What would cohomological localization be good for?

Question

An open problem in algebraic topology is whether arbitrary cohomological localizations of simplicial sets (or, equivalently, topological spaces) can be proven to exist in ZFC. It's provable in ZFC that arbitrary homological localizations exist, but for cohomological localizations I believe the best result is that they follow from the existence of a proper class of supercompact cardinals. (In particular, they follow from Vopenka's principle, but according to BCMR "Definable orthogonality classes in accessible categories are small", a proper class of supercompacts is enough since the class of cohomology equivalences is $\Sigma_2$.)

I would like to know how important this open problem is. Are there other open problems that could be solved if cohomological localizations existed? Have any important theorems been proven using cohomological localizations assuming the appropriate large-cardinal hypotheses for their existence? Are there particular cohomology theories that we would like to localize at for particular applications, but we don't know how to do so without large cardinals? Or is the open problem of mainly theoretical interest (e.g. perhaps most cohomology theories arising in practice can be localized at using less general techniques)?

Answer 1

I do not know any open problem related to this, but there is a situation where the theory would look quite different if this problem was solved because it is built on such cohomological localizations and everything turns out to be OK because the localization actually exists for a big enough class of objects of interest.

In Bertrand Toën's paper Champs affines. Selecta Math. (N.S.) 12 (2006), no. 1, 39–135 (arXiv), he wants to define a functor wished by Grothendieck in Pursuing stacks, the schematization of a space (you want, for each ring $k$, a kind of algebro-geometric gadget associated to a space $X$, so that the quasi-coherent sheaves on the gadget are the locally contant sheaves of $k$-modules on $X$). If you look at Theorem 2.2.9 (and unpack what it means in Definition 2.2.8), Toën characterizes the main objects of study of his paper (affine stacks) as representable sheaves that are local objects with respect to cohomology with coefficients in the multiplicative group. An important operator in the paper is then the affinization of a sheaf, but he can only construct it in a bigger universe because we cannot prove existence of left Bousfield localization with respect to a cohomology. The fact that this is a purely set theoretic problem is spelled out in Proposition 2.3.2. Toën then proves that this affinization exists for constant sheaves (Cor. 2.3.3) but there is no proof in general. This theory of affine stacks give a nice extension of rational homotopy theory, and there is even a version with integral coefficients which gives schematized version of a theorem of Mandell (see Cor. 5.5 here). Having affinization of more general sheaves might provide twisted forms of Toën's theorems, at the very least.