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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
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• 10
• 120
• 293