论文标题
再加上构造，成分和独特的分解类别，并应用于图形和类似业的理论
Plus constructions, plethysm, and unique factorization categories with applications to graphs and operad-like theories
论文作者
论文摘要
Baez-Dolan类型加建筑具有三个主要目的： 他们（1）相对于多数产物而言是单型的核心分类双模型，（2）允许将函子定义为双模型单模素，从而将代数放在函子上，（3）提供了一个单调曲折的理论。 Unital（单型）双模型单型屈服（单体）类别和核心代表是针对类别的索引富集。原始的Baez--Dolan构造在Operads上建造了代数。我们在类别的一般环境和（对称）单体类别的一般环境中定义了其中几个结构，表明它们是功能的，并证明了其核心代表属性。一种应用是，当且仅当核心代表性FC是一个加分的结构时，才可以将FC（例如Oprads，Props等）提供的结构定义为PlethySm Monoids。在一个方向上，我们证明了这样的加构建是基于{simel cranistization类别}（UFC）的新概念。我们还证明，由此产生的FC具有特殊的特性，例如立方体。这就解释了为什么没有用于循环或模块化的作战或道具的单体配方，但是有用于驻地和求职者。使用双模模型的观点，我们证明，作为单模型的单模型，FC的特征是以下事实：构造自由代数的函子保留了强烈单体的特性。我们提供了本地演示文稿以及全局描述，并使用装饰的Groupoid彩色图进行了图形版本。全局介绍利用了来自两个类别或同等双重类别的粘贴图。在UFC的特殊情况下，我们还提供了带有群体彩色图的图形形式主义。这使我们能够将我们的加上构造识别为Baez-Dolan和构造的分步概括。
Baez-Dolan type plus constructions serve three main purposes: They (1) corepresent categorical bimodules that are monoids with respect to a plethysm product, (2) allow to define functors as bimodule monoids, and thereby algebras over functors, (3) provide a theory of twists of monads. Unital (monoidal) bimodule monoids yield (monoidal) categories and the corepresentation is for indexed enrichments of categories. The original Baez--Dolan construction constructed algebras over operads. We define several of these constructions in the general context of categories and (symmetric) monoidal categories, show that they are functorial, and prove their corepresentation properties. One application is that the structures corepresented by an FC, like operads, props, etc. can be defined as plethysm monoids if and only if the corepresenting FC is a plus construction. In one direction, we prove that such a plus construction is based on the new notion of {Unique Factorization Category} (UFC). We also prove that the resulting FC has special properties, like being cubical. This explains why there is no monoid formulation for cyclic or modular operads or props, but there is for operads and properads. Using the bimodule monoids point of view, we prove that as monoidal bimodule monoids FCs are characterized by the fact that the functor constructing free algebras preserves the property of being strongly monoidal. We give a local presentation, as well as a global description, and a graphical version using decorated groupoid colored graphs. The global presentation utilizes pasting diagrams from 2-categories or equivalently double categories. In the special case of a UFC, we also present a graphical formalism with groupoid colored graphs. This allows us to identify our plus constructions as the step-by-step generalization of the Baez-Dolan plus constructions.
