Categorified presheaves and sieves

TL;DR

This paper develops a new framework for ${f Cat}$-valued presheaves and sieves over categories, introducing a Yoneda embedding for such presheaves and constructing sieves over subcategories of topological categories.

Contribution

It proposes a novel approach to ${f Cat}$-valued presheaves and sieves, including a Yoneda embedding adaptation and constructions over subcategories of topological categories.

Findings

Established a Yoneda embedding for ${f Cat}$-valued presheaves.

Constructed ${f Cat}$-valued sieves over subcategories of topological categories.

Provided a framework to handle presheaves valued in non-concrete categories.

Abstract

Let $\mathcal C$ be a category of a set of (small) categories. This paper concerns with the ${\mathbf {Cat}}$-valued presheaves and sieves over category $\mathcal C.$ Since ${\mathbf {Cat}}$ is not a concrete category, existing definition of presheaves can not deal with the situation. This paper proposes a new framework for the purpose. The main result is a version of Yoneda embedding for ${\mathbf {Cat}}$-valued presheaves, and construction of the ${\mathbf {Cat}}$-valued sieves over the category ${\mathcal O}(\mathbf B)$ of subcategories of a given topological category $\mathbf B.$