Homotopy Type Theory universe > history (Rev #1)

There are several kinds of universes that one can consider in type theory.

Universes a la Russell

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

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

With universes a 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 book and by Coq?.

Universes a la Tarski

A universe a la Tarski 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 a propositional equality?,we may speak of a weakly a la Tarski universe.

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