论文标题
分类模糊的单体
Categorifying equivariant monoids
论文作者
论文摘要
在数学的许多分支中，模糊是非常重要的对象：它们结合了乘法的概念和组动作的概念。在本文中，我们将构建类别，通过将道具和概率的理论与交叉简单群体的理论相结合，编码由单体和小组作用编码的结构。道具和概率是用于编码由对称和编织的单体类别中对象承担的类别的类别，而交叉的简单组是编码Unital，关联乘法和兼容组动作的类别。我们将生产其代数类别等于使用对称和辫子交叉的简单基团的扩展，其代数类别等同于单粒，共肌和双膜素的类别。我们将使用丝带编织的简单组将该理论扩展到平衡的编织单体类别。最后，我们将使用高二十字体的简单组来编码具有兼容组动作的涉及单肌的结构。
Equivariant monoids are very important objects in many branches of mathematics: they combine the notion of multiplication and the concept of a group action. In this paper we will construct categories which encode the structure borne by monoids with a group action by combining the theory of PROPs and PROBs with the theory of crossed simplicial groups. PROPs and PROBs are categories used to encode structures borne by objects in symmetric and braided monoidal categories respectively, whilst crossed simplicial groups are categories which encode a unital, associative multiplication and a compatible group action. We will produce PROPs and PROBs whose categories of algebras are equivalent to the categories of monoids, comonoids and bimonoids with group action using extensions of the symmetric and braid crossed simplicial groups. We will extend this theory to balanced braided monoidal categories using the ribbon braid crossed simplicial group. Finally, we will use the hyperoctahedral crossed simplicial group to encode the structure of an involutive monoid with a compatible group action.
