Univalent categories constitute a well-behaved and useful notion of category in univalent foundations. The notion of univalence has subsequently been generalized to bicategories and other structures in (higher) category theory. Here, we zoom in on monoidal categories and study them in a univalent setting. Specifically, we show that the bicategory of univalent monoidal categories is univalent. Furthermore, we construct a Rezk completion for monoidal categories: we show how any monoidal category is weakly equivalent to a univalent monoidal category, universally. We have fully formalized these results in UniMath, a library of univalent mathematics in the Coq proof assistant.
翻译：单一类别构成一种良好和有用的非象素基础类别概念。 单象概念后来被普遍地推广到( 高) 类理论中的二类和其他结构。 在这里, 我们放大一分子类别, 并在一个非象素环境中研究它们。 具体地说, 我们显示, 单象单象类别的双类是非象素的。 此外, 我们为一分子类别构建了一种Rezk 补全法 : 我们显示任何单象体类别都与一个非象素的单象素类别普遍相当。 我们已经在UniMath正式确定了这些结果, UniMath是科克验证助理的一个非象素数学图书馆。
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