论文标题
代数理论的$ω$分类
The $ω$-categorification of Algebraic Theories
论文作者
论文摘要
Batanin和Leinster在Globular Operads上的工作提供了许多弱$ω$类别的潜在缺陷之一。通过球状攻击的语言，他们构建了一个单子，其代数编码弱$ω$ - 类别。这项工作的目的是展示如何构建类似的单子，这将使我们能够制定任何方程代数理论的弱$ω$分类。我们首先回顾了经典的作业和专业理论。然后，我们介绍了如何通过对收藏类别的分类富集扩展到伦斯特的球状进场。然后显示出称为球形的过程如何使我们从经典的Pro P A Globular Pro构造其代数为P的代数，该代数是严格的$ω$类别和严格的$ω$ functors类别的内部代数。然后将伦斯特在球形oprad上的收缩结构的概念扩展到这种球状优点的环境，以建立一个单子，其代数是Classica Pro的球形弱点。其中这些弱点是最初的弱化，其代数是通过构造完全弱化的$ω$分类的代数，该代数由Algebraic理论编码的Alggebraic P. ectecorp.
Batanin and Leinster's work on globular operads has provided one of many potential defnitions of a weak $ω$-category. Through the language of globular operads they construct a monad whose algebras encode weak $ω$-categories. The purpose of this work is to show how to construct a similar monad which will allow us to formulate weak $ω$-categorifications of any equational algebraic theory. We first review the classical theory of operads and PROs. We then present how Leinster's globular operads can be extended to a theory of globular PROs via categorical enrichment over the category of collections. It is then shown how a process called globularization allows us to construct from a classical PRO P a globular PRO whose algebras are those algebras for P which are internal to the category of strict $ω$-categories and strict $ω$-functors. Leinster's notion of a contraction structure on a globular operad is then extended to this setting of globular PROs in order to build a monad whose algebras are weakenings of the globularization of the classica PRO P. Among these weakenings is the initial weakining whose algebras are by construction the fully weakened $ω$-categorifications of the algebraic theory encoded by P.
