Renormalization groupoids in algebraic topology

拓扑基本广群. I

Topological fundamental groupoid. I

本文证明局部道路连通且半局部单连通空间的基本广群可被赋予自然拓扑成为拓扑广群，论证了假设的必要性与最小性，并推翻了其非etale及非变换广群的传统信念。

The Fundamental Infinity-Groupoid of a Parametrized Family

Given an infinity-category C, one can naturally construct an infinity-category Fam(C) of families of objects in C indexed by infinity-groupoids. An ordinary categorical version of this construction was used by Borceux and Janelidze in the study of generalized covering maps in categorical Galois theory. In this paper, we develop the homotopy theory of such "parametrized families" as generalization of the classical homotopy theory of spaces. In particular, we study homotopy-theoretical constructions that arise from the fundamental infinity-groupoids of families in an infinity-category. In the sa…

Morita theory for Hopf algebroids and presheaves of groupoids

Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff, the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of presheaves of groupoids with respect to a Grothendieck topology on Aff give rise to equivalences of categories of sheaves in that topology. We then show using faithfully flat descent that an internal equivalenc…

A bigroupoid's topology (or, Topologising the homotopy bigroupoid of a space)

The fundamental bigroupoid of a topological space is one way of capturing its homotopy 2-type. When the space is semilocally 2-connected, one can lift the construction to a bigroupoid internal to the category of topological spaces, as Brown and Danesh-Naruie lifted the fundamental groupoid to a topological groupoid. For locally relatively contractible spaces the resulting topological bigroupoid is locally trivial in a way analogous to the case of the topologised fundamental groupoid.

无开集的代数拓扑：极限空间中同伦理论的一种网方法

Algebraic Topology Without Open Sets: A Net Approach to Homotopy Theory in Limit Spaces

本研究基于网理论探讨收敛空间，简化连续收敛描述并应用于同伦理论，在极限空间中定义基本广群并证明了 Seifert-van Kampen 定理的广群版本。

The $\infty$-groupoid generated by an arbitrary topological $λ$-model

The lambda calculus is a universal programming language. It can represent the computable functions, and such offers a formal counterpart to the point of view of functions as rules. Terms represent functions and this allows for the application of a term/function to any other term/function, including itself. The calculus can be seen as a formal theory with certain pre-established axioms and inference rules, which can be interpreted by models. Dana Scott proposed the first non-trivial model of the extensional lambda calculus, known as $ D_\infty$, to represent the $λ$-terms as the typical functio…