nLab
type of types

Idea

In type theory, a type of (small) types – usually written $Type$ – is a type whose terms are themselves types. Thus, it is a universe of (small) types, a universe in type theory.

One also speaks of $Type$ as being a reflection of the type system in itself (e.g. MartinLöf 74, p. 6, Palmgren, pp. 2-3, Rathjen, p. 1, Luo 11, section 2.5, Luo 12, p. 2, Stanf. Enc. Phil.), following the reflection principle in set theory.

In homotopy type theory a type of (small) types is what in higher categorical semantics is interpreted as a (small) object classifier. Thus, the type of types is a refinement of the type of propositions which only contains the (-1)-truncated/h-level-1 types (and is semantically a subobject classifier).

In the presence of a type of types a judgement of the form

$$\vdash A : Type$$

says that $A$ is a term of type $Type$, hence is a (small) type itself. More generally, a hypothetical judgement of the form

$$x : X \vdash A(x) : Type$$

says that $A$ is an $X$-dependent type.

In homotopy type theory the type of types $Type$ is often assumed to satisfy the univalence axiom. This is a reflection of the fact that in its categorical semantics as an object classifier is part of an internal (∞,1)-category in the ambient (∞,1)-topos: the one that as an indexed category is the small codomain fibration.

Per Martin-Lof‘s original type theory contained a type of all types, which therefore in particular contained itself, i.e. one had $Type : Type$. But it was pointed out by Jean-Yves Girard that this was inconsistent; see Girard's paradox. Thus, modern type theories generally contain a hierarchy of types of types, with $Type_0 : Type_1$ and $Type_1 : Type_2$, etc.

Formalizations

Type Universe à la Russell

A universe à la Russell is a type whose terms are types. In the presence of a separate judgment “$A \;type$”, this can be formulated as a deduction rule of the form

$$\frac{A:U}{A \;type}$$

Thus, the type formers have rules saying which universe they belong to, such as:

$$\frac{A:U\quad B:A\to U}{\Pi\, A\, B : U}$$

With universes à la Russell, we can also omit the judgment “$A\; type$” and replace it everywhere by a judgment that A is a term of some universe. This is the approach taken by the HoTT textbook and by Coq.

Type Universe à la Tarski

A universe à la Tarski (Hofmann, section 2.1.6, Luo 12, Gallozzi 14, p. 40) is a type $U$ together with an “interpretation” operation allowing us to regard its terms as codes or names for actual types. Thus we have a rule such as

$$\frac{A:U}{El(A)\;type}$$

saying that for each term $A$ of the type universe $U$ there is an actual type $El(A)$. (Conversely, with notation as used at object classifier in an (infinity,1)-topos, one might write $A = 'El(A)'$ to indicate that $A$ is the name of the type $El(A)$ in the type universe.)

We usually also have operations on the universe corresponding to (but not identical to) type formers, such as

$$\frac{A:U\quad B:A\to U}{\pi(A, B) : U}$$

with an equality $El(\pi(A,B))=\Pi \, El(A)\, El(B)$. Usually this latter equality (and those for other type formers) is a judgmental equality. If it is only an equivalence (i.e. we have a rule which gives us a canonical term of the equivalence type), we may speak of a weakly à la Tarski universe (Gallozzi 14, p. 49-50).

We can give a slightly different definition of weakly à la Tarski universe using propositional equality and a larger universe. More precisely, we can consider two (or many) universes $U$ and $U'$ with the usual rules for the relative reflection $el(a):U'$ for any $a:U$, a choice of weakly or strongly a la Tarski computation rules for the reflections $El$ and $El'$, and a computation rule for the relative reflection el of $U$ inside $U'$ based on propositional equality, which gives us canonical elements of the identity types $Id_{U'}(\pi'(el(a),el(b)),el(\pi(a,b)))$ and similarly for the other type formers.

If the containing universe is univalent the two definitions turn out to coincide.

Universes defined internally via induction-recursion are (strongly) à la Tarski. Weakly à la Tarski universes are easier to obtain in semantics (see below): they are somewhat more annoying to use, but probably suffice for most purposes.

Properties

Universe enlargement

Both Coq and Agda support universe polymorphism to deal with the issue of universe enlargement. Moreover, Coq supports typical ambiguity.

Categorical semantics

Univalent type universes à la Russell have been shown to be interpreted in type-theoretic model categories presenting the base (∞,1)-topos ∞Grpd

(Kapulkin-Lumsdaine-Voevodsky 12) and more generally presenting (∞,1)-toposes of (∞,1)-presheaves over elegant Reedy categories (Shulman 13).

Discussion for general (∞,1)-toposes (of (∞,1)-sheaves) that should have implementation weakly à la Tarski (Gallozzi 14, p. 49-50) is in (Gepner-Kock 12).

For more on this see the respective sections at relation between type theory and category theory.

Related concepts

• resizing axiom

• universe polymorphism

• univalence

• Prop, the type of propositions,

  • subobject classifier

• object classifier

• parametric polymorphism

• Girard's paradox

• Awodey's conjecture

References

Some of the text above is adapted from the entry universe at the homotopy type theory web.

Type universes in Martin-Löf type theory originate around

• Per Martin-Löf, An intuitionistic theory of types: predicative part, In Logic Colloquium (1973), ed. H. E. Rose and J. C. Shepherdson (North-Holland, 1974), 73-118. (web)

Basic discussion of the syntax of type universes is in

• Erik Palmgren, On Universes in Type Theory (pdf)

Definition of type universes weakly à la Tarski is in

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

• Zhaohui Luo, Notes on Universes in Type Theory, 2012 (pdf)

• Cesare Gallozzi, Constructive Set Theory from a Weak Tarski Universe, MSc thesis (2014) (arXiv:1411.5591)

Detailed discussion of the type of types in Coq is in

• Adam Chlipala, Certified programming with dependent types – Library Universes

See also around slide 8 of the survey

• Frade, Calculus of inductive constructions (2008/2009) (pdf)

A formal proof in homotopy type theory that the type of homotopy n-types is not itself a homotopy $n$-type (it is an $(n+1)$-type) is in

• Nicolai Kraus, C. Sattler, The universe $\mathcal{U}_n$ is not an $n$-type May 2013 (pdf)

(∞,1)-Categorical semantics for univalent type universes is discussed in

• Chris Kapulkin, Peter LeFanu Lumsdaine, Vladimir Voevodsky, The Simplicial Model of Univalent Foundations (arXiv:1211.2851)

• Michael Shulman, The univalence axiom for elegant Reedy presheaves (arXiv:1307.6248)

• David Gepner, Joachim Kock, Univalence in locally cartesian closed ∞-categories (arXiv:1208.1749)

Relation to injective types:

• Martín Escardó, Injectives types in univalent mathematics (arXiv:1903.01211)

See also

• Michael Rathjen, The strength of Martin-Löf type theory with superuniverse. Part I pdf

• Stanford Encyclopedia of Philosophy, Type theory – Extensions of type systems, Polymorphism, Paradoxes

• Zhaohui Luo, Contextual analysis of word meanings in type-theoretical semantics, in Pogodalla, Prost (eds.) Logical Aspects of Computational Linguistics, 2011 (pdf)