从格罗滕迪克上余纤维化到分解系统：一个形式2-单子论的阐述
From Grothendieck cofibrations to factorization systems: a formal 2-monadic account
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中文总结 AI 辅助
该论文以2-单子论为框架，通过2-范畴方法建立格罗滕迪克上余纤维化与分解系统的联系，证明不同类型上余纤维化对应特定2-单子代数，实现两类结构的统一刻画。
中文摘要 AI 辅助
格罗滕迪克上余纤维化描述了在基上变化的范畴中的传输，而分解系统将范畴的箭头组织为两个互补类。我们对从前述结构到后者的过渡给出了完全2-范畴的阐述。箭2-范畴上的全局逗号2-单子编码了格罗滕迪克传输，而平方2-单子编码了分解。我们证明分裂上余纤维化恰好是逗号2-单子的严格代数，包含其1-胞腔和2-胞腔，且正则分裂上余纤维化恰好是其正规伪代数。从逗号2-单子到平方2-单子的典范余 lax 态射随后将余笛卡尔传输转化为总范畴中箭头的余笛卡尔-垂直分解。在严格层面，这产生了由指定余笛卡尔箭头和垂直箭头构成的严格分解系统；在相干层面，这产生了正交分解系统，其左类由所有余笛卡尔箭头组成，右类由在基中被映为同构的箭头组成。我们还将保留相干平凡基作用的无限制全局伪代数与对应任意 cleavages 的固定基伪代数区分开，并记录了纤维化的对偶严格结果。这将经典的上余纤维化-分解相互作用在所有这些变体中置于单一的2-单子变换构造内，并将其直接关联到现有的纤维化和分解文献。
英文摘要
Grothendieck cofibrations describe transport in a category varying over a base, while factorization systems organize the arrows of a category into two complementary classes. We give a fully 2-categorical account of the passage from the former structure to the latter. The global comma 2-monad on the arrow 2-category encodes Grothendieck transport, whereas the squaring 2-monad encodes factorizations. We prove that split cofibrations are precisely the strict algebras for the comma 2-monad, including their 1-cells and 2-cells, and that normally cloven cofibrations are precisely its normal pseudoalgebras. A canonical colax morphism from the comma 2-monad to the squaring 2-monad then turns cocartesian transport into the cocartesian-vertical factorization of arrows in the total category. At the strict level, this yields the strict factorization system of designated cocartesian and vertical arrows; at the coherent level, it yields the orthogonal factorization system whose left class consists of all cocartesian arrows and whose right class consists of the arrows sent to isomorphisms in the base. We also separate unrestricted global pseudoalgebras, which retain a coherently trivial base action, from fixed-base pseudoalgebras, which correspond to arbitrary cleavages, and record the dual strict result for fibrations. This places the classical cofibration-factorization interaction, in all these variants, within a single change-of-2-monads construction and relates it directly to the existing fibrational and factorization literature.
