Large categories vs. $\mathrm{U}$-categories: why is the loss of category-theoretic information inessential?

Question

I've asked a related question about nine months ago here, however, apparently, I lacked expertise to ask the precise question I want to ask here, as I wish to revisit the matter of universes. I hope it will not count as a duplicate.

One way to deal with set-theoretic difficulties when studying large categories are Grothendieck universes. In practice, it means that instead of studying, for example, the category of all abelian groups $\mathrm{Ab}$ we "fix" an arbitrary universe $\mathrm{U}$ and study the category $\mathrm{U\text{-}Ab}$ of all abelian groups which belong to $\mathrm{U}$. As $\mathrm{U}$ was arbitrary, everything we prove about $\mathrm{U\text{-}Ab}$ will be true for all categories $\mathrm{V\text{-}Ab}$ for any universe $\mathrm{V}$.

The thing that bothers me is: how can we circumvent the issue that we're not working with all abelian groups. Sure, adopting the axiom of universes ("every set is contained in some universe"), for every abelian group $G$ we have a an inverse $\mathrm{U}$ containing it, and that means that we have a category $\mathrm{U\text{-}Ab}$ which contains $G$ an object. But it doesn't solve all issues, at least not right away.

Mike Shulman in his notes says:

But then I don't understand why we can get away with this. Before reading this I though that any property of a $\mathrm{U}$-category can be directly translated to the "full" category, but it appears to be not the case. Why can we study, for examply, the category of $\mathrm{U}$-small algebraic varieties in algebraic geometry instead of the "full" category of all algebraic varieties? Or the category of compactly generated $\mathrm{U}$-small topological spaces in algebraic topology? Or the category of $\mathrm{U}$-small vector spaces over a field $F$?

Also, in this comment to his answer to my aforementioned question HeinrichD said that

"You won't be able to do many constructions with this "entire" category. And no, we don't really have to look at them. Also, it is common practice to just write "groups" when one uses U-groups if the context permits this."

My question here is why doesn't one need to study the "full" categories of schemes, topological spaces, vector spaces, symmetric spectra (and of any set satisfying a property $\phi(x)$, for that matter) together with a property for what it means to be a morphism between $x$ and $y$ such that $\phi(x)$ and $\phi(y)$, and can get away with studying they $\mathrm{U}$-versions (of $\mathrm{U}$-small sets $x$ satisfying $\phi(x)$ and so on...).

For any large category $\mathrm{C}$, does its version "restricted" to (an arbitrary) $\mathrm{U}$ suffice?

$\mathrm{P.S.}$ In the linked notes, apparently, Mike Shulman wrote a section concerning those limitations of universe, but his exposition is well over my head and seems to be aimed at logicians or at least logic students, besides, he seems to propose another foundation for large categories, from what I understand, an extension of $\mathrm{ZFC\text{+}U}$.

Answer 1

Let me try to answer as a set theorist, rather than as a category theorist, since I think that your question concerns at bottom a matter often considered in set theory.

Namely, the essence of your question, to my way of thinking, revolves around the fact that Grothendieck universes (or Grothendieck-Zermelo universes, as one might call them) need not all agree with each other or with the background ambient set-theoretic universe $V$ on mathematical assertions. Something can be true in one universe and false in another.

First of all, let me point out that indeed, this is the case. For example, if $\kappa$ is the least inaccessible cardinal, then $V_\kappa$ is the smallest Grothendieck universe, and one of the statements that is true in $V_\kappa$ and not true in any larger Grothendieck universe is the assertion, "there are no Grothendieck universes." This will cause certain kinds of limits to behave differently in $V_\kappa$ than in larger universes. For example, Easton-support limits are the same as inverse limits in this $V_\kappa$, but not in any larger universe, and I believe that one can translate this to category-theoretic instances which would make this distinction important.

Similarly, if $\kappa_1$ is the next inaccessible cardinal, then $V_{\kappa_1}$ will think that there is precisely one Grothendieck universe, but no other Grothendieck universe will think that this statement is true, and so again the truth differs.

In order to resolve this issue, what you really want is not merely a hierarchy of Grothendieck universes, but an elementary chain of universes $$V_{\gamma}\prec V_{\delta}\prec\cdots\prec V_{\lambda}\prec\cdots$$ where $\prec$ refers to the relation of elementary submodel. What $V_\gamma\prec V_\delta$ means is that any statement $\varphi(a)$ expressible in the first-order language of set theory that is true in $V_\gamma$ will also be true in $V_\delta$. Ultimately, with an elementary chain, all the universes agree on the truth of any particular assertion, and furthermore they agree with the full ambient background universe $V$ on such truths.

