We introduce the notion of a categorical join, which can be thought of as a categorification of the classical join of two projective varieties. This notion is in the spirit of homological projective duality, which categorifies classical projective duality. Our main theorem says that the homological projective dual category of the categorical join is naturally equivalent to the categorical join of the homological projective dual categories. This categorifies the classical version of this assertion and has many applications, including a nonlinear version of the main theorem of homological projective duality.

中文翻译：

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分类连接

我们引入了分类连接的概念，它可以被认为是对两个射影变体的经典连接的分类。这个概念符合同调射影二元性的精神，它对经典射影二元性进行了分类。我们的主要定理说定论连接的同调射影对偶范畴自然等价于同调射影对偶范畴的定论连接。这对这个断言的经典版本进行了分类，并有许多应用，包括同调射影对偶主要定理的非线性版本。