Accessible categories with a class of limits

TL;DR

This paper characterizes accessible categories with specific limits using a new notion of companion for classes of weights, unifying various existing theorems and extending to weakly sound classes.

Contribution

Introduces the concept of companion for classes of weights and characterizes accessible categories with limits as models of sketches, unifying several existing theories.

Findings

Characterizes accessible categories with specified limits via companions.

Unifies theorems for locally presentable, multipresentable, and polypresentable categories.

Extends framework to weakly sound classes and weakly locally presentable categories.

Abstract

In this paper we characterize those accessible $\mathcal V$-categories that have limits of a specified class. We do this by introducing the notion of companion $\mathfrak C$ for a class of weights $\Psi$, as a collection of special types of colimit diagrams that are compatible with $\Psi$. We then characterize the accessible $\mathcal V$-categories with $\Psi$-limits as those accessibly embedded and $\mathfrak C$-virtually reflective in a presheaf $\mathcal V$-category, and as the $\mathcal V$-categories of $\mathfrak C$-models of sketches. This allows us to recover the standard theorems for locally presentable, locally multipresentable, and locally polypresentable categories as instances of the same general framework. In addition, our theorem covers the case of any weakly sound class $\Psi$, and provides a new perspective on the case of weakly locally presentable categories.