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Suppose $C$ is an essentially small $\infty$-category (HTT.5.4.1), $\kappa$ is a regular cardinal, and let $C(\kappa)$ denote the category obtained by adding $\kappa$-small colimits to $C$ (HTT.5.3.6). Why is $C(\kappa)$ still essentially small?

This is used in the third paragraph of HA.1.4.4.2's proof.

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While there is likely a more conceptual argument, there is a fairly direct argument using an explicit construction of this cocompletion. We can realize this extension of $\mathcal{C}$ as a full subcategory of $\mathrm{Pr}(\mathcal{C})$, namely the smallest full subcategory of $\mathrm{Pr}(\mathcal{C})$ closed under $\kappa$-small colimits and containing the image of $y : \mathcal{C} \to \mathrm{Pr}(\mathcal{C})$.

It suffices to show that this subcategory is essentially small, but it can be realized as a union of a chain of $\kappa^+$-many essentially small full subcategories of $\mathrm{Pr}(\mathcal{C})$. Explicitly, we start with the image of $y$ (essentially small as it is $\mathcal{C})$, then take the full subcategory of objects built out of $\kappa$-small colimits of representables (small because there is only a set's worth of $\kappa$-small colimits in any small category) and then proceeding inductively.

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    $\begingroup$ You never specifically use that it lives in $Pr(C)$ :D Your second paragraph simply argues that for any inclusion C -> D, the smallest subcategory of D containing C and closed under k-small colimits is small (which applies to C -> C(k) without needing to know what C(k) "looks like") $\endgroup$
    – Maxime Ramzi
    Commented Aug 12 at 14:10
  • $\begingroup$ @MaximeRamzi Good point! I guess I had it in my head merely to make sense of what it means to "form a category built whose objects are $\kappa$-colimits of objects in $\mathcal{C}$" $\endgroup$
    – daniel gratzer
    Commented Aug 12 at 16:11

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