Re: [HoTT] A small observation on cumulativity and the failure of initiality

[Thomas Streicher <stre...@mathematik.tu-darmstadt.de>]

unnotated cumulativity just means that we syntactically omit the
inclusions of U_n into U_{n+1} but semantically they are there
and have to be inserted when interpreting syntax

that's similar to universes `a la Russell which are just a shorthand
for universes `a la Tarski

but what is true is that there are syntaxes where terms don't have
unique types, but those always consider terms together with a type

but generally in CS and logic one distinguishes between typing `a la
Church and `a la Curry, the first is used in ML-like type theories,
the latter when typing terms of out of an untyped collection of
preterms

Thomas


On Fri, Oct 13, 2017 at 10:03:06AM +0200, Peter LeFanu Lumsdaine wrote:
> On Thu, Oct 12, 2017 at 8:43 PM, Dimitris Tsementzis <dtse...@princeton.edu
> > wrote:
> 
> > Dear all,
> >
> > Let???s say a type theory TT is *initial* if its term model C_TT is initial
> > among TT-models, where TT-models are models of the categorical semantics of
> > type theory (e.g. CwFs/C-systems etc.) with enough extra structure to model
> > the rules of TT.
> >
> 
> I like the examples, but I would give a different analysis of what they
> tell us.
> 
> The definition of ???initial??? presupposes that we have already defined what
> ???TT-models??? means ??? i.e. what the categorical semantics should be.  There
> is as yet no proposed general definition of this (as far as I know).
> 
> Heuristically, there???s certainly a large class of type theories where we
> understand what the categorical semantics are, and all clearly agree.  But
> rules like un-annotated cumulativity are *not* in this class.  It???s not
> clear what should correspond to un-annotated cumulativity, as a rule in
> CwA???s (or CwF???s, C-systems, etc).  A certain operation on terms?  An
> operation, plus the condition that it???s mono?  An assumption that terms of
> one type are literally a subset of terms of the other?  Some of these will
> make initiality clearly false; others may make it true but very
> non-obviously so (that is, more non-obviously than usual).
> 
> So I don???t think we can say ???These theories aren???t initial.??? ??? but more
> like ???We???re not sure what the correct initiality statement is for these
> theories, and some versions one might try are false.???  But I definitely
> agree that they show
> 
> >  the claim that e.g. Book HoTT or 2LTT is initial cannot be considered
> obvious
> 
> ???p.
> 
> 
> 
> Then we have the following, building on an example of Voevodsky???s.
> >
> 
> 
> 
> 
> > *OBSERVATION*. Any type theory which contains the following rules
> > (admissible or otherwise)
> >
> > ?? |- T *Type*
> > ????????????????????????  (C)
> > ?? |- B(T) *Type*
> >
> > ?? |- t : T
> > ????????????????????????  (R1)
> > ?? |- t : B(T)
> >
> > ?? |- t : T
> > ????????????????????????  (R2)
> > ?? |- p(t) : B(T)
> >
> > together with axioms that there is a type T0 in any context and a term t0
> > : T0 in any context, is not initial.
> >
> > *PROOF SKETCH.* Let TT be such a type theory. Consider the type theory
> > TT* which replaces (R1) with the rule
> >
> > ?? |- t : T
> > ????????????????????????  (R1*)
> > ?? |- q(t) : B(T)
> >
> > i.e. the rule which adds an ???annotation??? to a term t from T that becomes a
> > term of B(T). Then the category of TT-models is isomorphic (in fact, equal)
> > to the category of TT*-models and in particular the term models C_TT and
> > C_TT* are both TT-models. But there are two distinct TT-model homomorphisms
> > from C_TT to C_TT*, one which sends p(t0) to pq(t0) and one which sends
> > p(t0) to qp(t0) (where p(t0) is regarded as an element of Tm_{C_TT} (empty,
> > B(B(T0))), i.e. of the set of terms of B(B(T0)) in the empty context as
> > they are interpreted in the term model C_TT).
> >
> > *COROLLARY. *Any (non-trivial) type theory with a ???cumulativity" rule for
> > universes, i.e. a rule of the form
> >
> > ?? |- A : U0
> > ????????????????????????  (U-cumul)
> > ?? |- A : U1
> >
> > is not initial. In particular, the type theory in the HoTT book is not
> > initial (because of (U-cumul)), and two-level type theory 2LTT as presented
> > here <https://arxiv.org/abs/1705.03307> is not initial (because of the
> > rule (FIB-PRE)).
> >
> > The moral of this small observation, if correct, is not of course that
> > type theories with the guilty rules cannot be made initial by appropriate
> > modifications to either the categorical semantics or the syntax, but rather
> > that a bit of care might be required for this task. One modification would
> > be to define their categorical semantics to be such that certain identities
> > hold that are not generally included in the definitions of
> > CwF/C-system/???-gadgets (e.g. that the inclusion operation on universes is
> > idempotent). Another modification would be to add annotations (by replacing
> > (R1) with (R1*) as above) and extra definitional equalities ensuring that
> > annotations commute with type constructors.
> >
> > But without some such explicit modification, I think that the claim that
> > e.g. Book HoTT or 2LTT is initial cannot be considered obvious, or even
> > entirely correct.
> >
> > Best,
> >
> > Dimitris
> >
> > PS: Has something like the above regarding cumulativity rules has been
> > observed before ??? if so can someone provide a relevant reference?
> >
> >
> >
> >
> >
>