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 
>