论文标题
免费的前类别类别
Free precategories as presheaf categories
论文作者
论文摘要
前类别概括了严格的$ n $类别和sesquicategory的概念：它们的定义与严格的$ n $类别中的一个基本相同，只是我们不需要各种互换法律。这些被提出是一个框架，在该框架中，人们可以表达对弱的高级类别的半标题定义：在维度3中，灰色类别是其中的一个实例，已被证明与三曲曲面相同，并且已经提出，用作诸如Globular的证明助手的基础，并将半刻板性的四方面的定义用作。在本文中，我们对免费预定剂感兴趣。这些可以由发电机和关系提出，使用对测谎仪的概念（又称Computad）的适当变化，并且早期的作品表明，重写的理论可以推广到这种设置，并享受大多数基本构建和属性，这些基本结构和属性在传统的理论中可以找到严格的类别。我们在这里进一步研究了为什么是这种情况，通过提供几个结果，这些结果表明前类别及其相关的测谎仪具有确保我们对这些的良好语法的特性。特别是，我们表明，预制的测谎仪类别形成了预局部类别。
Precategories generalize both the notions of strict $n$-category and sesquicategory: their definition is essentially the same as the one of strict $n$-categories, excepting that we do not require the various interchange laws to hold. Those have been proposed as a framework in which one can express semi-strict definitions of weak higher categories: in dimension 3, Gray categories are an instance of them and have been shown to be equivalent to tricategories, and definitions of semi-strict tetracategories have been proposed, and used as the basis of proof assistants such as Globular. In this article, we are mostly interested in free precategories. Those can be presented by generators and relations, using an appropriate variation on the notion of polygraph (aka computad), and earlier works have shown that the theory of rewriting can be generalized to this setting, enjoying most of the fundamental constructions and properties which can be found in the traditional theory, contrarily to polygraphs for strict categories. We further study here why this is the case, by providing several results which show that precategories and their associated polygraphs bear properties which ensure that we have a good syntax for those. In particular, we show that the category of polygraphs for precategories form a presheaf category.
