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Suppose we use Tarski-Grothendieck set theory as a foundation for category theory (ZFC together with the assumption that every set is contained in a Grothendieck universe, or equivalently that there is a proper class of inaccessible cardinals). It is sometimes necessary when working in this theory to pass to a larger Grothendieck universe. However, given Grothendieck universes $\mathscr U\subseteq\mathscr U'$, it is possible that $\mathscr U$ is not an elementary substructure of $\mathscr U'$: there exists a formula $\phi(x_1,\dots,x_n)$ in the language of set theory with parameters $x_1,\dots,x_n\in\mathscr U$ such that $\phi^{\mathscr U}(x_1,\dots,x_n)\leftrightarrow \phi^{\mathscr U'}(x_1,\dots,x_n)$ does not hold. For example, if $\mathscr U$ is the smallest Grothendieck universe, and $\mathscr U'$ is the second smallest, then $\mathscr U$ thinks that Grothendieck universes do not exist, whereas $\mathscr U'$ thinks that exactly one Grothendieck universe exists.

Question. Does this "lack of agreement" between Grothendieck universes ever present problems in category theory?

Some thoughts

One could "fix" this problem by assuming something stronger than Tarski-Grothendieck set theory. For example, one could postulate that there is a proper class $I$ of inaccessibles $\kappa$ such that $V_{\kappa}$ is an elementary submodel of $V$. This idea is presented in the following answer by Joel David Hamkins. (Postulating this within the framework of ZFC takes a little work, due to complications relating to Tarski's theorem on the undefinability of truth, but it can be done.)

However, for most "down to earth" category theory, using a stronger foundation than Tarski-Grothendieck set theory seems unnecessary. To take an example, suppose $\mathscr U$ is a universe, and $\mathcal C$ is a category such that $\DeclareMathOperator{\Hom}{Hom}\Hom_{\mathcal C}(X,Y)$ is isomorphic to an element of $\mathscr U$ for all objects $X$ and $Y$ in $\mathcal C$. Consider the question of whether a functor $F$ from $\mathcal C^{\mathsf{op}}$ to the category of $\mathscr U$-sets is representable. This question does not depend on the choice of $\mathscr U$, in the following sense: given any universe $\mathscr U'$ such that $\mathscr U\subseteq\mathscr U'$, the functor $F$ is representable if and only if $F$ is representable when viewed as a functor into the category of $\mathscr U'$-sets. More generally, I believe the paper Universes for category theory by Zhen Lin shows that a number of universal constructions are not "universe dependent".

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    $\begingroup$ In principle, there could be a problem. But I think usually there is no problem if you are careful enough to keep track of the universe parameters. $\endgroup$ Commented 2 days ago
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    $\begingroup$ They are certainly universe-dependent prima facie. The whole point is to show that they are not universe dependent despite initial appearances. $\endgroup$ Commented 2 days ago
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    $\begingroup$ @ZhenLin: My concern is that if in the middle of establishing some theorem about category theory, one passes to a larger universe, then usually one tacitly assumes that things which "were" true "stay" true. This is perhaps a naive viewpoint, but I could certainly imagine someone with less experience having it. I suppose what I'm really looking for is a deeper explanation/intuition about which statements "should" be universe independent, and which are not. $\endgroup$ Commented 2 days ago
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    $\begingroup$ @SimonHenry: In the examples of changing universe I am familiar with, this is in some sense just a technical device employed to make our categories small. But my question is whether passing to a larger universe could cause some other "change" which is undesirable or unwanted. I give an example above in my comment to Zhen Lin. $\endgroup$ Commented 2 days ago
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    $\begingroup$ This paper is all about the situation described in the beginning of the question: [2009.07164] Categorical large cardinals and the tension between categoricity and set-theoretic reflection arxiv.org/abs/2009.07164. $\endgroup$ Commented yesterday

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There are plenty of interactions between category theory and large cardinal combinatorics which can lead to inconsistent behavior as one passes between Grothendieck universes. Whether these count as legitimate 'complications' is pretty unclear to me, but I still think they're worth documenting here. (As I've discussed before on MathOverflow, I heavily suspect that the vast majority of uses of Grothendieck universes in practice could be removed with the teensiest bit of cardinal arithmetic and on some level I feel like this explains why you don't see things like, say, cohomology groups depending on what Grothendieck universe is used to construct a relevant sheaf topos.)

