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It is true in a Heller triangulated category aka $\infty$-triangulated category (although strictly speaking one only needs a 3-triangulation for octahedra) that any morphism between the bases of octah …

There is a similar question here. It is asked in the particular case of the derived category of abelian groups and a counterexample is given by Tyler Lawson. There is also an answer of mine which all …

So if I understand correctly the question you wanted to ask was: Is it true that a triangle $$X \stackrel{u}{\to} Y \stackrel{v}{\to} Z \stackrel{w}{\to} \Sigma X$$ is split if and only if one of $u$, …

The answer is in general no - $K(R\text{-}\mathrm{Mod})$ can fail to be well generated even when $R$ is artinian. As you mention $K(R\text{-}\mathrm{Mod})$ is compactly generated if $R$ is of finite r …

The problem with respect to applications of the abelianization is that the abelian categories one produces are almost uniformly horrible. More precisely they are just too big to deal with. So using th …

Generally speaking a unique lifting does not exist and I believe it is open as to what the possible liftings can be. As an example of the non-uniqueness consider a slight variant of the particular ca …

For triangulated categories the notion of Bousfield localization is a "special case" of the notion of Verdier quotient. As you observe (and is shown in Lemma 3.1 of the paper you mention) any Bousfiel …

I just wanted to point out that this failure is quite standard rather than pathological. As a starting point it can go wrong more generally than Tyler points out. For instance there exist triangles wh …

The category of sheaves of $\mathbb{Q}$ vector spaces on $M$ is a Grothendieck abelian category. It follows that the derived category of such, $D(M)$ in your notation, is a well generated triangulated …

I would imagine it is false in general that given a triangulated category $T$ the category $T\otimes R$ is also triangulated. The following is a concrete counterexample. Consider $D^b(\mathbb{Z})$ an …

There are some things like what you ask for but as Tyler points out one needs restrictions on the categories one can consider. Any algebraic triangulated category which is well generated is equivalen …

The title pretty much sums it up - but let me give a little bit of background first. In the p-local stable homotopy category (basically one localizes away the torsion spectra which are not p-torsion) …

The two notions are related in the sense that they share a common generalization, namely the notion of torsion pair on a pre-triangulated category (this term has at least two meanings, here we mean a …

To address Hanno's question about checking that composition gives a graded-commutative ring structure on $End^{*}(\mathbb{Z}) = \oplus_i [\mathbb{Z}, \Omega^{-i} \mathbb{Z}]$ suppose first that $a \st …

One example is the following - suppose that $M$ is a non-compact connected manifold of dimension $\geq 1$. Then the unbounded derived category of chain complexes of sheaves of abelian groups on $M$ ha …