高等拓扑理论

On the algebraic K-theory of higher categories

We prove that Waldhausen K-theory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, K-theory spaces admit canonical (connective) deloopings, and the K-theory functor enjoys a universal property. Using this, we give new, higher categorical proofs of both the additivity and fibration theorems of Waldhausen. As applications of this technology, we study the algebraic K-theory of associative ring spectra and spectral Deligne-Mumford stacks.

Unifying Causal Inference and Reinforcement Learning using Higher-Order Category Theory

We present a unified formalism for structure discovery of causal models and predictive state representation (PSR) models in reinforcement learning (RL) using higher-order category theory. Specifically, we model structure discovery in both settings using simplicial objects, contravariant functors from the category of ordinal numbers into any category. Fragments of causal models that are equivalent under conditional independence -- defined as causal horns -- as well as subsequences of potential tests in a predictive state representation -- defined as predictive horns -- are both special cases of…

Yoneda Lemma for Elementary Higher Toposes

We prove the Yoneda lemma inside an elementary higher topos, generalizing the Yonda lemma for spaces.

有向algebraic topology与高维过渡系统

Directed algebraic topology and higher dimensional transition system

Cattani-Sassone的高维转移系统概念被解释为弱高维转移系统的局部可表示拓扑范畴的小正交类。特别地，与标号n-立方体相关联的高维跃迁系统原来是由一个n维跃迁生成的自由高维跃迁系统。作为这个构造的第一个应用，证明了高维转移系统范畴的局部化等价于标号对称预立方集范畴的局部可表示的反射满子范畴.第二个应用程序是米尔纳的演算的通信系统（CCS）：映射采取进程名称在CCS流是因式分解通过类别的高维过渡系统。该方法也适用于其他进程代数和拓扑模型的并发流以外。

在高拓扑论中近似纤维化

Approximate Fibrations in Higher Topos Theory

本文在更高拓扑论框架中研究近似纤维化，通过形状理论和拓扑论方法推广并特征化细胞状映射。

物理学中的高阶拓扑斯理论

Higher Topos Theory in Physics

本文为《数学物理百科全书》第2版撰写，简要阐述了高阶拓扑斯理论在数学物理中的意义。