We focus on the transfer of some known orthogonal factorization systems from $$\mathsf {Cat}$$ to the 2-category $${\mathsf {Fib}}(B)$$ of fibrations over a fixed base category B: the internal version of the comprehensive factorization, and the factorization systems given by (sequence of coidentifiers, discrete morphism) and (sequence of coinverters, conservative morphism) respectively. For the class of fibrewise opfibrations in $${\mathsf {Fib}}(B)$$ , the construction of the latter two simplify to a single coidentifier (respectively coinverter) followed by an internal discrete opfibration (resp. fibrewise opfibration in groupoids). We show how these results follow from their analogues in $$\mathsf {Cat}$$ , providing suitable conditions on a 2-category $${\mathcal {C}}$$ , that allow the transfer of the construction of coinverters and coidentifiers from $${\mathcal {C}}$$ to $${\mathsf {Fib}}_{{\mathcal {C}}}(B)$$ .

中文翻译：

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Fib(B) 中的离散和保守因式分解

我们专注于将一些已知的正交分解系统从 $$\mathsf {Cat}$$ 转移到固定基类别 B 上纤维化的 2-类别 $${\mathsf {Fib}}(B)$$：综合分解的内部版本，以及分别由 (coidentifiers 序列，离散态射) 和 (coinverters 序列，保守态射) 给出的分解系统。对于 $${\mathsf {Fib}}(B)$$ 中的纤维化光纤类，后两者的构造简化为单个 coidentifier（分别为 coinverter），然后是内部离散 opfibration（分别为 groupoids 中的 Fiberwise opfibration ）。我们展示了这些结果如何从 $$\mathsf {Cat}$$ 中的类似物得出，在 2 类别 $${\mathcal {C}}$$ 上提供合适的条件，