In Theorem 5.19 of Kelly's Basic Concepts of Enriched Category Theory, it is proven that a fully faithful functor $K \colon \mathcal A \to \mathcal C$ is dense if and only if $\mathcal C$ is the closure of $\mathcal A$ under colimits of a family of diagrams, by which is meant a class of pairs $(F \colon \mathcal L^\circ \to \mathcal V, P \colon \mathcal L \to \mathcal C)$ consisting of a weight and a diagram for that weight, and these colimits are $K$-absolute.
Typically, cocompletions of (enriched) categories are taken with respect to a class of weights, rather than a family of diagrams. However, I have found it difficult to find explicit comparisons between the expressivity of these approaches. I have several related questions.
- Is it possible to characterise those fully faithful functors $K \colon \mathcal A \to \mathcal C$ that exhibit $\mathcal C$ as a closure of $\mathcal A$ under $\Phi$-weighted colimits which are $K$-absolute (for some class of weights $\Phi$)? Is this concept strictly less general than density?
Assuming closure under weighted colimits is less general than closure under colimits of a family of diagrams:
- What are some natural examples of dense functors that are only captured by considering diagrams rather than weights?
- What are the difficulties in considering cocompletions with respect to families of diagrams, rather than cocompletions with respect to classes of weights?
I looked through several papers on these concepts, but couldn't find the answer in them, e.g. Albert–Kelly's The closure of a class of colimits; Kelly–Schmitt's Notes on enriched categories with colimits of some class; Kelly–Lack's On the monadicity of categories with chosen colimits.
