Elements of Topological Algebra. III. The Closed Category of Filters

Decorated marked surfaces III: The derived category of a decorated marked surface

We study the Ginzburg dg algebra $Γ_\mathbf{T}$ associated to the quiver with potential arising from a triangulation $\mathbf{T}$ of a decorated marked surface $\mathbf{S}_\bigtriangleup$, in the sense of Qiu. We show that there is a canonical way to identify all finite dimensional derived categories $\mathcal{D}_{fd}(Γ_\mathbf{T})$, denoted by $\mathcal{D}_{fd}(\mathbf{S}_\bigtriangleup)$. As an application, we show that the spherical twist group $\operatorname{ST}(\mathbf{S}_\bigtriangleup)$ associated to $\mathcal{D}_{fd}(\mathbf{S}_\bigtriangleup)$ acts faithfully on its space of stability…

Simple Skew Category Algebras Associated with Minimal Partially Defined Dynamical Systems

In this article, we continue our study of category dynamical systems, that is functors $s$ from a category $G$ to $\Top^{\op}$, and their corresponding skew category algebras. Suppose that the spaces $s(e)$, for $e \in \ob(G)$, are compact Hausdorff. We show that if (i) the skew category algebra is simple, then (ii) $G$ is inverse connected, (iii) $s$ is minimal and (iv) $s$ is faithful. We also show that if $G$ is a locally abelian groupoid, then (i) is equivalent to (ii), (iii) and (iv). Thereby, we generalize results by Öinert for skew group algebras to a large class of skew category algebr…

Filtered Frobenius algebras in monoidal categories

We develop filtered-graded techniques for algebras in monoidal categories with the main goal of establishing a categorical version of Bongale's 1967 result: A filtered deformation of a Frobenius algebra over a field is Frobenius as well. Towards the goal, we first construct a monoidal associated graded functor, building on prior works of Ardizzoni-Menini, of Galatius et al., and of Gwillian-Pavlov. Next, we produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form; this builds on work of Fuchs-Stigner. Thes…

Torsion pairs in categories of modules over a preadditive category

It is a result of Gabriel that hereditary torsion pairs in categories of modules are in bijection with certain filters of ideals of the base ring, called Gabriel filters or Gabriel topologies. A result of Jans shows that this bijection restricts to a correspondence between (Gabriel filters that are uniquely determined by) idempotent ideals and TTF triples. Over the years, these classical results have been extended in several different directions. In this paper we present a detailed and self-contained exposition of an extension of the above bijective correspondences to additive functor categori…

交换 Quantale 与局部化

Commutative Quantale and Localization

本文为 quantale 引入局部化构造，通过定义乘法滤子并证明类似“Spec R 是层”的定理，获得了贝尔纲定理及其代数版本等结果。

交换Hopf代数的光滑性

Smoothness of commutative Hopf algebras

本文研究了半单阿贝尔对称幺单范畴中交换Hopf代数的等变光滑性，证明了代数光滑性、余模代数光滑性与积态射单射性的等价条件，并分析了特征零与正特征下各类范畴中Hopf代数的光滑性质。