3.5 Nominal hypercoherence spaces
3.5.1 The category
To refine the previous model in order to avoid bad relations, one can use the notion of coher- ence, developed by Girard in the so-called coherence spaces, to express the fact that the bad relations, such as the R in equation 3.25 above, are incoherent. As this notion has since been refined, in [27] into hypercoherence, we present the latter. Furthermore, coherence removes a second downside of nominal linear polarised relations, their degeneracy. Nominal hypercoher- ence spaces differentiate between ⊗ and M, and between ⊕ and &.
All the definitions are taken from [27], just being modified to take into account the fact that our atomes are nominal, and polarised. More precisely, the elements of our nominal sets, renamed to “webs”, are, as before, separated annotated lists of polarised atomes. We write Pfin(A) for the set of finite subsets of any set A, and P∗fin(A), for the set of non-empty finite subsets. Similarly, we write w ⊆∗fin Ato mean w ∈ P∗fin(A).
Definition 3.24. A nominal hypercoherence space A= (|A|, Γ(A)) consists of:
• A nominal enumerable set |A|, called web, of annotated, polarised and separated lists. • A nominal subset Γ(A) ⊆ P∗
fin(|A|), such that all singletons are inΓ(A).
This structure on objects allows us to discriminate between relations, thanks to these definitions.
Definition 3.25. Let (|A|,Γ(A)) be a hypercoherence space. • A non-empty finite subset v ⊆∗
fin|A| is coherent when v ∈Γ(A).
• A set of elements R ⊆ |X| forms a clique when: 1. ∀v ⊆∗fin R , v is coherent.
2. R is closed under permutations.
3. R is linear: ∀x ∈ R ,ν(Neg(x))= ν(Pos(x)).
That is, a clique is a refinement of a nominal linear polarised relation. Cliques are the underlying structure behind the morphisms of the category of hypercoherence nominal spaces, that we shall define below. GivenΓ(A) ⊆ P∗fin(A), we writeΓ∗(A) for the subset ofΓ(A) of sets of cardinality greater than one, and say that such sets are strictly coherent. We writeΓ⊥(A) for the set of non-empty subsets that are incoherent:Γ⊥(A)= P∗fin(A) \Γ∗(A), andΓ⊥,∗(A) for those that are strictly incoherent, meaning incoherent and of cardinality greater than one. Finally, we set (Γ(A))⊥= {w⊥| w ∈Γ(A)}, where w⊥= {x⊥| x ∈ w}.
The dual A⊥of a hypercoherence space (|A|,Γ(A)) is defined as follows: • |A⊥|= |A|⊥
• Γ(A⊥)= P∗ fin(|A|
We now give the interpretations of the connectives of linear logic. In the sequel, for simplicity, we will write w A for w |A|. The tensor product of two hypercoherence spaces A ⊗ B is defined as follows, where ?polis the polarised separated product as defined in section 3.4.3.
• |A ⊗ B| = |A| ?pol|B|
• Γ(A ⊗ B) = {w ∈ P∗
fin(|A| ?pol|B|) | w A ∈ Γ(A) ∧ w B ∈ Γ(B)}
The unit of the tensor product is I = ({(•, 1)}, {{(•, 1)}}). We can define A M B through the De-Morgan duality (A M B) = (A⊥⊗ B⊥)⊥.
• |A M B| = |A| ?pol|B|
• Γ∗
(A M B) = {w ∈ P∗fin(|A| ?pol|B|) | (w A ∈ Γ∗(A) ∨ w B ∈ Γ∗(B))}
M has almost the same unit as the tensor product, except the atom is negated: ⊥= ({(•, −1)}, {{(•, −1)}}). As a result, we can define the connective (, with A ( B = A⊥M B.
• |A ( B| = |A|⊥? pol|B|
• Γ(A ( B) = {w ∈ P∗
fin(|A( B|) |(w |A| ⊥
∈Γ(A)⊥⇒ w |B| ∈ Γ(B)) ∧ (w |A|⊥∈ (Γ∗(A))⊥⇒ w |B| ∈ Γ∗(B)}
In the future, we will simply write w A for w A⊥, the fact that the polarities in A are reversed
being clear from the context. Finally, we define the additives. We start with A & B, writing ] for the disjoint union.
