Virtual concepts in the theory of accessible categories

TL;DR

This paper introduces virtual reflectivity and orthogonality to characterize enriched accessible categories, offering new insights and tools for understanding and recognizing these categories within the framework of enriched category theory.

Contribution

It proposes the concepts of virtual reflectivity and orthogonality as generalizations, providing a clearer characterization of enriched accessible categories and their limits.

Findings

Introduces virtual reflectivity and orthogonality in enriched categories.

Provides a new characterization of accessible $ u$-categories.

Shows the 2-category of accessible $ u$-categories has all flexible limits.

Abstract

We provide a new characterization of enriched accessible categories by introducing the two new notions of virtual reflectivity and virtual orthogonality as a generalization of the usual reflectivity and orthogonality conditions for locally presentable categories. The word virtual refers to the fact that the reflectivity and orthogonality conditions are given in the free completion of the $\mathcal V$-category involved under small limits, instead of the $\mathcal V$-category itself. In this way we hope to provide a clearer understanding of the theory as well as a useful way of recognizing accessible $\mathcal V$-categories. In the last section we prove that the 2-category of accessible $\mathcal V$-categories, accessible $\mathcal V$-functors, and $\mathcal V$-natural transformations has all flexible limits.