Just as the presheaf category is the free cocompletion of any small category, there is an analogous notion of free cocompletion for any small restriction category. In this paper, we extend the work on restriction presheaves to presheaves over join restriction categories, and show that the join restriction category of join restriction presheaves is equivalent to some partial map category of sheaves. We then use this to show that the Yoneda embedding exhibits the category of join restriction presheaves as the free cocompletion of any small join restriction category.

中文翻译：

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连接限制类别上的 Presheaves

正如前层范畴是任何小范畴的自由共同完成一样，任何小限制范畴都有一个类似的自由共同完成概念。在本文中，我们将约束预层的工作扩展到连接限制类别上的预层，并证明了连接限制预层的连接限制类别等价于一些部分映射类别的滑轮。然后，我们使用它来表明米田嵌入将连接限制 presheaves 的类别展示为任何小连接限制类别的自由协同完成。