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Tim Campion
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One remark about the standard theory of enriched accessible categories is that the enriching category $\mathcal{V}$ is typically assumed to itself be locally presentable. This ensures that for sufficiently large $\kappa$, there is a good notion of $\kappa$-small limits and $\kappa$-filtered colimits -- for instance, they should commute in $\mathcal{V}$ -- which makes for a good notion of "smallness" for $\mathcal{V}$-enriched categories.

In principle, though, one should be able to develop a theory of $(\mathcal{V},\Phi)$-accessible categories based on any good duality between a class $\Phi$ of $\mathcal{V}$-limits and a class $\Phi^\vee$ of $\mathcal{V}$-colimits that commute in $\mathcal{V}$. (By "good", I mean that a condition called soundness should be satisfied.) Conceptually, it's not necessary for $\mathcal{V}$ to be locally presentable or for $\Phi$ to be the class of $\kappa$-small $\mathcal{V}$-limits. For example, in the unenriched case, the duality between finite products and sifted colimits can be developed in perfect analogy to the theory of $\kappa$-accessible categories for fixed $\kappa$, and one recovers the usual theory of Lawvere theories. Some aspects of such a general theory of locally $(\mathcal{V},\Phi)$-presentable categories are touched on by Lack and Rosický.

What's missing from this story is an interesting notion of "raising the index of accessibility". In the standard enriched setup, where $\mathcal{V}$ is locally presentable, we have (just like the unenriched case) a hierarchy of $\Phi$'s to use -- $\kappa$-small $\mathcal{V}$-limits for arbitrarily large $\kappa$ -- and interesting theorems to state that rely on letting $\kappa$ vary. I simply don't know any good examples of such a hierarchy of $\Phi$'s besides the standard ones to motivate the development of the sort of general theory I'm driving at.

Tim Campion
  • 40664
  • 6
  • 97
  • 242