Given a well-generated tensor triangulated category , let the Bousfield class of an object , denoted , be the class . It was proven by Ohkawa that if is the stable homotopy category, , then the collection of all Bousfield classes is a set of cardinality at most . It was proven more generally by Iyengar and Krause that such a collection is always a set and not a proper class when is well-generated. This set has a partial ordering on it and the structure of a complete lattice. This lattice is called the Bousfield lattice of , denoted .
Note that, perhaps by some abuse of notation, , the collection of localizing subcategories, since every Bousfield class is a localizing subcategory. However, the question of whether or not every localizing subcategory is a Bousfield class is still open in general.
Within , there is a distributive lattice that is precisely all Bousfield idempotent objects of . That is, is precisely the objects such that . Because is an affine quantale, it follows that is a frame. In particular, is a distributive lattice, so by the work of Stone, it corresponds to a coherent space. For more on this, see Stone duality.