On direct product, semidirect product of groupoids and partial actions

作为对称单纯集的偏群

Partial groups as symmetric simplicial sets

本文将偏群刻画为对称单纯集的子范畴，利用显式反射计算偏群范畴的余极限，并引入了涵盖广群与偏群的偏广群概念。

Partial Category Actions on Sets and Topological Spaces

We introduce (continuous) partial category actions on sets (topological spaces) and show that each such action admits a universal globalization. Thereby, we obtain a simultaneous generalization of corresponding results for groups, by Kellendonk and Lawson, and for monoids, by Megrelishvili and Schroder. We apply this result to the special case of partial groupoid actions where we obtain a sharpening of a result by Gilbert, concerning ordered groupoids, in the sense that mediating functions between universal globalizations always are injective.

可嵌入的偏群

Embeddable partial groups

本文研究了偏群的嵌入条件，提出偏群oid嵌入群oid的判定方法，并探讨了非嵌入性例子的性质。

In Ehresmann's footsteps: from Group Geometries to Groupoid Geometries

For a smooth (locally trivial) principal bundle in Ehresmann's sense, the relation between the commuting vertical and horizontal actions of the structural Lie group and the structural Lie groupoid (isomorphisms between vertical fibers) is regarded as a special case of a symmetrical concept of conjugation between "principal" Lie groupoid actions, allowing possibly non-locally trivial bundles. A diagrammatic description of this concept via a symmetric "butterfly diagram" allows its "internalization" in a wide class of categories (used by "working mathematicians") whenever they are endowed with t…

置换2-群I：结构与可裂性

Permutation 2-groups I: structure and splitness

该研究引入对称群与任意2-群的wreath 2-积，证明有限型群胚的置换2-群等价于该类2-积的乘积，计算其同伦不变量并明确不可裂性的唯一来源，还给出多个置换2-群的显式计算结果。

Generalized Inversions and the Construction of Musical Group and Groupoid Actions

Transformational music theory is a recent field in music theory which studies the possible transformations between musical objects, such as chords. In the framework of the theory initiated by David Lewin, the set of all transformations forms a group which acts simply transitively on the set of musical objects. In particular, neo-Riemannian theory focuses on the action of the T/I group or the PLR group, which are both isomorphic to the dihedral group D24, on the set of major and minor triads. It has been shown recently that generalized neo-Riemannian groups of transformations can be built as gr…

Étale inverse semigroupoids - the fundamentals

In this article we will study semigroupoids, and more specifically inverse semigroupoids. These are a common generalization to both inverse semigroups and groupoids, and provide a natural language on which several types of dynamical structures may be described. Moreover, this theory allows us to precisely compare and simultaneously generalize aspects of both the theories of inverse semigroups and groupoids. We begin by comparing and settling the differences between two notions of semigroupoids which appear in the literature (one by Tilson and another by Exel). We specialize this study to inver…

C*-algebras from partial isometric representations of LCM semigroups

We give a new construction of a C*-algebra from a cancellative semigroup $P$ via partial isometric representations, generalising the construction from the second named author's thesis. We then study our construction in detail for the special case when $P$ is an LCM semigroup. In this case we realize our algebras as inverse semigroup algebras and groupoid algebras, and apply our construction to free semigroups and Zappa-Szép products associated to self-similar groups.

逆范畴的部分作用及其代数

Partial actions of inverse categories and their algebras

本文定义逆范畴在偏序集上的纤维与对称作用，研究 Bernoulli 作用、半直积、Szendrei 扩张及扩张的范畴等价性，并证明有限情形卷积代数的 Morita 等价。