(in category theory/type theory/computer science)
of all homotopy types
of (-1)-truncated types/h-propositions
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
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.
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
says that $A$ is a term of type $Type$, hence is a (small) type itself. More generally, a hypothetical judgement of the form
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.
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
Thus, the type formers have rules saying which universe they belong to, such as:
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.
A universe à la Tarski (Hofmann, section 2.1.6) is a type together with an “interpretation” operation allowing us to regard its terms as types (or “codes for types”). Thus we have a rule such as
We usually also have operations on the universe corresponding to (but not identical to) type formers, such as
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.
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 computational 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.
Both Coq and Agda have systems to manage universe sizes and universe enlargement automatically; Agda’s is more advanced (universe polymorphism), whereas Coq’s is good enough for many purposes but tends to produce “universe inconsistencies” when working with univalence.
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 is in (Gepner-Kock 12).
For more on this see the respective sections at relation between type theory and category theory.
Prop, the type of propositions,
Some of the text above is adapted from the entry universe at the homotopy type theory web.
Basic discussion of the syntax of type universes is in
Definition of type universes weakly à la Tarski is in
Detailed discussion of the type of types in Coq is in
See also around slide 8 of the survey
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
(∞,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)