Slices of groupoids are group-like

逆半群、étale群胚及其C*-代数的范畴论方法

A categorical approach to inverse semigroups, étale groupoids, and their C*-algebras

本文通过范畴论框架统一逆半群作用、étale群胚及其C*-代数，构造谱作用、变换群胚与C*-代数函子，并建立切片作用函子的右伴随关系。

Homotopy linear algebra

By homotopy linear algebra we mean the study of linear functors between slices of the $\infty$-category of $\infty$-groupoids, subject to certain finiteness conditions. After some standard definitions and results, we assemble said slices into $\infty$-categories to model the duality between vector spaces and profinite-dimensional vector spaces, and set up a global notion of homotopy cardinality à la Baez-Hoffnung-Walker compatible with this duality. We needed these results to support our work on incidence algebras and Möbius inversion over $\infty$-groupoids; we hope that they can also be of i…

A semi-model structure for Grothendieck weak 3-groupoids

In this paper we apply some tools developed in our previous work on Grothendieck $\infty$-groupoids to the finite-dimensional case of weak 3-groupoids. We obtain a semi-model structure on the category of Grothendieck 3-groupoids of suitable type, thanks to the construction of an endofunctor $\mathbb{P}$ that has enough structure to behave like a path object. This makes use of a recognition principle we prove here that characterizes globular theories whose models can be viewed as Grothendieck $n$-groupoids (for $0\leq n \leq \infty$). Finally, we prove that the obstruction in arbitrary dimensio…

A Categorification of Hall Algebras

In recent years, there has been great interest in the study of categorification, specifically as it applies to the theory of quantum groups. In this thesis, we would like to provide a new approach to this problem by looking at Hall algebras. It is know, due to Ringel, that a Hall algebra is isomorphic to a certain quantum group. It is our goal to describe a categorification of Hall algebras as a way of doing so for their related quantum groups. To do this, we will take the following steps. First, we describe a new perspective on the structure theory of Hall algebras. This view solves, in a uni…

On Group-Like Magmoids

A magmoid is a non-empty set with a partial binary operation; group-like magmoids generalize group-like magmas such as semigroups, monoids and groups. In this article, we first consider the many ways in which the notions of associative multiplication, identities and inverses can be generalized when the total binary operation is replaced by a partial binary operation. Poloids, groupoids, skew-poloids, skew-groupoids, prepoloids, pregroupoids, skew-prepoloids and skew-pregroupoids are then defined in terms of generalized associativity, generalized identities and generalized inverses. Some basic …

The Lorentz Boost-Link Is Not Unique. Relative velocity as a morphism in a connected groupoid category of null objects

The isometry-link problem is to determine all isometry transformations among given pair of vectors with the condition that if these initial and final vectors coincide, the transformation-link must be identity on entire vector space. In the first part of this essay we provide the complete solution for the link problem for arbitrary isometry, for any dimension and arbitrary signature of the invertible metric tensor. We apply these considerations for pure Lorentz transformations, for the Lorentz boost, parameterized by relative velocity, and we are showing that the isometric pure Lorentz transfor…