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

[Gabriel Scherer <gabriel...@gmail.com>]

In type theory, we care about coherence: if a judgment has
several derivations, then those derivations should have the same
meaning (in the denotational semantics we have in mind). (This is
a different requirement from the requirement that the untyped
semantics of the term, if it exists, be compatible with the
semantics of the typed derivation, which is also interesting.)
This means that having a derivation system with rules that are
not reflected in the term syntax give you *more* information, in
a sense, about the designer's intent: stronger constraints on
what would be a valid semantics for the type system.
(I am no expert of categorical semantics, but I suppose one could
regain initiality by restricting the class of models, adding
strictness conditions that guarantee that the coherence property
holds in any model of the class.)

In "Functors are Type Refinement Systems", 2015, Paul-André
Melliès and Noam Zeilberger give a categorical presentation of
the relation between the fully-typed derivations (or Church-style
fully-typed terms) and untyped
terms (or uni-typed derivations). They also reformulate
a coherence result from Reynolds, about subtyping, in their
framework -- coherence is obtained by relating the meaning
of any derivation to the meaning of the untyped term.

Finally, if a type theory comes with strong enough computation
rules (or in general equivalences), then the coherence property
may be shown internally, by showing that any two derivations of
the same term reduce to (or are equivalent to) a canonical
derivation. Then I would suppose that the very natural
requirement that models should respect these
computation (or equivalence) rules suffices to avoid initiality
issues. This would even work for higher-dimensional notions of
coherence -- seeing the computation rules as 2-cells in the model
instead of equalities between morphisms.

On Fri, Oct 13, 2017 at 6:23 PM, Michael Shulman <shu...@sandiego.edu> wrote:
> To my understanding, there are two different essentially-algebraic
> theories involved: type theory (or more precisely, its derivations) is
> essentially by its definition the initial object in one of them, but
> the one we're interested in (the appropriate sort of category) is a
> different theory with different operations.  For instance, in a
> category we have composition as a basic operation, but in type theory
> composition is admissible rather than primitive.  So there is always
> something to prove in relating the two, even when we know that both
> are essentially-algebraic.
>
> But my main point was that the essentially-algebraic theory to which
> the syntax belongs consists of derivations rather than terms, so it
> can be essentially-algebraic even if terms don't have unique types.
>
> On Fri, Oct 13, 2017 at 9:17 AM, Steve Awodey <steve...@gmail.com> wrote:
>>
>> 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> 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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