We survey Lawvere theories at the level of infinity categories, as an alternative framework for higher algebra (rather than infinity operads). From a pedagogical perspective, they make many key definitions and constructions less technical. They are also better-suited than operads for equivariant homotopy theory and its relatives. Our main result establishes a universal property for the infinity category of Lawvere theories, which completely characterizes the relationship between a Lawvere theory and its infinity category of models. Many familiar properties of Lawvere theories follow directly. As a consequence, we prove that the Burnside category is a classifying object for additive categories, as promised in an earlier paper, and as part of a more general correspondence between enriched Lawvere theories and module Lawvere theories.

中文翻译：

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高等劳维尔理论

我们在无穷范畴水平上调查 Lawvere 理论，作为高级代数（而不是无穷操作数）的替代框架。从教学的角度来看，它们使许多关键定义和结构的技术性降低。它们也比等变同伦理论及其相关的操作数更适合。我们的主要结果为 Lawvere 理论的无限范畴建立了一个普遍性质，它完全描述了 Lawvere 理论与其模型的无限范畴之间的关系。Lawvere 理论的许多熟悉的性质直接遵循。因此，我们证明了 Burnside 范畴是可加范畴的一个分类对象，正如在早期论文中所承诺的那样，并且是丰富的 Lawvere 理论和模块 Lawvere 理论之间更一般对应的一部分。