Let $F : \mathcal{D}_1 \rightarrow \mathcal{D}_2$ be a triangulated functor of triangulated categories $\mathcal{D}_1$ and $\mathcal{D}_2$, and let $\{ M_{\alpha}\}$ be a collection of object which generate $\mathcal{D}_1$, i.e. any full triangulated subcategory containing all objects $M_{\alpha}$ is equivalent to $\mathcal{D}_1$ (via the inclusion). I need to show that if $F$ induces isomorphisms \begin{equation} \tag{1} \text{Hom}(M_{\alpha}, M_{\beta}) \cong \text{Hom}(F(M_{\alpha}), F(M_{\beta})) \end{equation} then $F$ is fully faithful. The second part of the exercise is showing that if the collection $\{F(M_{\alpha})\}$ generates $\mathcal{D}_2$ then $F$ is an equivalence.

While this sounds perfectly easy intuitively, the problem is that I don't see how I should use the definition of the generating collection of objects to show that $(1)$ is true for all $X,Y \in \mathcal{D}_1$. Any help would be much appreciated.

  • $\begingroup$ Every object is a cone of a coproduct of suspensions of a generator. Induct on a decomposition in these terms. $\endgroup$ – Kevin Carlson Apr 13 '17 at 21:42
  • $\begingroup$ @KevinCarlson thanks! If you don't mind, why is every object is a coproduct of suspensions of a generator? $\endgroup$ – baltazar Apr 14 '17 at 0:16

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