In the set-thoretic model -- which is the simplest, most "standard" model one can think of -- the universes are indeed cumulative, and "coherence" is just the observation that conversion (definitional equality) of raw terms is preserved by the interpretation function.
In categorical models, strict equality between interpreted objects is perhaps a more subtle concept.
Still, it would seem to be a natural requirement of *ANY* class or flavor of semantics, that expressions which are definitionally equal in the object language, are evaluated to entities which are again (judgmentally) equal on the meta-level.
This really amounts to nothing more than asking that the semantics in question actually validate the conversion rule -- one of the structural rules of the theory.
Best,
Andrew
On Sunday, October 15, 2017 at 11:26:22 AM UTC+2, Thomas Streicher wrote:
> A quite detailed interpretation of type theory in set theory is presented
>
> in the paper of Peter Aczel "On Relating Type Theories and Set Theories".
>
> On can define directly the interpretation of lambda terms (provided that abstraction is
>
> typed), and using set theoretic coding, one can use a global application operation
>
> (so that the interpretation is total). One can then checked by induction on derivations
>
> that all judgements are valid for this interpretation.
Thanks, Thierry, for pointing this out. But Peter's method does not
extend to arbitrary split models of dependent type theory. What Peter
uses here intrinsically is that everything is a set since otherwise he
couldn't interpret type theoretic quantification.
My interpretation got partial on pseudoexpressions since pseudoterms
can't be understood as pseudo-type-expressions.
I don't see how Peter's method extends to interpretation of
realizability or (pre)sheaf models of dependent type theories not to
speak of arbitrary contextual cats or other split categorical models
of dependent type theories.
Thomas