Localization(s) of Categories

Question

I've been trying to read a paper by Krause and came across a strange (to me, of course) notion of localization. After looking around for a long time, and then finding this on his site, I see that there are two notions for localization, both with significant usage online. These are namely Verdier localization and Bousfield localization. Is there a strong motivation to use one over the other? A little bit of context:

I see that Bousfield localization is defined for model categories, and this includes the notion of modules over a ring, among many many others. I don't see a similar restriction for the Verdier localization.

Verdier localization uses the (standard for ''localization'') idea of a multiplicative set S of maps which are formally inverted by a functor Q from a category T to a new category denoted T/S. Hartshorne's Residues and Duality is a reference for this. (BTW, where does the assumption that the pullback of a multiplicative map is multiplicative come from?)

Bousfield localization is stated in several places (such as the Krause reference above) as a Verdier localization composed with a right adjoint for Q, which I understand to mean a functorial way of choosing objects in the isomorphism classes, and maps in the multiplicative subsets of each Hom(A,B). It is also stated in the generality of model categories as needing three distinguished collections of morphisms: namely quasi-isomorphisms and (co)fibrations. What bothers me more is the definition as given in Krause: an exact functor L from a triangulated category T to itself for which there exists a natural transformation η:Id-->L which commutes with L (ηL=Lη) and for which ηL is invertible. As a second, smaller, question, what is encoded by the commutative condition (what would be lost without it?)? I can come up with contrived examples (using the automorphisms of the objects LX) of course, but in what precise way does η really just encode L as a natural transformation?

Answer 1

One point to make is that the machinery of Bousfield localization is applicable to wider contexts, like model categories of spaces, simplicial rings, commutative ring spectra, et cetera, et cetera, and so it doesn't implicitly depend on a "stable" situation like you have in a triangulated category.

For example, you can use it to construct rationalizations and p-localizations of nilpotent spaces (as well as other unstable localizations), or functorially construct "Postnikov"-type decompositions in other situations by localizing with respect to maps that are highly connected.

Answer 2

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 Bousfield localization arises as the composition of a quotient functor and a right adjoint to the quotient functor. One can view the point of this as being that by general nonsense the right adjoint to a Verdier quotient is always fully faithful and so one can perform the localization by taking an appropriate subcategory of the category in which one started.

Indeed suppose we have a fully faithful exact functor of triangulated categories R -> S where lets say (just to keep everything easy) S is compactly generated, so in particular has all small products and coproducts, and R is localizing then the following statements are equivalent (this works more generally):
i) R -> S admits a right adjoint;
ii) The quotient functor S -> S/R admits a right adjoint

and in this case the right adjoint S/R -> S identifies the Verdier quotient with the full subcategory R^{\perp} of R-local objects. We also have that the right adjoint S -> R identifies R with the Verdier quotient S/R^{\perp}.

By composing these adjoint pairs we get two functors S -> S one for each adjoint pair which are the acyclization and localization functors. These "project" objects of S onto their parts in R and S/R respectively (in the sense that our pair of adjoints define functorial triangles for objects of S presenting them as extensions of objects in S/R by objects of R and we project onto the corresponding object in this triangle) and this localization is the Bousfield localization with respect to the class of maps whose cone lies in R. Every Bousfield localization arises in this way which is the content of Lemma 3.1 in the paper of Benson-Iyengar-Krause.
The conditions on the endofunctor L in Lemma 3.1 boil down to the fact that they are necessary for L to be idempotent and to arise as the monad associated to the unit of an adjunction.

If you wish I can provide references/more details later when I have a bit more time - essentially the point is that in a lot of situations we get Bousfield localizations from our quotient (for instance if R satisfies Brown representability and is localizing, or if it is localizing the quotient is essentially small and S satisfies Brown rep) and whether one uses the localization sequence (i.e. the pair of adjoints) or the localization and acyclization functors is a matter of which is most convenient and personal choice - the formalism tells you that they are equivalent.

Answer 3

The conceptual home of all these notions is best understood by realizing that whenever we have a category with weak equivalences it is best viewed as an (oo,1)-category: a category which has an oo-groupoid (aka Kan complex) of morphisms between each pair of objects.

An ordinary localization of a category at a collection of morphisms is a geometric embedding: a faithful inclusion functor that has a left exact left adjoint.

This abstract-nonsense statement has -- and that's the advantage of abstract nonsense -- a straightforward generalization to (oo,1)-categories: a localization of an (oo,1)-category is just a faithful inclusion into it that has a left exact left adjoint -- only that now all these terms are interpreted in the corresponding higher category theory context. For instance adjoint functor now means adjoint (oo,1)-functor and so on.

So this tells us what localization should be conceptually. To actually do something with it we usually pick concrete presentations of these higher structures by model categories.

Under this presentation, the abstractly-defined localization of (oo,1)-categories is presented by the Bousfield localization of the presented model categories.

Answer 4

You mention the idea of formally inverting a set S of maps in a category C to obtain a localization C → D, which is initial among functors from C which send the maps of S to isomorphisms/equivalences. But usually you're not just interested in working with plain old categories and functors, but categories with some extra structure or properties and functors which preserve these. This will affect your notion of localization. (Left) Bousfield localization is what you get in the case of model categories and left Quillen functors, but even better is to think of presentable (∞,1)-categories and functors which are left adjoints (equivalently, preserve all colimits).