The n-Category Café

So what do I mean by “the naive construction”? The easiest case is the category $\mathrm{Set}$ of small sets, i.e. sets belonging to some universe $U$. If $V$ is a larger universe with $U\in V$, then the obvious enlargement of $\mathrm{Set}$ to $V$ is the category $\mathrm{SET}$ of sets belonging to $V$.

(In the context of two universes $U\in V$, I like to speak of sets in $U$ as “small,” sets in $V$ as “large,” and sets not necessarily in $V$ as “very large.” Thus $\mathrm{Set}$ is the large category of small sets, and $\mathrm{SET}$ is the very large category of large sets. In the context of a third universe $W$ with $V\in W$, I would say “very large” for sets in $W$ and “extremely large” for sets not necessarily in $W$.)

Similarly, we can replace the large category $\mathrm{Top}$ of small topological spaces by the very large category $\mathrm{TOP}$ of large topological spaces, the large category $\mathrm{Grp}$ of small groups by the very large category $\mathrm{GRP}$ of large groups, and so on. In general, if $C$ is a large category whose objects are the “models of some theory” in $\mathrm{Set}$, then its obvious enlargement is the very large category of models of the same theory in $\mathrm{SET}$. Note that the theory in question could be “algebraic,” as in the case of groups, but it could also be a “higher-order” theory such as is required for topological spaces—all that matters is that we know how to talk about models of the theory in an elementary topos such as $\mathrm{Set}$ or $\mathrm{SET}$.

That approach covers pretty much any situation arising in practice, but as I said, for theoretical reasons it would be nice to have a construction that works on any large category. A very nice way to do this uses the Yoneda embedding, as described in Kelly’s book: for any large category $C$, the very large category $[C^{\mathrm{op}},\mathrm{SET}]$ contains $C$ as a full subcategory. Furthermore, $[C^{\mathrm{op}},\mathrm{SET}]$ is $\mathrm{SET}$-bicomplete, i.e. it has all large limits and colimits, whether or not $C$ was $\mathrm{Set}$-complete or cocomplete, and the embedding $C\hookrightarrow [C^{\mathrm{op}},\mathrm{SET}]$ preserves all limits that exist in $C$.

However, it preserves hardly any colimits (since it is a free cocompletion, after all). If we want an enlargement of $C$ which preserves some class $\Phi$ of colimits in $C$, then we can restrict to the full subcategory of $[C^{\mathrm{op}},\mathrm{SET}]$ consisting of the presheaves $C^{\mathrm{op}}\to \mathrm{SET}$ which preserve the colimits $\Phi$ (i.e. take them to limits in $\mathrm{SET}$). Let’s denote this category by ${\Uparrow }_{\Phi }C$. (Suggestions for better notation are welcome.) Since representable functors preserve all limits, the Yoneda embedding of $C$ factors through ${\Uparrow }_{\Phi }C$, and all the colimits in $\Phi$ are preserved by this restricted embedding.

Moreover, ${\Uparrow }_{\Phi }C$ is closed under limits in $[C^{\mathrm{op}},\mathrm{SET}]$, so it is itself $\mathrm{SET}$-complete. And as long as the (diagrams sizes of the) colimits in $\Phi$ are at most large, then (by theorems of Day etc.) ${\Uparrow }_{\Phi }C$ is reflective in $[C^{\mathrm{op}},\mathrm{SET}]$, and hence also $\mathrm{SET}$-cocomplete. (Usually, the only colimits we might care about in a large category $C$ are small, hence a fortiori also large. Occasionally one encounters large limits in a large category, such as in some forms of the adjoint functor theorem, but I don’t think I’ve ever seen any use of a very large limit in a large category.)

