On the abelianization of derived categories and a negative solution to Rosicky's problem

Monadic forgetful functors and (non-)presentability for $C^*$- and $W^*$-algebras

We prove that the forgetful functors from the categories of $C^*$- and $W^*$-algebras to Banach $*$-algebras, Banach algebras or Banach spaces are all monadic, answering a question of J.Rosický, and that the categories of unital (commutative) $C^*$-algebras are not locally-isometry $\aleph_0$-generated either as plain or as metric-enriched categories, answering a question of I. Di Liberti and Rosický. We also prove a number of negative presentability results for the category of von Neumann algebras: not only is that category not locally presentable, but in fact its only presentable objects are…

从阿贝尔化到切范畴

From Abelianization to Tangent Categories

本文证明群范畴在阿贝尔化诱导下构成切范畴，通过引入线性指派推广该构造，证明有限余完备正则幺元范畴均为切范畴，并提供了多个新示例。

An embedding theorem for tangent categories

Tangent categories were introduced by Rosicky as a categorical setting for differential structures in algebra and geometry; in recent work of Cockett, Crutwell and others, they have also been applied to the study of differential structure in computer science. In this paper, we prove that every tangent category admits an embedding into a representable tangent category---one whose tangent structure is given by exponentiating by a free-standing tangent vector, as in, for example, any model of Kock and Lawvere's synthetic differential geometry. The key step in our proof uses a coherence theorem fo…

A representability theorem for some huge abelian categories

We define quasi--locally presentable categories as big unions of coreflective subcategories which are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a quasi--locally presentable category taking values in abelian groups. We show that the abelianization of a well generated triangulated category is quasi--locally presentable and we obtain a new proof of Brown representability theorem. Examples of functors which are not representable are also given.

微分束作为自由模的函子

Differential bundles as functors from free modules

本文提出微分束与结构类别之间的等价关系，并通过函子构造方法证明其等价性。

切触范畴中的嘉当演算

Cartan calculus in tangent categories

研究切触范畴中构建嘉当演算所需结构，核心是由交换环对象\(R\)进行标量乘法，使切丛成\(R -\)模。在此基础上向量场李代数满足相关法则，各对象有李 - 莱因哈特形式的嘉当演算，还给出多种示例范畴。