语言∞-类别为表达高维数学中的许多结果提供了一种有见地的新方法，但对新手来说可能很有挑战性。来解释∞-类别is需要各种技术模型，这就提出了如何进行比较的问题。为了克服这一点，需要一种独立于模型的方法，这样，任何模型证明的定理都将适用于它们。这篇文章发展了生命周期理论∞-以独立于模型的方式，使用∞-宇宙，在其中的宇宙∞-类别作为对象存在。一∞-宇宙是形式范畴理论的肥沃土壤∞-类别，并以这种方式在∞-范畴理论与普通范畴理论的经典基础非常相似。本课程的第一部分是为学生和研究者提供了一个经典的1-范畴理论的基础。
Elements of ∞- Category Theory
The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.
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