Cool Examples of Localisation in Triangulated Cats Besides the Usual

Question

In the theory of triangulated categories there is a hefty literature on localisation -- the most common example in algebra being (variants of) localising the homotopy category of chain complexes over a ring $A$ with respect to the quasiisomorphisms, giving the derived category $\mathbf{D}(A)$, and similarly for $\mathcal{O}_X$-modules on a scheme $X$.

Although not all localisations are with respect to a multiplicative system, the above is (with respect to $H^0$). There are other examples that aren't: one curious and interesting one in the paper of Buan and Marsh, where they provide an example of triangulated category that is localised with regard to a system of morphisms that is not multiplicative, and out comes an abelian category.

Unlike localisation in commutative rings (or even noncommutative rings), I feel like there are many fewer examples of localisation that could be helpful, if only to practice computing in triangulated categories.

My question is: what are the nice, and perhaps "crucial" examples if any, of localisation of triangulated categories, other than the above mentioned ones?

I would be particularly interested in "toy examples" or curiosities that aren't necessarily deep but yet still slightly tricky and yet elucidate some curious ways to use localisation. Ideally, any examples should pertain to applications of triangulated categories and localisation to algebra (representation theory, homological algebra, rings), or algebraic geometry. For example, papers such as the one mentioned above would be very welcome. For instance one could localise with respect to the multiplicative system given by morphisms that are converted to isomorphisms via any $\mathrm{Hom}(A,-)$, because it is a cohomological functor---but I haven't yet figured out if this leads to anywhere interesting...

Answer 1

As Fernando points out, Verdier localization is the most general localization of a triangulated category at a triangulated subcategory (actually you can do it even if $C$ is not a thick subcategory, but the localization will kill exactly the thick subcategory generated by $C$). A good resource is this article by Krause. Verdier localization is also mentioned in Amnon Neeman's book on triangulated categories.

A special case of Verdier localization is when you take the functor $T \to T/C$ and then include back into $T$. This yields an endofunctor $L:T\to T$ which is augmented (i.e. there's a natural transformation $\eta: 1\to L$) and idempotent (there's a natural isomorphism $L\circ L \to L$). Such localizations are called Bousfield localizations. Again, the sources above are excellent. The nice thing about Bousfield localization is that it can often be defined on the model category or infinity category level if the triangulated category in question is a homotopy category (see e.g. the nLab article). I'm a homotopy theorist, so the cool localization examples I know are Bousfield localizations. Off the top of my head here are some highlights:

• Localizing the stable homotopy category at a homology theory (so the new weak equivalences are the maps $f$ such that $E_*(f)$ is an isomorphism
• Localizing to define the motivic stable homotopy category (e.g. via Hovey's general machinery). Here you have motivic spaces and you've formed a category of spectra with respect to the simplicial sphere. You do Bousfield localization at the maps $f$ such that $f \wedge T$ are weak equivalences where $T$ is the Tate sphere. The result is (equivalent to) Voevodsky's field-metal winning stable motivic category which helped solve Milnor's conjecture and may help in constructing Grothendieck's conjectural category of motives
• Dugger's Universal model categories and the fact that any combinatorial (or left proper and cellular) model category is Quillen equivalent to a Bousfield localization of a category of simplicial presheaves.
• Many algebraic examples of localization (and completion!) are examples of Bousfield localization of an Eilenberg-Mac lane spectrum. A great resource is this note by Jacob Lurie. EDIT: A nice reference for the fact that completion is a special case of Bousfield localization is this answer of Tom Goodwillie's.
• Hovey-Shipley-Smith passage from the levelwise model structure on symmetric spectra to the stable one
• Shipley's construction of a positive model structure on symmetric spectra makes use of Bousfield localization.

For more about which model categories admit Bousfield localization, see here.

Answer 2

If $\mathcal C$ is a thick subcategory of a triangulated category $\mathcal T$, then you can localize w.r.t. the maps $f\colon X\rightarrow Y$ whose mapping cone is in $\mathcal C$, and the localized category, usually denoted $\mathcal T/\mathcal C$ is called Verdier quotient. The 'projection' functor $\mathcal T\rightarrow \mathcal T/\mathcal C$ is universal among functors sending objects in $\mathcal C$ to trivial objects. The derived category of an abelian category $\mathcal A$ is $D(\mathcal A)=K(\mathcal A)/A(\mathcal A)$ where $A(\mathcal A)$ is the full subcategory of the homotopy category $K(\mathcal A)$ spanned by acyclic complexes.

Let me modify your last question, demanding a localization with respect to the maps $f\colon X\rightarrow Y$ such that $\hom(A[n],f)$ is an isomorphism for any $n\in\mathbb Z$. Then the localization is $\mathcal T/\mathcal C$ for $\mathcal C$ the full subcategory spanned by the objects $X$ such that $\hom(A[n],X)=0$ any $n\in\mathbb Z$, so it is triangulated. This localization is often called $A$-cellularization in homotopy theoretic contexts, since any object in $\mathcal T/\mathcal C$ can be built from $A$ by taking shifts, coproducts, and iterated homotopy colimits.