Title: Moment categories and operads Show Chinese title

Title: 矩类和操作符

Abstract: A moment category is endowed with a distinguished set of split idempotents, called moments, which can be transported along morphisms. Equivalently, a moment category is a category with an active/inert factorisation system fulfilling two simple axioms. These axioms imply that the moments of a fixed object form a monoid, actually a left regular band. Each locally finite unital moment category defines a specific type of operad which records the combinatorics of partitioning moments into elementary ones. In this way the notions of symmetric, non-symmetric and $n$-operad correspond to unital moment structures on $\Gamma$, $\Delta$ and $\Theta_n$ respectively. There is an analog of the plus construction of Baez-Dolan taking a unital moment category $\mathbb{C}$ to a unital hypermoment category $\mathbb{C}^+$. Under this construction, $\mathbb{C}$-operads get identified with $\mathbb{C}^+$-monoids, i.e. presheaves on $\mathbb{C}^+$ satisfying strict Segal conditions. We show that the plus construction of Segal's category $\Gamma$ embeds into the dendroidal category $\Omega$ of Moerdijk-Weiss. Show Chinese abstract

Abstract: 一个时刻范畴被赋予了一组特殊的分裂幂等元，称为时刻，这些时刻可以通过态射进行传输。 等价地，一个时刻范畴是一个具有主动/惰性分解系统的范畴，满足两个简单的公理。 这些公理表明，固定对象的时刻形成一个独异半群，实际上是一个左正则带。 每个局部有限的单位时刻范畴定义了一种特定类型的操作符，该操作符记录了将时刻分解为基本时刻的组合学。 这样，对称、非对称和$n$-操作符的概念分别对应于$\Gamma$、$\Delta$和$\Theta_n$上的单位时刻结构。 存在一个类似于 Baez-Dolan 的加法构造，它将一个单位时刻范畴$\mathbb{C}$转换为一个单位超时刻范畴$\mathbb{C}^+$。 在这种构造下，$\mathbb{C}$-操作符与$\mathbb{C}^+$-单子相一致，即在$\mathbb{C}^+$上的预层，满足严格的Segal条件。我们证明Segal范畴的加法构造$\Gamma$嵌入到Moerdijk-Weiss的树状范畴$\Omega$中。