We use a simplicial product version of Quillen’s Theorem A to prove classical Waldhausen Additivity of \(wS_\bullet \), which says that the “subobject” and “quotient” functors of cofiber sequences induce a weak equivalence \(wS_\bullet {\mathcal {E}}({\mathcal {A}},{\mathcal {C}},{\mathcal {B}}) \rightarrow wS_\bullet {\mathcal {A}}\times wS_\bullet {\mathcal {B}}\). A consequence is Additivity for the Waldhausen K-theory spectrum of the associated split exact sequence, namely a stable equivalence of spectra \({\mathbf {K}}({\mathcal {A}}) \vee {\mathbf {K}}({\mathcal {B}}) \rightarrow {\mathbf {K}}({\mathcal {E}}({\mathcal {A}},{\mathcal {C}},{\mathcal {B}}))\). This paper is dedicated to transferring these proofs to the quasicategorical setting and developing Waldhausen quasicategories and their sequences. We also give sufficient conditions for a split exact sequence to be equivalent to a standard one. These conditions are always satisfied by stable quasicategories, so Waldhausen K-theory sends any split exact sequence of pointed stable quasicategories to a split cofiber sequence. Presentability is not needed. In an effort to make the article self-contained, we recall all the necessary results from the theory of quasicategories, and prove a few quasicategorical results that are not in the literature.

中文翻译：

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Waldhausen可加性：古典和准分类

我们使用Quillen定理A的简单乘积形式来证明\（wS_ \ bullet \）的经典Waldhausen可加性，它表示光纤纤维序列的“子对象”和“商”函子引起了弱等价\（wS_ \ bullet { mathcal {E}}（{\ mathcal {A}}，{\ mathcal {C}}，{\ mathcal {B}}）\ rightarrow wS_ \ bullet {\ mathcal {A}} \ times wS_ \ bullet {\ mathcal {B}} \）。结果是相关拆分精确序列的Waldhausen K-理论谱具有可加性，即谱\（{\ mathbf {K}}（{\ mathcal {A}}）\ vee {\ mathbf {K} }（{\ mathcal {B}}）\ rightarrow {\ mathbf {K}}（{\ mathcal {E}}（{\ mathcal {A}}，{\ mathcal {C}}，{\ mathcal {B} }））\）。本文致力于将这些证明转移到准分类环境中，并开发Waldhausen准分类及其顺序。我们还给出了足以使分割的精确序列等同于标准序列的充分条件。这些条件始终被稳定的准分类所满足，因此Waldhausen K-理论将尖的稳定的准分类的任何分割精确序列发送到分割的cofiber序列。无需展示。为了使文章变得独立，我们回顾了准分类理论的所有必要结果，并证明了一些文献中没有的准分类结果。