$ there exists a distinguished triangle

    \[ A \to X \to B \to A[1] \]

    in $\mathcal{D}$ with $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ and $A \in \mathop{\mathrm{Ob}}\nolimits ({}^\perp \mathcal{B})$.

If this holds, then $\mathcal{B}$ is saturated (Definition 13.6.1) and if $\mathcal{B}$ is strictly full in $\mathcal{D}$, then $\mathcal{B} = ({}^\perp \mathcal{B})^\perp $.

Proof. Dual to Lemma 13.40.7. $\square$

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