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The adjoint functor theorem as stated here and the special adjoint functor theorem (which can also both be found in Mac Lane) are both very handy for showing the existence of adjoint functors. First …

One can prove that for any non-zero ring $R$ the category $R$-Mod$^{op}$ is not a category of modules. Indeed any category of modules is Grothendieck abelian i.e., has exact filtered colimits and a ge …

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 …

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 …

Suppose that $F\colon C\to D$ is a functor. Then there are many situations in which thinking of finding left and right adjoints to $F$ as solving approximation problems is very good intuition. So thes …

This follows from the Yoneda lemma - probably my favourite way of thinking about this fact is via coends in the way described here by Todd Trimble, which I think makes it quite clear what is going o …

In Appendix C (Corollary C.3.3 to be precise) of Neeman's book "Triangulated Categories" an example of an abelian category which is not well-powered is given. The actual counterexample is given by $A …

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 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 …

One can take the category of modules over a Laurent polynomial ring in one variable $\textrm{Mod}\;k[t,t^{-1}]$ and think of $k[t,t^{-1}]$ as the group algebra of $\mathbb{Z}$. The corresponding cocom …

I assume you are working in some fixed abelian category $\mathcal{A}$. It is not true in general that every short exact sequence in $\mathcal{A}$ will split. The problem is that although you can pick …

Dan Murfet has some notes on foundations for category theory which can be found here. They contain an introduction to Grothendieck universes as well as some references for learning about NBG class th …

I'm not sure if I constitute an expert or this constitutes a real answer but let me try. If I understand correctly your first question is whether it is open that every localizing subcategory of an al …

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 …