Cyclic structures for simplicial objects from comonads

Hochschild homology, lax codescent, and duplicial structure

We study the duplicial objects of Dwyer and Kan, which generalize the cyclic objects of Connes. We describe duplicial objects in terms of the decalage comonads, and we give a conceptual account of the construction of duplicial objects due to Bohm and Stefan. This is done in terms of a 2-categorical generalization of Hochschild homology. We also study duplicial structure on nerves of categories, bicategories, and monoidal categories.

The cyclic bar construction on A_\infty H-spaces

We set up a general framework for enriching a subcategory of the category of noncommutative sets over a category C using products of the objects of a non-Σoperad P in \C. By viewing the simplicial category as a subcategory of the category of noncommutative sets in two different ways, we obtain two generalizations of simplicial objects. For the operad given by the Stasheff associahedra we obtain a model for the 2-sided bar construction in the first case and the cyclic bar and cobar construction in the second case. Using either the associahedra or the cyclohedra in place of the geometric simplic…

循环Segal空间

Cyclic Segal Spaces

本文为代数拓扑领域综述，回顾循环范畴$\Lambda$的概念进路与交叉单纯群描述，给出循环集模型结构的新证明，梳理循环空间广义Reedy结构，还引入循环Segal空间与循环2-Segal空间的模型结构。

循环无穷算畴的模型

Models for cyclic infinity operads

本文在循环树状集与空间上构建了循环拟算畴与完备循环树状 Segal 空间的模型结构，证明其与单纯循环算畴 Quillen 等价，并推广至平面情形以支持 Walde 纲领。

范畴论笔记：以基础数学为例

Notes on Category Theory with examples from basic mathematics

该笔记以线性代数为前提，通过基础数学实例从零讲解范畴论，重点阐述米田引理、范畴与有向多重图的关系，并平等处理单子与余单子，现已成为专著。

Higher Segal spaces I

This is the first paper in a series on new higher categorical structures called higher Segal spaces. For every d > 0, we introduce the notion of a d-Segal space which is a simplicial space satisfying locality conditions related to triangulations of cyclic polytopes of dimension d. In the case d=1, we recover Rezk's theory of Segal spaces. The present paper focuses on 2-Segal spaces. The starting point of the theory is the observation that Hall algebras, as previously studied, are only the shadow of a much richer structure governed by a system of higher coherences captured in the datum of a 2-S…

2-Segal objects and algebras in spans

We define a category parameterizing Calabi-Yau algebra objects in an infinity category of spans. Using this category, we prove that there are equivalences of infinity categories relating, firstly: 2-Segal simplicial objects in C to algebra objects in Span(C); and secondly: 2-Segal cyclic objects in C to Calabi-Yau algebra objects in Span(C).

高阶Segal空间与偏群

Higher Segal spaces and partial groups

本研究探讨d-Segal条件与partial groups的相互作用，定义partial群的度并开发基于其作用离散几何的度计算工具，在穿孔外尔群等实例中验证了度与半单李代数阿贝尔子代数最大维数的关联。

A graphical category for higher modular operads

We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial presheaves for a certain category of undirected graphs. This new category of undirected graphs, denoted $\mathbf{U}$, plays a similar role for modular operads that the dendroidal category $Ω$ plays for operads. We carefully study properties of $\mathbf{U}$, including the existence of certain factorization systems. Related structures, such as cyclic operads and stab…