In a 2005 paper, Casacuberta, Scevenels and Smith construct a homotopy idempotent functor $E$ on the category of simplicial sets with the property that whether it can be expressed as localization with respect to a map $f$ is independent of the ZFC axioms. We show that this construction can be carried out in homotopy type theory. More precisely, we give a general method of associating to a suitable (possibly large) family of maps, a reflective subuniverse of any universe $\mathcal{U}$. When specialized to an appropriate family, this produces a localization which when interpreted in the $\infty$-topos of spaces agrees with the localization corresponding to $E$. Our approach generalizes the approach of [CSS] in two ways. First, by working in homotopy type theory, our construction can be interpreted in any $\infty$-topos. Second, while the local objects produced by [CSS] are always 1-types, our construction can produce $n$-types, for any $n$. This is new, even in the $\infty$-topos of spaces. In addition, by making use of universes, our proof is very direct. Along the way, we prove many results about "small" types that are of independent interest. As an application, we give a new proof that separated localizations exist. We also give results that say when a localization with respect to a family of maps can be presented as localization with respect to a single map, and show that the simplicial model satisfies a strong form of the axiom of choice which implies that sets cover and that the law of excluded middle holds.
翻译:Casacuberta、Scevenels和Smith在2005年的一篇论文中,Casacuberta、Scvenels和Smith在简单化组的类别上构建了一个同质化的一元真能真能真能真能真能真能真能真能真能真能真能真能真能的真能真能真能真能的真能真能。当专门为一个合适的家族而专门设计时,这产生了一种本地化,当用美元表示地图的本地化时,当用美元表示,当用美元表示时,美元表示的本地化与美元相对能表示的本地化的同一类型。我们的方法概括了[CS]的同一类型理论,我们用任何美元表示的本地化的本地化方法可以解释我们的构造,当我们用新式的本地化工具存在时, 我们的本地化工具是一个新的类型, 我们的本地化工具也可以产生美元表示的、 美元类型表示的直方能证明。