Let $\mathit{Pr}^L$ be the $\infty$-category of presentable $\infty$-categories and continuous functors in some universe. Is it presentable itself a larger universe?
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2 Answers
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When you pass to a larger universe, all categories that were in your old universe become small. A small category which is not a poset cannot be closed under colimits (think about taking coproducts with index sets larger than your category), and so cannot be presentable. All this applies equally well to $(\infty,1)$-categories, and ought to apply to any other reasonable generalized kind of category.
answered Jul 4, 2015 at 17:16
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2$\begingroup$ Was this answer downvoted because it is incorrect? It seems right to me. $\endgroup$– ZippyCommented Jul 4, 2015 at 19:45
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Let us fix a universe and use the words "large" and "small" with respect to that universe. Presentable $\infty$-categories are typically large $\infty$-categories (since, as the previous answer mentioned, small $\infty$-categories are rarely presentable). One might then expect that $Pr^L$ would be a "very large" $\infty$-category (like the $\infty$-category $\widehat{Cat}_\infty$ of possibly large $\infty$-categories). In that case $Pr^L$ would turn from being very large to just large upon increasing the universe, and it would be natural to ask if it then becomes presentable. However, the $\infty$-category $Pr^L$ is actually not "very large", but just large. This is because presentable $\infty$-categories, though being large, are actually determined by a small amount of data. To formally prove this one might consider, for example, for each cardinal $\kappa$, the subcategory $Pr^L_{\kappa} \subseteq Pr^L$ consisting of $\kappa$-compactly generated presentable $\infty$-categories and functors preserving $\kappa$-compact objects between them (see section 5.5.7 of higher topos theory). Proposition 5.5.7.10 loc. cit. shows that when $\kappa > \omega$ the $\infty$-category $Pr^L_{\kappa}$ is equivalent to the $\infty$-category of small $\infty$-categories admitting $\kappa$-small colimits, and hence $Pr^L_{\kappa}$ is large (but not very large). Consequently, $Pr^L$ is a large colimit of large $\infty$-categories, and hence large (but again not very large). It follows that if we increase the universe $Pr^L$ will become small, and it will not be so natural to ask if it is presentable. On the other hand, since $Pr^L$ is just large you might ask if it is presentable without increasing the universe. In principle the answer would have to be no, because then $Pr^L$ would contain itself, and we know that such stories do not end well (although I admit I do not have a direct proof in mind, and would like to see one). Morally, $Pr^L$ should not be a presentable $\infty$-category, but some $(\infty,2)$-version of the notion, similarly to how the $\infty$-category of $\infty$-topoi should be something like an $(\infty,2)$-topos (but not an $(\infty,1)$-topos). Unfortunately, I am not aware of these ideas being made precise anywhere.
Edit: a formal argument why $Pr^L$ is not presentable (in the current universe) is that it is not locally small, see comments below.
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2$\begingroup$ Even "very large" categories become small when you pass to a larger universe. For instance, if $V_\kappa$ is a universe, then the collection of all "large" categories in $V_\kappa$ has cardinality only $2^\kappa$, which is far smaller than any inaccessible larger than $\kappa$. $\endgroup$ Commented Jul 4, 2015 at 20:24
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1$\begingroup$ @EricWofsey, you're right, but after you made this enlargement it will not be interesting anymore to ask if the original $\infty$-category is presentable. Instead, you should stop when it is just large, and ask if it is presentable then. $\endgroup$ Commented Jul 4, 2015 at 20:32
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1$\begingroup$ Sure. I just wanted to correct your assertion that a very large category might still be large in a larger universe. $\endgroup$ Commented Jul 4, 2015 at 20:34
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6$\begingroup$ It just occurred to me that $Pr^L$ is not presentable simply because it is not locally small (the space of functors between two presentable $\infty$-categories is the maximal $\infty$-groupoid of a presentable $\infty$-category, and as such is large). However, when considered as an $(\infty,2)$-category, the mapping categories in $Pr^L$ between every two objects are presentable, and hence controlled by a small amount of data. This might be considered as evidence that $Pr^L$ should be something like $(\infty,2)$-presentable. $\endgroup$ Commented Jul 4, 2015 at 21:00
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2$\begingroup$ This definition is not correct, it should be the (not full) subcategory spanned by $\kappa$-compactly generated presentable $\infty$-categories and functors which preserve $\kappa$-small objects. This is the definition to which Proposition 5.5.7.10 applies to and that is indeed a (non-full) subcategory which is small. The argument appearing in the answer above still applies to show that $Pr^L$ is just large, and not very large. $\endgroup$ Commented Jul 10, 2015 at 18:58