If one has a proper class $I$ of cardinals $\delta$ that form such an elementary chain $V_\delta$ for $\delta\in I$, then you can use these Grothendieck universes and be confident that any statement true in one of them is true in all of them (or all of them that contain the objects about which the statement is made, if there are parameters in your statement). This feature would apply to any statement made in the language of set theory. (But it would not apply to statements made in the language with the class $I$, since of course, the least element of $I$ will again exhibit the phenomenon as before.)

The consistency strength of having such an elementary chain is a slight upgrade from the Grothendieck universe axiom, but still strictly below the existence of a Mahlo cardinal, which is still rather low in the large cardinal hierarchy. Let us formalize it with a definition.

Definition. The elementary-chain universe axiom is the assertion that there is a proper class $I$ of inaccessible cardinals that form a chain of elementary substructures, in the sense that the truth of any particular formula $\varphi(a)$ with parameters is the same in all $V_\gamma$ for $\gamma\in I$.

It follows by a standard result in model theory theory that every $V_\gamma$ also agrees with truth in $V$ for these assertions $\varphi(a)$.

The axiom is stated as a scheme, with a separate assertion for each $\varphi$. One should think of this axiom as a large cardinal axiom, and we can place it into the hierarchy of large cardinal strength as follows.

Theorem. The following theories are equiconsistent over ZFC.

1. The elementary-chain universe axiom.

2. The assertion "Ord is Mahlo", which asserts that every definable closed unbounded class of ordinals contains a regular cardinal.

Proof. ($1\to 2$) If we have an elementary chain and $C$ is a definable club in the ordinals, then above the parameters used in the definition, $C$ will be unbounded in every element of $I$ and therefore contain all elements of $I$ above those parameters. Thus, $C$ will contain a regular cardinal.

($2\to 1$) If ZFC + "Ord is Mahlo" is consistent, then consider the theory $T$ asserting $I$ is an unbounded class of inaccessible cardinals and $V_\delta\prec V$ for each $\delta\in I$. The assertion $V_\delta\prec V$ is asserted as a scheme, with a separate statement asserting agreement between $V_\delta$ and $V$ for each formula $\varphi$ separately. This theory is finitely consistent, since every formula reflects on a club of ordinals, which must contain an unbounded proper class of inaccessible cardinals by the assumption "Ord is Mahlo". Thus, it is consistent. And so we have a model of statement 1. $\Box$

It follows as an immediate corollary that if $\kappa$ is a Mahlo cardinal, then there is a class $I\subset\kappa$ making $\langle V_\kappa,\in,I\rangle$ a model of the elementary chain universe axiom. Thus, the strength of this axiom is strictly less in consistency strength than the existence of a Mahlo cardinal.

A stronger version of the elementary chain universe axiom would be to assert that there is a unbounded class $I$ of inaccessible cardinals for which $V_\gamma\prec V_\delta$ for any $\gamma<\delta$ in $I$. This can be stated as a single axiom, not a scheme, in Gödel-Bernays set theory GBC. It is very similar to the axiom above, but strictly stronger, since it implies the existence of a truth-predicate for first-order truth. The difference between this axiom and the one above is whether or not you get elementarity $V_\gamma\prec V_\delta$ in nonstandard models of the theory, that is, with respect to nonstandard formulas.

Conclusion. If you assume the elementary-chain universe axiom, and prove a set-theoretically expressible statement true in an arbitrary Grothendieck universe, then it will be true in the full ambient set-theoretic universe as well.

Answer 2

Zhen Lin Low's paper Universes for category theory (arXiv:1304.5227) is perhaps a great start. The abstract is:

The Grothendieck–Verdier universe axiom asserts that every set is a member of some set-theoretic universe U that is itself a set. One can then work with entities like the category of all U-sets or even the category of all locally U-small categories, where U is an “arbitrary but fixed” universe, all without worrying about which set-theoretic operations one may legitimately apply to these entities. Unfortunately, as soon as one allows the possibility of changing U, one also has to face the fact that universal constructions such as limits or adjoints or Kan extensions could, in principle, depend on the parameter U. We will prove this is not the case for adjoints of accessible functors between locally presentable categories (and hence, limits and Kan extensions), making explicit the idea that "bounded" constructions should not depend on the choice of U.

What this means is that for large categories that are built in a sensible way from small categories, certain universal constructions (like limits) in a 'truncated' version of the large category relative to a universe keep their universal property after being included to a larger truncation.

