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In Remark 5.4.2.15 in Higher Topos Theory, Lurie explains in what sense an accessible functor $F:\mathcal{C}\rightarrow \mathcal{D}$ between accessible $\infty$-categories is "determined by small data". In particular he wants to show that $F$ is induced by a functor $F':\mathcal{C}^\kappa\rightarrow \mathcal{D}^\kappa$, where $\mathcal{C}^\kappa$ refers to the $\kappa$-compact objects of $\mathcal{C}$.

He argues as follows. We begin by picking a regular cardinal $\tau$ such that $\mathcal{C}$, $\mathcal{D}$ and $F$ are $\tau$-accessible. Now $\mathcal{C}\simeq \mathrm{Ind}_\tau(\mathcal{C}^\tau)$, and so the universal property tells us that $F$ is induced by a functor $F':C^\tau\rightarrow \mathcal{D}$.

This is where I lose the plot. Lurie then seems to claim that remark 5.4.2.13 implies that the image of $F'$ is contained in the $\tau'$-compact objects of $\mathcal{D}$ for some other $\tau'$. However all that remark 5.4.2.13 says is that given any regular cardinal $\tau'$, $\mathcal{D}^{\tau'}$ is small. We also know that the image of $F'$ is small, but nevertheless I don't see why this is enough to conclude what Lurie requires. So the question is the following: Why does the image of $F'$ land inside the $\tau'$-compact objects of $\mathcal{D}$ for some regular cardinal $\tau'$?

Ps. After concluding that the image of $F'$ lands in $\mathcal{D}^{\tau'}$, Lurie claims that we can pick $\kappa>> \tau'$ and obtain our conclusion. But again I fail to understand how we still know that $F(\mathcal{C}^\kappa)\subset \mathcal{D}^\kappa$.

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    $\begingroup$ 5.4.2.13 also tells you that $\mathcal C^\tau$ is (essentially) small. So let $\tau'$ be a regular cardinal bigger than the presentability rank of $F(c)$ for any $c \in \mathcal C^\tau$. $\endgroup$
    – Tim Campion
    Commented Mar 9, 2021 at 17:58
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    $\begingroup$ Hi @Sil, welcome to MO! $\endgroup$
    – Alessandro Codenotti
    Commented Mar 10, 2021 at 11:29
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    $\begingroup$ @AlessandroCodenotti Hello Alessandro!! $\endgroup$
    – Sil Linskens
    Commented Mar 10, 2021 at 12:35

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He is applying 5.4.2.13 to $C$ and not to $D$: Because $C^\kappa$ is essentially small, and each $F(c)$ is $\lambda_c$-compact for some $\lambda$; there is a $\kappa = \sup_{c \in C^\kappa} \lambda_c$ such that each $F(c)$ is $\kappa'$-compact (well, rather, take some regular cardinal bigger than $\sup_{c \in C^\kappa} \lambda_c$, and by $c \in C^\kappa$, I really, mean for $c$ running in a small set of object representing all isomorphisms class of objects in $C^\kappa$.

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  • $\begingroup$ So every object in an accessible category is $\kappa$-compact for some $\kappa$? Could you point towards where this is proven in HTT, or am I missing something obvious. $\endgroup$
    – Sil Linskens
    Commented Mar 9, 2021 at 20:15
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    $\begingroup$ I don't know if/where this is stated exactly in HTT, but this follows from the fact that an accessible category is generated under colimits by $\kappa$-compact objects and the a small colimits of $\kappa$-compact object is $\kappa'$ compact for any regular cardinal $\geqslant \kappa$ and $>$ the size of the colimit diagram. $\endgroup$
    – Simon Henry
    Commented Mar 9, 2021 at 20:22

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