## November 23, 2010

### Universe Enlargement

#### Posted by Mike Shulman

When dealing with categories of more than one size—that is, belonging to more than one set-theoretic universe—we are sometimes faced with the need to replace some given category by a version of “the same” category defined at a different size level. Usually, it is fairly obvious how to do this, but for theoretical reasons one would like a general construction that always works. One very nice way to perform such an enlargement is described in sections 3.11–3.12 of Kelly’s book, using Day convolution. But he doesn’t mention that in many cases, this general construction actually gives the same result as the naive one.

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

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

Similarly, we can replace the large category $Top$ of small topological spaces by the very large category $TOP$ of large topological spaces, the large category $Grp$ of small groups by the very large category $GRP$ of large groups, and so on. In general, if $C$ is a large category whose objects are the “models of some theory” in $Set$, then its obvious enlargement is the very large category of models of the same theory in $SET$. 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 $Set$ or $SET$.

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 $[C^{op},SET]$ contains $C$ as a full subcategory. Furthermore, $[C^{op},SET]$ is $SET$-bicomplete, i.e. it has all large limits and colimits, whether or not $C$ was $Set$-complete or cocomplete, and the embedding $C \hookrightarrow [C^{op},SET]$ 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 $[C^{op},SET]$ consisting of the presheaves $C^{op}\to SET$ which preserve the colimits $\Phi$ (i.e. take them to limits in $SET$). 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 $[C^{op},SET]$, so it is itself $SET$-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 $[C^{op},SET]$, and hence also $SET$-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 $\mathbf{U}$. Then there is a small cardinal $\lambda$ and a small category $A$ with $\lambda$-small colimits such that $C$ is equivalent to the category $\lambda Cts(A^{op},Set)$ of $\lambda$-limit–preserving functors from $A^{op}$ to $Set$. Evidently, the “naive” enlargement of $C$ is then $C' \coloneqq \lambda Cts(A^{op},SET)$, which is $\mathbf{V}$-locally-presentable, and we have a full embedding $C\hookrightarrow C'$, 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 $C'$. Thus, every object of $C$ is $\kappa$-presentable in $C'$, where $\kappa$ is the size of the universe $\mathbf{U}$. Conversely, since $C'$ is locally $\lambda$-presentable (relative to $\mathbf{V}$), any $\kappa$-presentable object in it is a $\kappa$-small colimit of $\lambda$-presentable objects—so since $C$ is closed under small colimits in $C'$, it must consist exactly of the $\kappa$-presentable objects. However, since $C'$ is locally $\lambda$-presentable, it is also locally $\kappa$-presentable, so this implies that $C' \simeq \kappa Cts(C^{op},SET)$. But $\kappa Cts(C^{op},SET)$ 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 $\Uparrow Set = SET$, $\Uparrow Grp = GRP$, and so on. And importantly, if $C=Sh(S)$ is the topos of small sheaves on a small site, then $\Uparrow C = SH(S)$ 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 $Set$-complete or cocomplete, in which case its naive enlargement will almost surely not be $SET$-complete or cocomplete, whereas $\Uparrow C$ is always both. But the two completions can also differ when $C$ is complete and cocomplete; for instance, $\Uparrow Top \neq TOP$. Of course, $Top$ 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 $Set$-cocomplete, the generic enlargement $\Uparrow C$ is always $\mathbf{V}$-locally-presentable, being the category of $\kappa$-limit-preserving $SET$-valued presheaves on a $\mathbf{V}$-small category with $\kappa$-small limits (namely, $C$) for some $\mathbf{V}$-small cardinal $\kappa$ (namely, the cardinality of $\mathbf{U}$). But if $C$ is not $\mathbf{U}$-locally-presentable, one would be surprised if its naive enlargement were $\mathbf{V}$-locally presentable.

It can happen, however. Suppose that $C$ is $\mathbf{U}$-cocomplete and has a $\mathbf{U}$-small dense subcategory, but is not $\mathbf{U}$-locally-presentable. The existence of such a $C$ is equivalent to saying that $\mathbf{U}$ 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 $\mathbf{V}$-cocomplete and has a $\mathbf{V}$-small dense subcategory, but $\mathbf{V}$ is a Vopenka cardinal, then the naive enlargement of $C$ will be $\mathbf{V}$-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.

Posted at November 23, 2010 10:30 PM UTC

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### Re: Universe Enlargement

I like your compromise on the notation for the enlargement - the ‘thick up arrow’ does give a more intuitive meaning once explained than my own ‘ordinary up arrow’ version I proposed at the nForum. Also the subscript $\Phi$ is good.

Posted by: David Roberts on November 23, 2010 11:34 PM | Permalink | Reply to this

### Re: Universe Enlargement

I’m glad you like it. (For those who weren’t present at the forum discussion, David suggested $\uparrow C$ to mean “a bigger $C$,” thus denoting the operation of enlargement by $\uparrow$. My thought was just that $\Uparrow$ is a little less overloaded.)

Posted by: Mike Shulman on November 23, 2010 11:39 PM | Permalink | Reply to this

### Re: Universe Enlargement

Thanks, Mike. It’s good that you nail this down.

Just so I am sure I understand (at this time of night): this is a reformulation of the argument in HTT 6.3.5.17, differing in that there the argument uses the characterization of locally presentable categories as accessible reflective localizations of presheaf categories (left exact localization is mentioned but not actually usedd in the argument, if I see correctly) while your argument works with the equivalent characterization by limit sketches.

We just need to remember that whatever symbol for universe enlargement we settle on, we still need to add it here and there at shape/coshape of an $\infty$-topos and polish huge $\infty$-sheaf $\infty$-toposes, preferably by entirely replacing it by a more general discussion along the lines you just gave.

But I’ll call it quits for today. And for the rest of the week I need to focus on some derived loop spaces. Hopefully you or somebody else finds the time to bring the Lab across the univeeerse .

Posted by: Urs Schreiber on November 24, 2010 12:41 AM | Permalink | Reply to this

### Re: Universe Enlargement

I'm glad that the nLab universe entry is good enough now that you can link to it!

Posted by: Toby Bartels on November 26, 2010 4:40 AM | Permalink | Reply to this

### Re: Universe Enlargement

There is something else you could do, which is to freely adjoin small-filtered large colimits (so look at small-flat functors rather than small-continuous ones). My impression is that this would give the same as the naive construction for any small-accessible category as well as any locally small-presentable one.

Posted by: Richard Garner on November 26, 2010 5:50 AM | Permalink | Reply to this

### Re: Universe Enlargement

Though obv. it is still garbage for topological spaces etc.

Posted by: Richard Garner on November 26, 2010 5:52 AM | Permalink | Reply to this

### Re: Universe Enlargement

That’s an excellent point! I had the feeling that something like this should work for accessible categories.

Posted by: Mike Shulman on November 29, 2010 5:39 AM | Permalink | Reply to this

### Re: Universe Enlargement

I have now nlabified this discussion, including Richard’s suggestion.

Posted by: Mike Shulman on December 2, 2010 4:48 AM | Permalink | Reply to this

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