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Let $\mathcal{T}$ be a triangulated category and take $\mathcal{S}$ a triangulated subcategory. Consider the Verdier quotient $\mathcal{T} \left/ \mathcal{S} \right.$, morphisms in this category are constructed as roofs $E \leftarrow H \rightarrow F$ where $\text{cone}(H \rightarrow E) \in \mathcal{S}$ modulo a certain equivalence relation. In general it's not easy to compute morphisms in a general Verdier quotient because one needs to replace either E or F with a "resolution" in terms of objects which are in the left, respectively right, orthogonal to $\mathcal{S}$. An example to keep in mind is the derived category of an abelian category, in which case we replace $E$ with a projective resolution or $F$ with an injective resolution. However, in the case $D(\mathcal{A})$, where $\mathcal{A}$ is an abelian category, one has some spectral sequences converging to Homs in $D(\mathcal{A})$ (I'm thinking of $Hom_{D(\mathcal{A})} ( E, H^q(F)[p]) \implies Hom_{D(\mathcal{A})}(E,F[p+q])$ for example).

My question is whether there exist tools like such (spectral sequences or any kind of approximation) to compute morphisms in $\mathcal{T} / \mathcal{S}$. Even a way to relate them to morphisms in $\mathcal{T}$ would be fine. I hardly think such a thing exists in general, but I am mainly interested in the case $\mathcal{T} = D(\mathcal{A})$.

Thank you in advance for the help.

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    $\begingroup$ I agree that this kind of computations may be tricky but, if you are happy to restrict a little bit your setting, then computations can be done. In particular, two assumptions may help: 1) that your triangulated category is the homotopy category of a (stable, simplicial, maybe even combinatorial) model category, and 2) that the Verdier quotient functor has an adjoint, so that you have a Bousfield localization. In this case, the quotient $\mathcal T/\mathcal S$ should be the homotopy category of a Bousfield localization of the original model category, so you have descriptions for homs. $\endgroup$ – Simone Virili Jan 17 '20 at 13:28
  • $\begingroup$ @Simone Virili Can you give me a reference? Those conditions should be satisfied as I'm considering $T = D(A)$ and thus the quotient factor has a right adjoint by Brown representability. $\endgroup$ – Federico Barbacovi Jan 17 '20 at 14:06
  • $\begingroup$ I think the standard reference for Bousfield localization of Model categories is "Model Categories and Their Localizations" by Philip S. Hirschhorn. To get the idea and to understand if that is what you are looking for, you may have a look at the nLab page about "Bousfield localization of model categories". $\endgroup$ – Simone Virili Jan 18 '20 at 12:40

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