Any modality in homotopy type theory gives rise to an orthogonal factorization system of which the left class is stable under pullbacks. We show that there is a second orthogonal factorization system associated with any modality, of which the left class is the class of ○-equivalences and the right class is the class of ○-étale maps. This factorization system is called the modal reflective factorization system of a modality, and we give a precise characterization of the orthogonal factorization systems that arise as the modal reflective factorization system of a modality. In the special case of the n-truncation, the modal reflective factorization system has a simple description: we show that the n-étale maps are the maps that are right orthogonal to the map $${\rm{1}} \to {\rm{ }}{{\rm{S}}^{n + 1}}$$ . We use the ○-étale maps to prove a modal descent theorem: a map with modal fibers into ○X is the same thing as a ○-étale map into a type X. We conclude with an application to real-cohesive homotopy type theory and remark how ○-étale maps relate to the formally etale maps from algebraic geometry.

中文翻译：

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模态下降

同伦类型理论中的任何模态都会产生一个正交分解系统，其中左类在回调下是稳定的。我们表明，存在与任何模态相关的第二个正交分解系统，其中左类是 ○-等价类，右类是 ○-étale 映射类。这种分解系统称为模态反射分解系统，我们给出了作为模态反射分解系统出现的正交分解系统的精确表征。在特殊情况下n-truncation，模态反射分解系统有一个简单的描述：我们证明n-étale 地图是与地图正交的地图$${\rm{1}} \to {\rm{ }}{{\rm{S}}^{n + 1}}$$. 我们使用 ○-étale 映射来证明模态下降定理：将模态纤维映射到 ○X与 ○-étale 映射到类型是一样的X. 我们以实凝聚同伦类型理论的应用作为结束，并说明 ○-étale 映射如何与代数几何中的正式 etale 映射相关。