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).

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

References in corpus (3)

• Higher Topos Theory
• Notes on enriched categories with colimits of some class
• Cauchy completeness and causal spaces

Cited by in corpus (8)

• Virtual concepts in the theory of accessible categories
• Accessible categories with a class of limits
• On continuity of accessible functors
• Dualities in the theory of accessible categories
• Flatness, weakly lex colimits, and free exact completions
• More on soundness in the enriched context
• Towards enriched universal algebra
• Rewriting techniques for relative coherence