Flat vs. filtered colimits in the enriched context

Abstract

The importance of accessible categories has been widely recognized; they can be described as those freely generated in some precise sense by a small set of objects and, because of that, satisfy many good properties. More specifically finitely accessible categories can be characterized as: (a) free cocompletions of small categories under filtered colimits, and (b) categories of flat presheaves on some small category. The equivalence between (a) and (b) is what makes the theory so general and fruitful. Notions of enriched accessibility have also been considered in the literature for various bases of enrichment, such as $\mathbf{Ab},\mathbf{SSet},\mathbf{Cat}$ and $\mathbf{Met}$. The problem in this context is that the equivalence between (a) and (b) is no longer true in general. The aim of this paper is then to: (1) give sufficient conditions on $\mathcal{V}$ so that (a) $\iff$ (b) holds; (2) give sufficient conditions on $\mathcal{V}$ so that (a) $\iff$ (b) holds up to Cauchy completion; (3) explore some examples not covered by (1) or (2).

arXiv commentRevised version: major changes to the introduction, added some words at the beginning of Sect. 3 and 4. To appear on Advances in Mathematics

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Cited by in corpus (10)

• Virtual concepts in the theory of accessible categories
• Adjoint functor theorems for homotopically enriched categories
• Accessible categories with a class of limits
• On continuity of accessible functors
• Flatness, weakly lex colimits, and free exact completions
• Dualities in the theory of accessible categories
• Filtered bicolimit presentations of locally presentable linear categories, Grothendieck categories and their tensor products
• Rewriting techniques for relative coherence
• Towards enriched universal algebra
• More on soundness in the enriched context