nLab
Grothendieck universe

Context

Foundations

Universes

Contents

Idea

Problems of set theory arise by unjustified recursion of the naive notion of “collection of things.” If “Col”s are one notion of collections (such as “set” or “class”), then the notion “Col of all Cols” is in general problematic, since it is subject to the construction of Russell-style paradoxes (although it’s not the only source of such paradoxes).

One way out is to consider a hierarchy of notions of collections: postulate that the collection of all “Col”s is not a “Col” itself, but is another notion of collection, a “Col+”: the “Col+ of all Cols”. Similarly, the collection of all “Col+”-type collections may be taken to be a “Col++”, and so on.

One formalization of this idea is that of a Grothendieck universe: this is defined to be a set UU which behaves like a “set+ of all sets” in that all the standard operations of set theory (union, power set, etc.) can be performed on its elements.

Although developed for application to category theory, the definition is usually given in a form that only makes sense in a membership-based set theory. On this page we consider only that version; for a form that makes sense in structural set theory, see universe in a topos.

I don't really like how this page looks now, since it focuses once more on the ‘evil’ set theory. I think that I'll rewrite it to talk about strongly inaccessible cardinal numbers first, which is a reasonable approach from any perspective, and then look at how you can say this more directly in both material and structural set theory. That means that I'll restore some text that was removed (not much of that was moved to universe in a topos), but this time I'll verify the correctness of the structural material (using universe in a topos as a guide) as I go. —Toby

Mike: I’m not sure I agree. Firstly, I actually think it is better to define what you really want—in this case, a collection of sets closed under the operations we need—and only later observe that it may be equivalent to something in terms of inaccessible cardinals. I also expect that the equivalence between universes and inaccessibles requires the axiom of choice, so wouldn’t it be better to separate them? Finally, this page is called “Grothendieck universe,” not inaccessible cardinal, so I think that here we should take the universes as primary, not their cardinals.

Toby: I disagree that this is is ‘what [we] really want’; the definition below is only what we want if we're using material set theory, which for the most part we aren't in the nLab. So it's really out of place to focus on that.

But I'm not sure that you understand my intention, either. In explaining the point of universe to Urs, or more generally in explaining the point in a structural way, I find the cardinal number the easiest way to get at what matters. It is, as I said before, the bottom line for any proposed (re)definition. So I intend to define Grothendieck universes, certainly, although I intend to define them in terms of cardinal numbers. (Or rather, in terms of isomorphism classes of sets, but using cardinal arithmetic, so I'll call those cardinal numbers.) The discussion of inaccessible cardinals would be only lemmatic (if that's a word); if I get around to doing it, then you'll see what I mean.

But here's another possibility: Maybe we should reserve this page for the strict notion in material set theory and make another page, say universe, for the non-evil concept. Then most (if not all) links here would really want to go there. I'm not sure that I like this, since ‘universe’ has other meanings, but maybe there's another term that we could use that doesn't conflict with this term? In any case, if you want something like that, then I can go along with it. What I really don't want is a bunch of links on the lab implying that category theorists deal with size issues using something that's fundamentally part of material set theory.

Mike: I can definitely get behind that last sentence. “Grothendieck universe” seems to be used pretty much everywhere with the material version in mind, but this is probably just because people don’t understand or trust structural set theory, and of course the structural version would be just as good for the purposes people use them for. So I think it would be okay if we include both the material and structural versions on this page. (I definitely don’t think we should include only the structural version.)

Of course the definition below is only correct in material set theory, but there is also a straightforward structural version that you wrote down, in terms of families, that makes no reference to cardinality. What I don’t see is why the cardinal number is “what matters” or “the bottom line for any proposed (re)definition.” It seems to me that what’s important for category theory is that we have a collection of sets, called “small,” which are closed under various constructions (power sets, indexed unions, etc.), so that the resulting category SetSet of small sets behaves the way we want it to. It’s completely irrelevant whether “small” is defined to mean “of cardinality less than κ\kappa” for some κ\kappa, or defined in some other way. If you assume the axiom of choice, then any collection of small sets closed under enough constructions will consist precisely of the sets of cardinality <κ\lt\kappa for some κ\kappa, but in the absence of choice I see no reason for that to be true. All of this is equally true materially and structurally.