There are a couple of (arguably) positive statements that are equivalent to the non-existence of a measurable cardinal:

That means that there can be a Grothendieck universe $V_\kappa$ such that these statements hold but fail to hold in the next smallest Grothendieck universe $V_{\lambda}$ (specifically if $\kappa$ is the smallest measurable cardinal and $\lambda$ is the least inaccessible cardinal greater than $\kappa$).

There are statements (some positive and some negative) that are equivalent (or nearly equivalent) to the existence of a proper class of some kind of large cardinal.

When relativized to a Grothendieck universe $V_\kappa$, these statements correspond to $\kappa$ being an inaccessible limit of the relevant kind of large cardinal. This implies that the pattern of which Grothendieck universes satisfy conditions (1) and (2) can be quite complicated. For example:

  • There is no Grothendieck universe in which (1) and (2) both hold.
  • For any finite number $n$, it is consistent with Tarski-Grothendieck set theory that there are exacty $n$ Grothendieck universe $V_\kappa$ in which (2) holds.

Finally, Vopěnka's principle is famously equivalent to a lot of statements about the theory of locally presentable categories being particularly smooth (Adámek-Rosicky Chapter 6):

  • A category is locally presentable iff it is complete and has a small dense subcategory.
  • Each full subcategory of a locally presentable category closed under limits (resp. colimits) is reflective (resp. coreflective).
  • Each full embedding of an accessible category into a locally presentable category is accessible.
  • Every orthogonality class in a locally presentable category is a small orthogonality class.
  • Every subfunctor of an accessible functor is accessible.

Call all these statements (3). (3) holding in a Grothendieck universe $V_\kappa$ is now equivalent to $\kappa$ being a Vopěnka cardinal. Vopěnka cardinals are limits of extendible cardinals and extendible cardinals are limits of strongly compact cardinals, so again we get some complexity in the behavior of Grothendieck universes:

  • There is no Grothendieck universe in which (1) and (3) both hold. (Incidentally, I am not nearly competent enough at category theory to have intuition about whether this fact is obvious as stated.)
  • For any Grothendieck universe $V_\kappa$, if $V_\lambda$ is the next smallest Grothendieck universe, then in $V_\lambda$, (1) holds but (2) and (3) do not hold.
  • If there is a Grothendieck universe $V_\kappa$ in which (3) holds, then there are $\kappa$-many smaller Grothendieck universes in which (1) but not (2) and (3) hold and $\kappa$-many smaller Grothendieck universes in which (2) but not (1) and (3) hold. Moreover, it can consistently be the case that (3) fails to hold in all smaller Grothendieck universes (i.e., if $\kappa$ is the least Vopěnka cardinal), or that (3) (and therefore (2) but not (1)) holds for precisely $n$ smaller Grothendieck universes for some given finite $n$, or that (3) also holds for $\kappa$-many smaller Grothendieck universes.
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    $\begingroup$ Hi James, regarding your "incidentally" parenthetical, yes, the inconsistency of the category-theoretic phrasings of the two statements can be shown without any large-cardinal assumptions: if you has "complete and small dense subcategory implies l.p." along with "$\mathsf{Set}^{\mathrm{op}}$ has a small dense subcategory", then of course $\mathsf{Set}$ would be both l.p. and co-l.p., but that would make $\mathsf{Set}$ a thin category, by a theorem of Gabriel-Ulmer given as 1.64 in Adamek-Rosicky. $\endgroup$ Commented 14 hours ago
  • $\begingroup$ @KevinCarlson Is the inconsistency of (1) and (2) similarly easy to show? $\endgroup$ Commented 14 hours ago
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    $\begingroup$ No, not that I can see. $\mathsf{Set}^{\mathrm{op}}$ has no nontrivial accessibly embedded accesible subcategories at all, nor are any of its accessibility, co-accessibility, well-poweredness, or co-well-poweredness open to foundational question, so it seems rather surprising that these conditions affect each other from a CT viewpoint. $\endgroup$ Commented 12 hours ago

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