nLab universe enlargement

Idea

Let $\mathbf{U}$ and $\mathbf{V}$ be set-theoretic universes (such as Grothendieck universes or universes in an ambient topos) with $\mathbf{V}$ “bigger than” $\mathbf{U}$, and let $C$ be a category “in the world of $\mathbf{U}$,” i.e. whose objects are $\mathbf{U}$-small and which itself is $\mathbf{U}$-large. Its universe enlargement is supposed to be a category “in the world of $\mathbf{V}$” which is “the same” or at least similar to $C$, and perhaps better behaved.

Notation

An enlargement of a specific category, such as Set, Cat, Grp, or Top, is often denoted by writing its name in all capitals: SET, CAT, GRP, TOP.

As a notation for the operation on a general category $C$, the notation $\Uparrow C$ has been proposed.

Definitions

Since the enlargement is not well-specified by the above intuitive description, there are several different definitions, which are often equivalent but not always.

Henceforth we fix two universes $\mathbf{U}$ and $\mathbf{V}$ with $\mathbf{U}\in \mathbf{V}$. We will refer to sets in $\mathbf{U}$ as small, sets in $\mathbf{V}$ as large, and sets not necessarily in $\mathbf{V}$ (or in the next larger universe $\mathbf{W}$) as very large.

Models of a theory

If $C$ is the category of small models of some theory, then we can take $\Uparrow C$ to be the category of large models of the same theory. For instance:

• the large category Set of small sets is the category of small models of the theory with one sort and no operations or relations. The category of large models of this theory is just the very large category $SET$ of large sets.

• Similarly, from categories such as Grp, Cat, and Top of small groups, categories, and topological spaces, we obtain the categories $GRP$, $CAT$, and $TOP$ of large groups, categories, and topological spaces.

Note that the theory could be algebraic, as for $Grp$, or essentially algebraic, as for $Cat$, or even higher-order, as for $Top$. We will refer to this as the logical enlargement. In practice, this enlargement usually suffices, but for theoretical reasons it would be nice to have a construction which works on any large category.

Locally presentable enlargement

For any large category $C$, the very large presheaf 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 Yoneda embedding $C \hookrightarrow [C^{op},SET]$ preserves all limits that exist in $C$.

However, it hardly preserves 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 preserved limit the colimits $\Phi$ (i.e. take them to limits in $SET$). Let’s denote this category by $\Uparrow_\Phi C$. 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. Since usually, the only colimits we might care about in a large category $C$ are small, a natural choice for $\Phi$ is the class of all small colimits existing in $C$.

We will refer to this $\Uparrow_\Phi C$ as the locally presentable enlargement, since it is a locally presentable category relative to $\mathbf{V}$. (In fact, it is locally $\kappa$-presentable, where $\kappa$ is the cardinality of the first universe $\mathbf{U}$.) Moreover, we have the following (where $\lambda Cts(A,B)$ denotes the category of $\lambda$-small-limit preserving functors).

Theorem

Suppose $C$ is locally presentable relative to $\mathbf{U}$, so that there is a small cardinal $\lambda$ and a small category $A$ with $\lambda$-small colimits such that $C\simeq \lambda Cts(A^{op},Set)$. If $\Phi$ denotes the class of all small colimits in $C$, then $\Uparrow_\Phi C \simeq \lambda Cts(A^{op},SET)$.

Note that in this case, $\lambda Cts(A^{op},SET)$ is the category of large models of the theory (namely $A^{op}$) of which $C$ is the category of small ones. Thus, the theorem says that when $C$ is locally presentable, the locally presentable enlargement is the same as the logical one.

Proof

Write $C' \coloneqq \lambda Cts(A^{op},SET)$. Then 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 $\Uparrow_\Phi C$.

However, if $C$ is not locally presentable, then its logical and locally-presentable enlargements can disagree. In fact, they will usually disagree in this case, since if $C$ is not locally presentable relative to $\mathbf{U}$, its logical enlargement will usually not be locally presentable relative to $\mathbf{V}$ either, whereas its locally-presentable enlargement is, by definition, locally presentable relative to $\mathbf{V}$. For example, the logical and locally presentable enlargements of $Top$ are quite different.

Accessible enlargement

There is a simple modification of the locally-presentable enlargement which makes it agree with the logical enlargement for all accessible categories, not just locally presentable ones. Namely, instead of $\kappa Cts(C^{op},SET)$, we consider $\kappa Flat(C^{op},SET)$, the category of $\kappa$-flat functors (functors that “would preserve all $\kappa$-small limits if they existed”). We call this the accessible enlargement.

Essentially the same proof as for the theorem above shows that if $C$ is accessible relative to $\mathbf{U}$, hence $C\simeq \lambda Flat(A^{op},Set)$, then the accessible enlargement is equivalent to $\lambda Flat(A^{op},SET)$, which is of course the logical enlargement.

Note that if $C$ has all small limits, then its accessible and locally-presentable enlargements are the same, since a functor defined on a category with $\kappa$-small limits is $\kappa$-continuous iff it is $\kappa$-flat. However, if $C$ does not have small limits, then its locally-presentable enlargement does have all large limits, whereas its accessible enlargement does not.

Thus, when $C$ is “poorly behaved,” its accessible enlargement is “closer to it” (in particular, identical to its logical enlargement if $C$ is accessible) and thus equally poorly behaved, whereas its locally-presentable enlargement is further from it, but better behaved (complete and cocomplete). For some purposes one may want one of these behaviors; for other purposes one may want the other.

Monoidal enlargements

One may further ask for an enlargement which inherits whatever additional structure $C$ has, such as monoidal structure. The logical enlargement will usually inherit such structure trivially, while the Day convolution theorem implies that when $C$ is monoidal, so is its locally-presentable enlargement—and moreover, the latter is closed even if $C$ is not!

Examples

• inter-universal Teichmüller theory

Related concepts

• universe polymorphism

References

• Sections 2.6 and 3.11–3.12 of Basic Concepts of Enriched Category.

• Blog post

• Zhen Lin Low, MO comment on universe enlargement