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

[Steve Awodey <steve...@gmail.com>]

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> On Oct 13, 2017, at 11:50 AM, Michael Shulman <shu...@sandiego.edu> wrote:
> 
> On Thu, Oct 12, 2017 at 5:09 PM, Steve Awodey <steve...@gmail.com <mailto:steve...@gmail.com>> wrote:
>> in order to have an (essentially) algebraic notion of type theory, which
>> will then automatically have initial algebras, etc., one should have the
>> typing of terms be an operation, so that every term has a unique type. In
>> particular, your (R1) violates this. Cumulativity is a practical convenience
>> that can be added to the system by some syntactic conventions, but the real
>> system should have unique typing of terms.
> 
> I'm not convinced of that.  When we define the syntactic model, a
> morphism from A to B (say) is defined to be a term x:A |- t:B, where
> the types A and B are given.  So it's not clear that it matters
> whether the same syntactic object t can also be typed as belonging to
> some other type.  I thought that the fundamental structure that we
> induct over to prove initiality is the *derivation* of a typing
> judgment, which includes the type that the term belongs to: two
> derivations of x:A |- t:B and x:A |- t:C will necessarily be different
> if B and C are different.  In an ideal world, a judgment x:A |- t:B
> would have at most one derivation, so that we could induct on
> derivations and still consider the syntactic model to be built out of
> terms rather than derivations.  If not, then we need a separate step
> of showing that different derivations of the same judgment yield the
> same interpretation; but still, it's not clear to me that the
> simultaneous derivability of x:A |- t:C is fatal.

well, good luck with that : - )

I’m just saying that if you want to represent type theory in an essentially algebraic form — so that you automatically know you have free algebras, finitely-presented ones, products, sheaves of algebras, etc. — then typing of terms should be an operation.

sure, it may be that there are other ways to get the syntactic category to be initial w/resp. to some other notion of morphisms, but the algebraic approach is how it’s done for other categorical logics, like topos, CCC, coherent category, etc.  I think Peter Dybjer has also shown explicitly that this works for CwFs, too.

Steve

> 
> Moreover, I'm not an expert in this, but my understanding is that type
> theorists often think of typing as having two "modes": type checking,
> in which t and B are both given and a derivation of t:B is to be
> found, and type synthesis or inference, in which t is given and B has
> to be found along with a derivation of t:B.  Which mode you are in at
> which point in an algorithm depends on the structure of t and B.  This
> is not irrelevant to the question of initiality, since this sort of
> "bidirectional type checking" can also be encoded in the judgmental
> structure.
> 
> Mike
> 
>> 
>> Steve
>> 
>> 
>> On Oct 12, 2017, at 2:43 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.
>> 
>> Then we have the following, building on an example of Voevodsky’s.
>> 
>> OBSERVATION. Any type theory which contains the following rules (admissible
>> or otherwise)
>> 
>> Γ |- T Type
>> ————————  (C)
>> Γ |- B(T) Type
>> 
>> Γ |- t : T
>> ————————  (R1)
>> Γ |- t : B(T)
>> 
>> Γ |- t : T
>> ————————  (R2)
>> Γ |- 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
>> 
>> Γ |- t : T
>> ————————  (R1*)
>> Γ |- 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
>> 
>> Γ |- A : U0
>> ————————  (U-cumul)
>> Γ |- 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 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?
>> 
>> 
>> 
>> 
>> 
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