平衡范畴论

On coslices and commas of locally finitely presentable categories

We give an explicit description of the generator of finitely presented objects of the coslice of a locally finitely presentable category under a given object, as consisting of all pushouts of finitely presented maps under this object. Then we prove that the comma category under the direct image part of a morphism of locally finitely presentable category is still locally finitely presentable, and we give again an explicit description of its generator of finitely presented objects. We finally deduce that 2-category $\LFP$ has comma objects computed in $\Cat$.

平衡范畴论II

Balanced Category Theory II

在第一部分中，我们进一步研究了强平衡分解范畴C [Pisani，2008]中的范畴论，C是一个具有两个非稳定分解系统的完备范畴，使得X \to 1在M中当且仅当它在M '中。特别是一些方面有关的“内部”（合作）限制和柯西完整性被认为是。在第二部分中，我们认为拓扑的某些方面也可以有效地综合在一个（弱）平衡分解范畴T中，其对象应该被认为是可能的“无穷小”和适当的“正则”拓扑空间。在C中，类M和M'扮演离散纤维化和光纤维化的角色，而在T中，它们扮演局部同胚和完美映射的角色，因此X \to 1在M中（分别是）。M '）当且仅当它是一个离散的（resp.紧凑）空间。因此，人们在主体之间获得了一种直接的抽象联系，具有互利性。例如，根据C中的（E，M）和（E '，M'）作为x：1 \to X的第二因子获得的切片投影X/x \to X和共切片投影x \X \to X在T中对应于x \in X的“无穷小”邻域和x的闭包。此外，开闭互补（推广到互反稳定性）成为内部处理的关键工具，在一个连贯的方式，一些范畴概念（如（共）限制的预层），这是经典的对偶关系。

关于群中交叉模范畴的非平衡性

On the non-balanced property of the category of crossed modules in groups

一个代数范畴$\mathcal{C}$称为平衡的，如果$\mathcal{C}$的任一对象的上三重上同调在内射系数模上的正维数中为零.平衡和非平衡范畴的重要例子出现在文献中。本文证明了群中的交叉模范畴是非平衡的。

The balanced tensor product of module categories

The balanced tensor product M (x)_A N of two modules over an algebra A is the vector space corepresenting A-balanced bilinear maps out of the product M x N. The balanced tensor product M [x]_C N of two module categories over a monoidal linear category C is the linear category corepresenting C-balanced right-exact bilinear functors out of the product category M x N. We show that the balanced tensor product can be realized as a category of bimodule objects in C, provided the monoidal linear category is finite and rigid.

幺半范畴的Morita $(\infty,2)$-范畴作为$2$-复合单纯集

The Morita $(\infty,2)$-category of a monoidal category as a $2$-complicial set

本文为满足极简条件的幺半范畴构造了Morita $(\infty,2)$-范畴，将其实现为$3$-余骨架的$2$-复合单纯集，并明确编码了幺半群、双模及其映射和复合结构。

Balanced pairs and recollements in extriangulated categories with negative first extensions

A notion of balanced pairs in an extriangulated category with a negative first extension is defined in this article. We prove that there exists a bijective correspondence between balanced pairs and proper classes $ξ$ with enough $ξ$-projectives and enough $ξ$-injectives. It can be regarded as a simultaneous generalization of Fu-Hu-Zhang-Zhu and Wang-Li-Huang. Besides, we show that if $(\mathcal A ,\mathcal B,\mathcal C)$ is a recollement of extriangulated categories, then balanced pairs in $\mathcal B$ can induce balanced pairs in $\mathcal A$ and $\mathcal C$ under natural assumptions. As a a…