There are also slides of a talk by Low on the same topic that are a bit more ... accessible. I want to point out the concept of "weak universe" suggested at the end of the talk is more or less the same thing as some stage $V_\alpha$ in the cumulative hierarchy where $\alpha$ is some limit ordinal (and also not too different to a model of ETCS - although I'm deliberately being vague here). Such a $V_\alpha$ is a more general (weaker) notion than the $\mathbf{U}$ in your question. No results are given regarding weak universes, but keep them in mind for the rest of this answer.

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From the point of view of algebraic geometry, the Stacks Project goes to great lengths to make sure everything is a set. In particular the proof of the Reflection principle is unwound (or reverse engineered) to show how instances of it apply in all the cases of interest for categorical constructions involving schemes. Note that the Reflection principle applies involves the sets $V_\alpha$ for limit $\alpha$, hence weak universes. The trick is to write down an explicit (super)exponential function $Bound$ in cardinals (like $\kappa^\kappa$ or $\kappa^{2^\kappa}$) that bounds the growth of the cardinality of the sets necessary to build the various scheme-theoretic constructions on wants, and then use this to show the (small!) category $Sch_\alpha := Sch_{V_\alpha}$ is closed under said constructions (see eg Tag 000R). So for example, one can get existence of pullbacks and $\leq\lambda$-sized colimits (for a regular cardinal $\lambda$ - the notes prove the case of $\lambda = \aleph_0$; a simple modification of the bounding function allows for larger but fixed $\lambda$). Also treated are sites, which of course need to be small to make sheaf categories locally small.

All of the above would be in vain, but for the fact that all the constructions one wants to show are in your small category of schemes are in fact performed in the large category $Sch$ of all schemes, and then the theorem shows that these are actually in the (essential) image of $Sch_\alpha \hookrightarrow Sch$, for some appropriate $\alpha$. Given any concrete problem at hand (for instance, discussing the geometry of the compactified moduli stack of $n$-pointed genus-$g$ curves) one only ever starts with a set's worth of data, and so there exists some limit ordinal $\alpha$ such that no consideration of anything other than $Sch_\alpha$ is necessary. In particular, not even full universes are needed in this instance, although using universes might make the book-keeping of the proofs simpler.

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If you are more interested in differential geometry, then the category of all finite-dimensional manifolds (of your preferred flavour) is already essentially small. This is due to the Whitney embedding theorem.

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From a topological point of view, many interesting categories of spaces (even if not excessively general) are already essentially small, while being closed under a reasonable class of (co)limits. For instance, second-countable paracompact Hausdorff spaces all embed in the Hilbert cube $[0,1]^{\aleph_0}$, hence the category of such is essentially small. Polish spaces are another example of spaces embeddable in the Hilbert cube. One could perform a similar analysis to the Stacks Project for topological spaces in any given setting to find out which constructions allow one to stay entirely within a small category.

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Lastly, it should be worth pointing out that in the category of sets, one actually has reasonably canonical constructions of lots of things, for instance a limit can be given as a specific set of natural transformations, rather than just positing that such a set exists with some universal property. Likewise for colimits; given powersets (which in ZF(C) are unique, for instance) there is a construction of a colimit using Separation as a subset of an appropriate powerset. Then the universal property follows from the construction, so the comparison to arbitrary-sized sets is no longer a problem (in the sense that, going up a universe cannot destroy the universal property). Then for various categories of algebras for monads on $Set$, unless things are really weird (laptop is running low on battery, I've not time to remind myself if it's always ok: you might need the monad to have a rank) these type of results lift, if I'm not mistaken.

Answer 3

I think there can be no easy answer to your question or to be exact I'm afraid that there are many different reasons for using universes instead of sticking with just one foundational set theory.

I will limit myself to provide two such reasons.

For start when your doing your math (algebra, topology, algebraic geometry...) using a set theory (for instance ZFC) as underlying theory you are actually working with small objects for a fixed universe: this universe is a model of ZFC axioms where the underlying sets of your structures live.

As a second answer allow me to point out that there are lots of categorical results and constructions that work only with small-categories.

Examples of these results are for instance:

• presheaves categories are closed under small-colimits/limits
• every small-category can be embedded in $\mathbf{Set}$
• every presheaf over a small-category is a (small)-colimit of representable presheaves.

The only proofs I've seen of these results require the smallness condition. By working with $U$-small categories these results and constructions generalize to these categories. So if you consider only $U$-small categories these results and constructions become available for all categories.

Clearly, if one can live without these results then they can drop the smallness condition, it's a matter of what do you prefer, but again so does choosing the axioms of your foundations.

Hope this helps.