Title: Spectral sequences via linear presheaves Show Chinese title

Title: 通过线性预层的谱序列

Abstract: We study homotopy theory of the category of spectral sequences with respect to the class of weak equivalences given by maps which are quasi-isomorphisms on a fixed page. We introduce the category of extended spectral sequences and show that this is bicomplete by analysis of a certain linear presheaf category modelled on discs. We endow the category of extended spectral sequences with various model category structures, restricting to give the almost Brown category structures on spectral sequences of our earlier work. One of these has the property that spectral sequences is a homotopically full subcategory. By results of Meier, this exhibits the category of spectral sequences as a fibrant object in the Barwick-Kan model structure on relative categories, that is, it gives a model for an infinity category of spectral sequences. We also use the presheaf approach to define two d\'ecalage functors on spectral sequences, left and right adjoint to a shift functor, thereby clarifying prior use of the term d\'ecalage in connection with spectral sequences. Show Chinese abstract

Abstract: 我们研究在给定页面上为拟同构的映射所定义的弱等价类下的谱序列范畴的同伦理论。 我们引入扩展谱序列的范畴，并通过分析一种基于圆盘的线性预层范畴来证明该范畴是双完备的。 我们将各种模型范畴结构赋予扩展谱序列的范畴，并限制以在我们早期工作中提出的谱序列的几乎布朗范畴结构。 其中有一个结构具有谱序列是一个同伦满子范畴的性质。 根据Meier的结果，这展示了谱序列的范畴作为相对范畴上的Barwick-Kan模型结构中的一个纤维对象，即为谱序列的无穷范畴提供了一个模型。 我们还使用预层方法在谱序列上定义了两个decalage函子，分别作为移位函子的左伴随和右伴随，从而澄清了在与谱序列相关联的情况下对decalage一词的先前使用。