Reasonable triangulated categories have filtered enhancements

Homotopy categories and idempotent completeness, weight structures and weight complex functors

This article provides some basic results on weight structures, weight complex functors and homotopy categories. We prove that the full subcategories K(A)^{w < n}, K(A)^{w > n}, K(A)^- and K(A)^+ (of objects isomorphic to suitably bounded complexes) of the homotopy category K(A) of an additive category A are idempotent complete, which confirms that (K(A)^{w <= 0}, K(A)^{w >= 0}) is a weight structure on K(A). We discuss weight complex functors and provide full details of an argument sketched by M. Bondarko, which shows that if w is a bounded weight structure on a triangulated category T that ha…

Filtered objects in extriangulated categories

Let $R$ be an artin ring and $Θ=\{Θ(1),Θ(2),\cdots,Θ(n)\}$ be a family of objects in an artin extriangulated $R$-category $(\cal C,\mathbb{E},\mathfrak{s})$ such that $\mathbb{E}(Θ(j),Θ(i))=0$ for all $j\geq i$. In this paper, we show that the class $\cal P(Θ)$ of the $Θ$-projective objects is a precovering class and the class $\cal I(Θ)$ of the $Θ$-injective objects is a preenveloping one in $\cal C$. Furthermore, if $\cal C$ has enough projectives and enough injectives, we show that the subcategory $\cal F(Θ)$ of $Θ$-filtered objects is functorially finite in $\cal C$. As an appliacation, th…

Bondarenko 范畴的三角化结构

Triangulated structure for Bondarenko's Categories

本文针对 Bondarenko 范畴非阿贝尔但与 gentle 代数导出范畴密切相关的问题，在其商范畴上构造了三角范畴结构，为运用三角范畴技术研究偏序集表示提供了新工具。

三角范畴之间的可定义函子

Definable functors between triangulated categories

该文系统发展紧生成三角范畴间的可定义函子理论，刻画凝聚函子并给出可定义函子的等价条件，还将结果应用于Ziegler谱的函子性研究。

Hochschild 上同调与三角范畴的扩张

Hochschild cohomology and extensions of triangulated categories

本文定义增强三角范畴的一阶范畴形变，证明其与 HH² 类一一对应，并用于解释 dg 代数弯曲形变及构造经典导出范畴的范畴解消。

Derived categories for Grothendieck categories of enriched functors

The derived category $D[C,V]$ of the Grothendieck category of enriched functors $[C,V]$, where $V$ is a closed symmetric monoidal Grothendieck category and $C$ is a small $V$-category, is studied. We prove that if the derived category $D(V)$ of $V$ is a compactly generated triangulated category with certain reasonable assumptions on compact generators or $K$-injective resolutions, then the derived category $D[C,V]$ is also compactly generated triangulated. Moreover, an explicit description of these generators is given.

关于反常层等变导出范畴

On the Equivariant Derived Category of Perverse Sheaves

本文将 Beilinson 实现形式体系推广到伪函子与伪自然框架，证明代数群作用下反常层导出范畴与构造导出范畴等价的等变版本，并验证其他非 D-模结论的等变类比。

Calabi–Yau 范畴中的计数，及其在 Hall 代数和纽结多项式中的应用

Counting in Calabi--Yau categories, with applications to Hall algebras and knot polynomials

本文为 evenCY 范畴定义同伦基数，并据此构造 oddCY dg-范畴的内在 Hall 代数、证明其与 Drinfeld double 的关系，同时证明 Legendrian 纽结 ruling polynomial 的增广范畴公式猜想。