Now what does this have to do with the naive version? Suppose that $C$ is locally presentable relative to $U$. Then there is a small cardinal $\lambda$ and a small category $A$ with $\lambda$-small colimits such that $C$ is equivalent to the category $\lambda \mathrm{Cts}(A^{\mathrm{op}},\mathrm{Set})$ of $\lambda$-limit–preserving functors from $A^{\mathrm{op}}$ to $\mathrm{Set}$. Evidently, the “naive” enlargement of $C$ is then $C\prime ≔\lambda \mathrm{Cts}(A^{\mathrm{op}},\mathrm{SET})$, which is $V$-locally-presentable, and we have a full embedding $C\hookrightarrow C\prime$, which preserves limits and colimits since they are calculated in the same way in both categories.

Now every object of $C$ is a small colimit of objects of $A$, and the objects of $A$ are all $\lambda$-presentable in $C$ and in $C\prime$. Thus, every object of $C$ is $\kappa$-presentable in $C\prime$, where $\kappa$ is the size of the universe $U$. Conversely, since $C\prime$ is locally $\lambda$-presentable (relative to $V$), any $\kappa$-presentable object in it is a $\kappa$-small colimit of $\lambda$-presentable objects—so since $C$ is closed under small colimits in $C\prime$, it must consist exactly of the $\kappa$-presentable objects. However, since $C\prime$ is locally $\lambda$-presentable, it is also locally $\kappa$-presentable, so this implies that $C\prime \simeq \kappa \mathrm{Cts}(C^{\mathrm{op}},\mathrm{SET})$. But $\kappa \mathrm{Cts}(C^{\mathrm{op}},\mathrm{SET})$ is precisely ${\Uparrow }_{\Phi }C$, where $\Phi$ is the class of all small colimts in $C$.

Thus, for locally presentable categories, the general universe-enlargement agrees with the naive one: e.g. we have $\Uparrow \mathrm{Set}=\mathrm{SET}$, $\Uparrow \mathrm{Grp}=\mathrm{GRP}$, and so on. And importantly, if $C=\mathrm{Sh}(S)$ is the topos of small sheaves on a small site, then $\Uparrow C=\mathrm{SH}(S)$ is the category of large sheaves on the same site—which can also be identified with the category of large sheaves on $C$ itself with its canonical topology.

For non-locally-presentable categories, however, the two enlargements are generally different. This is most obvious if $C$ is not $\mathrm{Set}$-complete or cocomplete, in which case its naive enlargement will almost surely not be $\mathrm{SET}$-complete or cocomplete, whereas $\Uparrow C$ is always both. But the two completions can also differ when $C$ is complete and cocomplete; for instance, $\Uparrow \mathrm{Top}\ne \mathrm{TOP}$. Of course, $\mathrm{Top}$ is not locally presentable. I would like to be able to say that locally presentable is “the weakest hypothesis” one can hope for under which the two enlargements agree. For if $C$ is $\mathrm{Set}$-cocomplete, the generic enlargement $\Uparrow C$ is always $V$-locally-presentable, being the category of $\kappa$-limit-preserving $\mathrm{SET}$-valued presheaves on a $V$-small category with $\kappa$-small limits (namely, $C$) for some $V$-small cardinal $\kappa$ (namely, the cardinality of $U$). But if $C$ is not $U$-locally-presentable, one would be surprised if its naive enlargement were $V$-locally presentable.

It can happen, however. Suppose that $C$ is $U$-cocomplete and has a $U$-small dense subcategory, but is not $U$-locally-presentable. The existence of such a $C$ is equivalent to saying that $U$ is not a Vopenka cardinal. Then if the cocompleteness and small-dense-subcategory of $C$ are for reasons which “enlarge” to imply that its naive enlargement is $V$-cocomplete and has a $V$-small dense subcategory, but $V$ is a Vopenka cardinal, then the naive enlargement of $C$ will be $V$-locally-presentable. I don’t know whether in such a case, the naive enlargement can end up being equivalent to the general construction!

In any case, though, this seems contrived enough not to be particularly interesting in practice. So it still feels to me as though local presentability is the only reasonable hypothesis under which to expect this equivalence. In other words, another reason locally presentable categories are nice is that they have a unique and well-behaved enlargement to any bigger universe.