Accessible categories with a class of limits

Abstract

In this paper we characterize those accessible V-categories that have limits of a specified class. We do this by introducing the notion of companion C for a class of weights Ψ, as a collection of special types of colimit diagrams that are compatible with Ψ. We then characterize the accessible V-categories with Ψ-limits as those accessibly embedded and C-virtually reflective in a presheaf V-category, and as the V-categories of 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 Ψ, and provides a new perspective on the case of weakly locally presentable categories.

Original language	English
Article number	107444
Journal	Journal of Pure and Applied Algebra
Volume	228
Issue number	2
Early online date	1 Jun 2023
DOIs	https://doi.org/10.1016/j.jpaa.2023.107444
Publication status	Published - 1 Feb 2024

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10.1016/j.jpaa.2023.107444

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title = "Accessible categories with a class of limits",

abstract = "In this paper we characterize those accessible V-categories that have limits of a specified class. We do this by introducing the notion of companion C for a class of weights Ψ, as a collection of special types of colimit diagrams that are compatible with Ψ. We then characterize the accessible V-categories with Ψ-limits as those accessibly embedded and C-virtually reflective in a presheaf V-category, and as the V-categories of 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 Ψ, and provides a new perspective on the case of weakly locally presentable categories.",

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AB - In this paper we characterize those accessible V-categories that have limits of a specified class. We do this by introducing the notion of companion C for a class of weights Ψ, as a collection of special types of colimit diagrams that are compatible with Ψ. We then characterize the accessible V-categories with Ψ-limits as those accessibly embedded and C-virtually reflective in a presheaf V-category, and as the V-categories of 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 Ψ, and provides a new perspective on the case of weakly locally presentable categories.

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