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A triangulated category is an additive category equipped with the additional structure of an autoequivalence (called the translation functor) and a class of of triangles satisfying certain axioms.

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Does stabilization preserve the property of being a full subcategory?

Suppose $F:\mathcal{C}\hookrightarrow \mathcal{D}$ is a fully faithful symmetric monoidal functor of symmetric monoidal $\infty$-categories. Suppose $x\in\mathcal{C}$. Is it true that $\mathcal{C}[x^{ …