(As an aside, I’m not so sure that on the nLab we “aren’t” using material set theory; rather, I think that practically everything we do is completely agnostic as to whether the foundation is material or structural. In fact, if you assume the axioms of choice and foundation, then the two are completely equivalent—a model of ZFC can be reconstructed, up to isomorphism, from its category of sets, and that category of sets is determined, up to equivalence, by its well-founded-set-objects.)

Mike: In fact, you yourself wrote at foundations:

I understand all these large cardinals much better in terms of their categories of small sets.

Definition

A Grothendieck universe UU is a pure set UU such that: 1. for all uUu \in U and tut \in u, tUt \in U (so UU is transitive); 1. for all uUu \in U, the power set P(u)UP(u) \in U; 1. the empty set U\empty \in U; 1. for all IUI \in U and functions u:IUu: I \to U, the union i:Iu iU\cup_{i: I} u_i \in U.

Some authors leave out (3), which allows \empty itself to be a Grothendieck universe. Other authors use the set \mathbb{N} of natural numbers in place of \empty in (3), which prevents the countable set V ωV_\omega of hereditarily finite sets from being a Grothendieck universe.

Consequences

From the definition above, one can prove additional closure properties of a universe UU, including the usual codings in pure set theory of function sets and cartesian products and disjoint unions of sets, using these lemmata:

Lemma

If tt is a subset of uu and uUu \in U, then tUt \in U.

Proof

By (2), P(u)UP(u) \in U; tP(u)t \in P(u), so tUt \in U by (1).

Lemma

If u,vUu, v \in U, then uvUu \cup v \in U.

Proof

Since U\empty \in U by (3), so are =P()\star = P(\empty) and TV=P()TV = P(\star) by (2). Even in constructive mathematics, 2={,}2 = \{\bot, \top\} is a subset of TVTV, so 2U2 \in U in by Lemma 1. Then (u,v)(\bot \mapsto u, \top \mapsto v) is a function 2U2 \to U, so the union uvu \cup v in UU by (4).

Then using their usual encodings in set theory: * the nullary cartesian product \star is P()P(\empty) as in the previous proof; * the binary cartesian product u×vu \times v is a subset of P(P(uv))P(P(u \cup v)); * the general cartesian product i:Iu i\prod_{i: I} u_i is a subset of P(I× i:Iu i)P(I \times \bigcup_{i: I} u_i); * the nullary disjoint union is \empty; * the binary disjoint union uvu \uplus v is a subset of 2×(uv)2 \times (u \cup v); * the general disjoint union i:Iu i\biguplus_{i: I} u_i is a subset of I× i:Iu iI \times \bigcup_{i: I} u_i; * the set of functions uvu \to v is a subset of P(u×v)P(u \times v).

Terminology: small/large

Given a universe UU, an element of UU is called a UU-small set, while a subset of UU is called UU-moderate. Every UU-small set is UU-moderate by requirement (1) of the definition. If the universe UU is understood, we may simply say small and moderate.

The term UU-large is ambiguous; it sometimes means ‘not small’ but sometimes means the same as ‘moderate’ (or ‘moderate but not small’). The reason is that language that distinguishes ‘small’ from ‘large’ in terms of sets and proper classes translates fairly directly into terms of UU-small and UU-moderate sets. To be precise, if we redefine “set” to mean “UU-small set,“ then every proper class in this new world of sets will be represented by a UU-moderate set (a subset of UU). Those sets that are not even UU-moderate are ‘too large’ to be translated into language of proper classes.

