# If $\mathcal C$ is a $\kappa$-accessible $\infty$-category, then is $Mor \mathcal C$ $\kappa$-accessible?

If $$\mathcal C$$ is a $$\kappa$$-accessible 1-category, then the category of morphisms $$Mor \mathcal C$$ is a $$\kappa$$-accessible 1-category, with the $$\kappa$$-presentable objects being those morphisms whose domains and codomains are each $$\kappa$$-presentable.

In the context of $$\infty$$-categories, the best result I know of is HTT Proposition 5.4.4.3, which shows that if $$\mathcal C$$ is a $$\kappa$$-accessible $$\infty$$-category and $$\kappa \ll \tau$$ (meaning that $$\lambda < \tau \Rightarrow \kappa^\lambda < \tau$$ and $$\kappa < \tau$$), then $$Mor \mathcal C$$ is $$\tau$$-accessible.

Lurie's proof, via HTT Lemma 5.4.4.2 (note that this lemma's proof has been revised since the printed edition), seems to really use the full strength of the assumption $$\kappa \ll \tau$$. Can this be improved to $$\kappa = \tau$$? Or at least to the "sharply below" relation $$\kappa \triangleleft \tau$$ familiar from the theory of accessible 1-categories?

This boils down to asking: if $$\mathcal C$$ is $$\kappa$$-accessible, then is every morphism of $$\mathcal C$$ a levelwise $$\kappa$$-filtered colimit of morphisms between $$\kappa$$-presentable objects?

In the case of 1-categories, a follow-your-nose argument works: you just take colimiting diagrams for the domains and codomains and factor the original map through stages of the colimit. I suspect that the same must be true in $$\infty$$-categories, with the same argument in principle working. But the question seems to be much more subtle $$\infty$$-categorically.

• By the way -- Lurie systematically uses the relation $\kappa \ll \tau$ in place of the relation $\kappa \triangleleft \tau$ familiar from accessible 1-categories. I don't know whether this is the only place this matters, but I think it does here (in Lemma 5.4.4.2's construction of the $K(\alpha)$'s) Aug 30 '20 at 3:37
• $Mor(\mathcal C)$ is also $\kappa$-accessible by HTT Prop. 5.3.5.15. Aug 30 '20 at 8:07
• @MarcHoyois Thanks -- just a few pages from where I was looking, too! Aug 30 '20 at 12:20