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The following nicely does the trick I think... Lemma Every monomorphism in a triangulated category splits. Proof: Let $T$ be a triangulated category and suppose that $f\colon x\to y$ is a monomorphis …

As Hailong said in his comment this only happens in the Gorenstein case; here is a sketch of an argument. Suppose $X$ is a quasi-compact quasi-separated scheme with a dualising complex $D$ and let us …

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 …

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 …

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 …

I don't know of a counterexample but I can tell you some more situations in which it is true. Ballard has extended Orlov's result (in Equivalences of derived categories of sheaves on quasi-projective …

As Eric and Karol have noted it is usually not the case that there exists such an $S$ (which I'll take to be a dga; I don't know what happens off the top of my head if one asks for $S$ to be an honest …

The original action takes the form of an additive functor $A \times B \to B$ with notation as in the question (and appropriate coherence conditions giving compatibility with the monoidal structure on …

Provided $R$ has a dualizing complex an answer is given in this (very nice) preprint of Alonso, Jeremias, and, Saorin in terms of certain filtrations on the spectrum of $R$. Corollary 6.11 is the resu …

One can also view model structures as a solution to the problem of when is a localization of a category locally small. In other words when one wants to invert a collection of morphisms the morphisms i …

I'm not sure if this category has a particular name - usually until someone cares enough to give one of these a name or nice notation they just have long unwieldy names. I can suggest some notation th …

I wouldn't go as far as to say I understand, but what I get from it is the following: One can roughly consider the vanishing cycles functor as coming from gluing data for the recollement associated to …