(Note, though, that not all UU-moderate sets represent proper classes in the language of set theory relative to the world of UU-small sets, only those that are first-order definable from UU-small sets. In fact, if κ\kappa is the cardinality of the universe UU, then there are only κ\kappa proper classes relative to UU, but there are 2 κ2^\kappa UU-moderate sets.)

As defined above, these concepts are evil, since two sets may be isomorphic yet have different properties with respect to UU. However, a set which is isomorphic to a UU-small or UU-moderate set is called essentially UU-small or UU-moderate; these concepts are non-evil.

Axiom of universes

If UU is a Grothendieck universe, then it is easy to show that UU is itself a model of ZFC (minus the axiom of infinity unless you modify (3) to rule out countable universes). Therefore, one cannot prove in ZFC the existence of a Grothendieck universe containing \mathbb{N}, and so we need extra set-theoretic axioms to ensure that uncountable universes exist. Grothendieck’s original proposal was to add the following axiom of universes to the usual axioms of set theory:

  • For every set ss there exists a universe which contains ss, i.e. such that sUs \in U.

This way whenever any operation leads one outside of a given Grothendieck universe (see applications below), there is guaraneteed to be a bigger Grothendieck universe in which one lands. In other words, every set is small if your universe is large enough!

Later, Mac Lane pointed out that often, it suffices to assume the existence of one uncountable universe. In particular, any discussion of ‘small’ and ‘large’ that can be stated in terms of sets and proper classes can also be stated in terms of a single universe UU (with ‘large’ meaning UU-moderate but not UU-small).

Large cardinals

If UU is a Grothendieck universe, then one can prove in ZFC that it must be of the form V κV_\kappa where κ\kappa is a (strongly) inaccessible cardinal (Williams). Here V κV_\kappa is the κ\kappa-th set in the von Neumann hierarchy of pure sets. Conversely, every such V κV_\kappa is a Grothendieck universe. Thus, the existence of Grothendieck universes is equivalent to the existence of inaccessible cardinals, and so the axiom of universes is equivalent to the “large cardinal axiom” that “there exist arbitrarily large inaccessible cardinals.”

It is worth noting, for those with foundational worries, that the axiom of universes is much, much weaker than many large cardinal axioms which are routinely used, and believed to be consistent, by modern set theorists. Of course, one cannot prove the consistency of any large cardinal axiom (if it really is consistent) except by invoking a stronger one.

Structural version

An equivalent concept (at least for the purposes of category theory) can also be defined in structural set theories (like ETCS). See universe in a topos.

Examples

The set V ωV_\omega of hereditarily finite sets (finite sets of finite sets of …) is a Grothendieck universe, unless you phrase axiom (3) in the definition to specifically rule it out. In this way, the axiom of infinity can be seen as a simple universe axiom (stating that at least one universe exists), and Mac Lane's axiom that an uncountable universe exists is merely one step further.

If you refrain from using the axiom of universes (except perhaps once, to get \mathbb{N} as above), then the set of all sets (or cardinal numbers) that you can actually construct is a Grothendieck universe. Of course, you cannot possibly have proved that this universe exists, but the intuition that you ought be able to form the collection of ‘everything that we've used so far’ is the justification for the axiom of universes.

Similarly, if you use the axiom of universes at most nn times, then the set of all sets that you can construct with this restriction is a Grothendieck universe. Thus, we can find a sequence U 1U 2U 3U_1\in U_2\in U_3\in \dots of universes. The axiom of replacement then allows us to form the union (a directed colimit) nU n\bigcup_n U_n. This will not be a universe (it violates (4), by definition), but we can use the axiom of universes again to show that it is in some universe U ωU_\omega. Proceeding in this way, we can construct a tower of universes indexed by the ordinal numbers.

The set of all sets that can be constructed using the axioms of ZFC together with the axiom of universes is, if it exists, again a universe which contains all the U αU_\alpha constructed above. Of course, it cannot be shown to exist using only ZFC and the axiom of universes; the axiom of universes is not the final word on large cardinal axioms by any means.

