nLab Bousfield lattice

Definition

Given a well-generated? tensor triangulated category $(T,\otimes, 1)$, let the Bousfield class of an object $X$, denoted $\langle X\rangle$, be the class $\{ Y\in obj(T): X\otimes Y=0\}$. It was proven by Ohkawa that if $T$ is the stable homotopy category, $(\mathcal{S},\wedge,S)$, then the collection of all Bousfield classes is a set of cardinality at most $\beth_2$. It was proven more generally by Iyengar and Krause that such a collection is always a set and not a proper class when $T$ 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 $T$, denoted $B_T$.

Note that, perhaps by some abuse of notation, $B_T\subseteq\mathbf{Loc}(T)$, 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.

The Distributive Lattice of the Bousfield Lattice

Within $B_T$, there is a distributive lattice $DL_T$ that is precisely all Bousfield idempotent objects of $T$. That is, $DL_T$ is precisely the objects $X$ such that $\langle X\otimes X\rangle=\langle X\rangle$. Because $B_T$ is an affine quantale, it follows that $DL_T$ is a frame. In particular, $DL_T$ is a distributive lattice, so by the work of Stone, it corresponds to a coherent space. For more on this, see Stone duality.

References

• S. B. Iyengar and H. Krause, The Bousfield Lattice of a Triangulated Category and Stratification (arXiv:1105.1799)