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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 SetSet of small sets, i.e. sets belonging to some universe U\mathbf{U}. If V\mathbf{V} is a larger universe with UV\mathbf{U}\in\mathbf{V}, then the obvious enlargement of SetSet to V\mathbf{V} is the category SETSET of sets belonging to V\mathbf{V}.

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

Similarly, we can replace the large category TopTop of small topological spaces by the very large category TOPTOP of large topological spaces, the large category GrpGrp of small groups by the very large category GRPGRP of large groups, and so on. In general, if CC is a large category whose objects are the “models of some theory” in SetSet, then its obvious enlargement is the very large category of models of the same theory in SETSET. 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 SetSet or SETSET.

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 CC, the very large category [C op,SET][C^{op},SET] contains CC as a full subcategory. Furthermore, [C op,SET][C^{op},SET] is SETSET-bicomplete, i.e. it has all large limits and colimits, whether or not CC was SetSet-complete or cocomplete, and the embedding C[C op,SET]C \hookrightarrow [C^{op},SET] preserves all limits that exist in CC.

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

Moreover, ΦC\Uparrow_\Phi C is closed under limits in [C op,SET][C^{op},SET], so it is itself SETSET-complete. And as long as the (diagrams sizes of the) colimits in Φ\Phi are at most large, then (by theorems of Day etc.) ΦC\Uparrow_\Phi C is reflective in [C op,SET][C^{op},SET], and hence also SETSET-cocomplete. (Usually, the only colimits we might care about in a large category CC 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 CC is locally presentable relative to U\mathbf{U}. Then there is a small cardinal λ\lambda and a small category AA with λ\lambda-small colimits such that CC is equivalent to the category λCts(A op,Set)\lambda Cts(A^{op},Set) of λ\lambda-limit–preserving functors from A opA^{op} to SetSet. Evidently, the “naive” enlargement of CC is then CλCts(A op,SET)C' \coloneqq \lambda Cts(A^{op},SET), which is V\mathbf{V}-locally-presentable, and we have a full embedding CCC\hookrightarrow C', which preserves limits and colimits since they are calculated in the same way in both categories.

Now every object of CC is a small colimit of objects of AA, and the objects of AA are all λ\lambda-presentable in CC and in CC'. Thus, every object of CC is κ\kappa-presentable in CC', where κ\kappa is the size of the universe U\mathbf{U}. Conversely, since CC' is locally λ\lambda-presentable (relative to V\mathbf{V}), any κ\kappa-presentable object in it is a κ\kappa-small colimit of λ\lambda-presentable objects—so since CC is closed under small colimits in CC', it must consist exactly of the κ\kappa-presentable objects. However, since CC' is locally λ\lambda-presentable, it is also locally κ\kappa-presentable, so this implies that CκCts(C op,SET)C' \simeq \kappa Cts(C^{op},SET). But κCts(C op,SET)\kappa Cts(C^{op},SET) is precisely ΦC\Uparrow_\Phi C, where Φ\Phi is the class of all small colimts in CC.

Thus, for locally presentable categories, the general universe-enlargement agrees with the naive one: e.g. we have Set=SET\Uparrow Set = SET, Grp=GRP\Uparrow Grp = GRP, and so on. And importantly, if C=Sh(S)C=Sh(S) is the topos of small sheaves on a small site, then C=SH(S)\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 CC itself with its canonical topology.

For non-locally-presentable categories, however, the two enlargements are generally different. This is most obvious if CC is not SetSet-complete or cocomplete, in which case its naive enlargement will almost surely not be SETSET-complete or cocomplete, whereas C\Uparrow C is always both. But the two completions can also differ when CC is complete and cocomplete; for instance, TopTOP\Uparrow Top \neq TOP. Of course, TopTop 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 CC is SetSet-cocomplete, the generic enlargement C\Uparrow C is always V\mathbf{V}-locally-presentable, being the category of κ\kappa-limit-preserving SETSET-valued presheaves on a V\mathbf{V}-small category with κ\kappa-small limits (namely, CC) for some V\mathbf{V}-small cardinal κ\kappa (namely, the cardinality of U\mathbf{U}). But if CC is not U\mathbf{U}-locally-presentable, one would be surprised if its naive enlargement were V\mathbf{V}-locally presentable.

It can happen, however. Suppose that CC is U\mathbf{U}-cocomplete and has a U\mathbf{U}-small dense subcategory, but is not U\mathbf{U}-locally-presentable. The existence of such a CC is equivalent to saying that U\mathbf{U} is not a Vopenka cardinal. Then if the cocompleteness and small-dense-subcategory of CC are for reasons which “enlarge” to imply that its naive enlargement is V\mathbf{V}-cocomplete and has a V\mathbf{V}-small dense subcategory, but V\mathbf{V} is a Vopenka cardinal, then the naive enlargement of CC will be V\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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8 Comments & 0 Trackbacks

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 Φ\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 C\uparrow C to mean “a bigger CC,” 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, 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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