Presheaves over a join restriction category

TL;DR

This paper extends the concept of presheaves to join restriction categories, demonstrating that the category of join restriction presheaves serves as the free cocompletion of any small join restriction category, linking it to sheaf categories.

Contribution

It introduces join restriction presheaves and proves their category is the free cocompletion for small join restriction categories, generalizing existing restriction presheaf theory.

Findings

Join restriction presheaves form a category equivalent to a partial map category of sheaves.

The Yoneda embedding characterizes join restriction presheaves as the free cocompletion.

Extension of restriction presheaves to join restriction categories broadens the theoretical framework.

Abstract

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.