nLab universe

This entry is about the notion in mathematics/logic/type theory. For the notion of the same name in physics see at observable universe.

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Idea

A universe is a realm within which (some conceived part, naively and virtually all, of) mathematics may be thought of as taking place. Universes can be purely metamathematical, but we can also reflect upon them and bring them into mathematics. There are several different kinds of ‘universes’. For a physical notion of universe see observable universe.

Categories as universes

Much of ordinary mathematics can be thought of as taking place inside “the archetypical category SET of sets”. Typically, the properties of $SET$ are formulated in first-order logic using a set theory such as ZFC or (more directly) ETCS.

We can generalise this from $SET$ to any other category $C$. Without further assumptions on the category, there is in general very little mathematics that can be formulated inside it, but a few extra properties and structures are usually sufficient to provide something interesting. This is the general topic of internalisation. The attitude to take is that any specific category is merely one model, while a general class of categories is a theory (really a doctrine, or 2-theory).

We can also use higher categories instead of mere categories here. Even for ordinary mathematics, this means starting with $\infty$-GRPD instead of $SET$.

Universes as classes

The idea of the large category $SET$ as the universe of mathematics has an analogue in class theory. In a material class theory, the von Neumann universe $V$ is the class of all well-founded pure sets.

More explicitly: for every ordinal number $\alpha$, we have a set $V_\alpha$ (the von Neumann universe of rank $\alpha$), defined recursively using the operations of power set and (material) union as

$$V_\alpha \coloneqq \bigcup_{\beta \lt \alpha} \mathcal{P}V_\beta .$$

Then $V$ itself is the union of all of the $V_\alpha$.

A similar but more complicated definition allows us to define the universe $L$ of constructible sets, called the constructible universe.

See also a Wikipedia article written largely by Toby Bartels in another lifetime.

In a structural class theory, such as the category with class structure found in algebraic set theory, a universe is a class $U$ with a monic class map $\mathcal{P}(U) \hookrightarrow U$ from the powerclass of $U$ to $U$. In particular, every universal class is a universe.

Grothendieck universes

A Grothendieck universe is a set $U$ such that the following properties hold for $U$:

1. if $x \in a \in U$, then $x \in U$;
2. if $a, b \in U$, then $\{a, b\}$ and $a\times b$ are elements of $U$;
3. if $a \in U$, then $\cup a$ and $P(a)$ are elements of $U$;
4. the set $\omega$ of all natural numbers is an element of $U$;
5. if $f:a \to b$ is surjective with $a \in U$ and $b \subseteq U$, then $b \in U$.

Universes inside $SET$

If set theory is the foundation of mathematics, then the universes above are part of metamathematics rather than mathematics itself. However, we can also look for small categories or sets that exist within set theory and have the properties of a universe.

There is already a hint of this in the hierarchy $V_\alpha$ out of which the von Neumann universe is built. For some values of $\alpha$, $V_\alpha$ might be a sufficiently large and complete collection of sets in which to do ordinary mathematics. From the nPOV, we can instead look at the category $Set_\alpha$ of these sets and the functions between them, although it's more common to think about $Set_\kappa$ for some cardinal number $\kappa$.

An infinite set is a simple example, as finite mathematics can be done inside it. Going beyond this, a Grothendieck universe is a set within which other infinite sets exist but which is complete under the operations of material set theory that are needed for ordinary mathematics. The structural analogue is a universe in the topos $SET$.

In general, we need some axioms to state the existence of such universes; we can think of these as large cardinal axioms. (The existence of an infinite set is the axiom of infinity; the existence of a Grothendieck universe is the existence of an inaccessible cardinal.)

The use of such universes is convenient when we want to work with large categories “as if they were small.” That is, if we redefine “small” to mean “element of some fixed universe,” then the categories of all small structures of some sort will still be sets (rather than proper classes) in the bigger ambient universe, and thus we can for instance form functor categories between them freely. We do sometimes then need a way to “move a category” from one universe to another; see universe enlargement.

Reflection

In logic, we use reflection principles (see Wikipedia) to systematically identify features of our meta-universe and see what is needed to prove the existence of an internal universe with these features.

This follows the following outline:

1. We assume that we understand a priori what it means to use first-order logic (or some other finitary base logic).

2. Using this base logic, we can formulate a foundational set theory that describes a meta-universe such as $SET$.

3. Of course, $SET$ cannot be described from inside itself. But there may be objects in $SET$ (internal categories) that look like $SET$ itself.

4. We add an additional axiom to our set theory stating that such objects, the internal universes, exist. These are then models of our original set theory.

5. Now we are using a new, stronger set theory; repeat.

Universes in type theory

Set theory is not the only foundation of mathematics. For example, there are also various foundational type theories, closely related to structural set theory. Then we have a meta-universe of all types, and we can also add axioms for the existence of internal type universes.

Related concepts

• constructible universe

• universe enlargement

• universe polymorphism

• type of types

  • subobject classifier, object classifier, partial map classifier

• reflective subuniverse

• G-universe

References

General

For universes in class theory and algebraic set theory, see

• Steve Awodey. Notes on algebraic set theory, Notes for lectures given at the Summer School on Topos Theory, Haute-Bodeux, Belgium. May 29 to June 5, 2005. Carnegie Mellon University Technical Report No. CMU-PHIL-170. June 2005. (pdf)

In type theory

• Martin Hofmann, section 2.1.6 of Syntax and semantics of dependent types (web)