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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 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 …

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 …

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 …

It is the minimal separator in the sense that it corepresents the forgetful functor Grp $\rightarrow$ Set, but this uses sets so probably isn't what you are after. In fact it is the same as the statem …

To answer the first question provided one has, as you say, (small) products and equalizers the notion of sheaf makes sense as one has the right diagram corresponding to any cover. But we can just say …

I'm not completely sure if this is the sort of thing you are after, but the telescope conjecture (conjecture isn't a great word as it is known to be false for some categories) springs to mind as somet …

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 …

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 …

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 …

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 …

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 …

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 …

There are still strictly speaking elements floating around in the following since we are using indexing sets but maybe it is better? Consider for a set $S$ and an abelian group $A$ the isomorphisms $$ …

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 …