Flat vs. filtered colimits in the enriched context

TL;DR

This paper investigates conditions under which enriched accessible categories can be characterized as free cocompletions or categories of flat presheaves, extending classical results to enriched contexts.

Contribution

It provides sufficient conditions on the base of enrichment for the equivalence between free cocompletions and flat presheaves to hold in enriched categories.

Findings

Identifies conditions on the enriching category for equivalence to hold

Establishes criteria for the equivalence up to Cauchy completion

Explores examples outside the main conditions

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) $\Leftrightarrow$ (b) holds; (2) give sufficient conditions on $\mathcal V$ so that (a) $\Leftrightarrow $ (b) holds up to Cauchy completion; (3) explore some examples not covered by (1) or (2).