Applications

Let USetU Set be the category of UU-small sets, a full subcategory of Set. It is common, especially when UU is understood, to redefine SetSet to mean USetU Set; here we keep the distinction for clarity. But when SetSet means USetU Set, sometimes SETSET is used to mean the category of all sets.

A category whose set of morphisms is (essentially) UU-small may be called a UU-small category; it can also be thought of as an internal category in USetU Set. A category whose hom-sets are all (essentially) UU-small may be called locally UU-small; it can also be thought of as an enriched category over USetU Set. Every UU-small category is locally UU-small.

A category whose set of morphisms is UU-moderate may be called a UU-moderate category; again ‘UU-large’ may mean not UU-small, UU-moderate, or both. In practice, most UU-moderate categories are locally UU-small, and vice versa, but there is no theorem that this must be true. Note that USetU Set itself is UU-moderate and locally UU-small but not UU-small.

All notions of category theory that reference size, such as completeness and local presentability, must then be relativized to UU. In order to move from a category defined in one universe to another, we need a procedure of universe enlargement.

Presheaf categories

Let CC be a UU-small category. Then the category of UU-presheaves on CC (the functor category [C op,USet][C^{op}, U Set]) is also UU-moderate and locally UU-small but not UU-small unless CC is empty. (USetU Set itself is the special case of this where CC is the point.) These arguments go as follows:

  • UPSh(C)U PSh(C) is UU-moderate: An upper bound for the size of [C op,USet][C^{op}, U Set], hence of the set Obj([C op,USet])Obj([C^{op},U Set]) is the size of {F:Obj(C)×Mor(C)U}\{ F : Obj(C)\times Mor(C) \to U \} where both Obj(C)Obj(C) and Mor(C)Mor(C) are in USetU Set. So we are looking at the cardinal number |U| |u|×|v||U|^{|u| \times |v|} where u=Obj(C)u = Obj(C) and v=Mor(C)v = Mor(C). Use the fact that any Grothendieck universe must be infinite (since it has \emptyset, P()P(\emptyset), etc) and the result follows from cardinal arithmetic that κ λ=κ\kappa^\lambda = \kappa when λ<κ\lambda \lt \kappa and κ\kappa is infinite.

  • UPSh(C)U PSh(C) is locally UU-small: An upper bound for the size of the set of morphisms between two functors F,G:C opUSetF,G : C^{op} \to U Set is the disjoint union indexed by the objects cc of CC over the UU-sets G(c) F(c)G(c)^{F(c)}. Now G(c) F(c)UG(c)^{F(c)} \in U since it is a function set and cObj(C)G(c) F(c)\cup_{c \in Obj(C)} G(c)^{F(c)} by the assumption that unions stay in UU.

Now let CC be a UU-moderate category (and not small). Then the category of UU-presheaves on CC is not even locally UU-small, nor is it even UU-moderate (it is ‘too large’). However, it is locally UU-moderate. Also, it is quite possible, if CC is a UU-moderate site, that the category of UU-sheaves on CC is UU-moderate and locally UU-small.

Note: Here we are considering presheaves on CC with values in UU-small sets. In many cases, a more appropriate notion of “UU-small presheaf” is that discussed at small presheaf, namely a presheaf that is a UU-small colimit of representables.

Alternative approaches

  • A different, potentially much more elegant and natural proposal for solving the problem to be solved by Grothendieck universes is that described at category of all sets. Don’t get your hopes up too high, though; even if it works, it isn’t quite the category theory you’re used to.

References

The proof that a Grothendieck universe is equivalently a set of κ\kappa-small sets for κ\kappa an inaccessible cardinal is in

  • N. H. Williams, On Grothendieck universes, Compositio Mathematica, tome 21 no 1 (1969) (numdam)

Revised on September 11, 2012 23:40:30 by Urs Schreiber (131.174.190.72)