Accessible categories with a class of limits

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.

arXiv commentJournal version. Some references added, as well as 4.10, 4.11, 4.18, 4.20(4)

References in corpus (9)

• Limits of small functors
• Notes on enriched categories with colimits of some class
• Enriched weakness
• Enriched Regular Theories
• Flat vs. filtered colimits in the enriched context
• Virtual concepts in the theory of accessible categories
• Adjoint functor theorems for homotopically enriched categories
• Accessibility and presentability in 2-categories
• On continuity of accessible functors

Cited by in corpus (4)

• Adjoint functor theorems for lax-idempotent pseudomonads
• Flatness, weakly lex colimits, and free exact completions
• Dualities in the theory of accessible categories
• More on soundness in the enriched context