• |A & B| =inl(|A|) ]inr(|B|) • Γ(A & B) = {w ∈ P∗
fin(|A & B|) | (winl(A)= ∅ ⇒ w inr(B) ∈inr(Γ(B))) ∧ (w inr(B)=
∅ ⇒ w inl(A) ∈inr(Γ(A)))}
We finish with A ⊕ B, that is the dual of A & B, in the sense that (A ⊕ B)= (A⊥⊕ B⊥)⊥ • |A ⊕ B| =inl(|A|) ]inr(|B|)
• Γ(A ⊕ B) = {w ∈ P∗
fin(|A ⊕ B|) | (w A = ∅ ∧ w B ∈ Γ(B)) ∨ (w B = ∅ ∧ w A ∈ Γ(A))}
The empty set acts as units for the two additive connectives. We therefore have the following denotation function:
• ~1 = I, ~⊥ = ⊥. • ~> = ~0 = (∅, ∅).
• We define the interpretation of an atomic type ~X = (|X|, Γ(X)) as being the following hypercoherence space:
– |X|= {(a, 1) | a ∈ AX}
– Γ(X) = {{(a, 1)} | a ∈ AX}
In the following, we simply write X for ~X.
Definition 3.26. NomHypCoh is the category of nominal linear polarised hypercoherent re- lations (often abbreviated nominal hypercoherent relations, or cliques) that has as objects the
smallest set such that I, X, 0 ∈Obj(NomHypCoh) and
A, B ∈Obj(NomHypCoh) ⇒ A ⊗ B ∈Obj(NomHypCoh) A, B ∈Obj(NomHypCoh) ⇒ A ⊕ B ∈Obj(NomHypCoh) A ∈Obj(NomHypCoh) ⇒ (A)⊥∈Obj(NomHypCoh)
Morphisms A → B ofNomHypCohare cliques of A( B.
The composition of cliques inNomHypCohcomes from the composition of nominal linear polarised relations. The rest of this section is devoted to check that the composition is well defined. More precisely, we have to ensure that the resulting linear nominal polarised relation is indeed a clique. To do that, we introduce the notation dΓ(A), to denote the appropriate notion of coherence when we close will close the cliques under strict substitution. That is, dΓ(A) ⊆ P∗
fin( b|A|), and ( b|A|,Γ(A)) forms a hypercoherence space. Given a clique R , after closure, bd R is
not a subset of A( B anymore but a subset of [A( B, and we must consider a new notion of coherence for this space. Given a formula F, dΓ(F) is defined exactly as Γ(F), except that we consider cartesian product instead of separated product. This is defined formally below. Definition 3.27. The categoryLaxHypCohis the category of lax nominal linear polarised hy- percoherent relations (often abbreviated lax nominal hypercoherent relations, or lax cliques), that has as set of objects the smallest set containing the elements( b|A|,Γ(A)) subsequently de-d fined, and closed under the operations defined below.
• bI = INomHypCoh,b⊥= ⊥NomHypCoh, b0= 0NomHypCoh. • Xb= XNomHypCoh, cX⊥= XNomHypCoh⊥ .
• [A ⊗ B= ( b|A| × b|B|,Γ(A) ×d Γ(B))d
• [A M B = ( b|A| × b|B|,Γ∗(A M B) = {w ∈ P[ ∗
fin( b|A| × b|B|) | w b|A| ∈ [Γ
∗(A) ∨ w b|B| ∈ [Γ∗(B)}.
• [(A & B)= (inl( b|A|) ]inr( b|B|), {w ∈ P∗fin( [|A & B|) | w ⊆inl( b|A|) ⇒ w ∈inl( [Γ(A)) ∧ w ⊆inr( b|B|) ⇒ w ∈inr( [Γ(|B|))} • [A ⊕ B= (inl( b|A| ]inr( b|B|,inl( dΓ(A)) ]inr( dΓ(B)).
• cA⊥= (| ˆA|⊥, P∗ fin(| ˆA|
⊥) \ {w⊥| w ∈ [Γ∗(A)})
The morphisms A → B are cliques of A⊥M B, where seeing A has a formula of linear logic, A⊥ is the lax hypercoherence space that corresponds to its negation.
Note that the definitions are compatible with the De-Morgan formula, that is [Γ(A⊥)=
d Γ(A)⊥. Indeed, these hold for the base cases, and the definitions of A M B and A⊕ B are settled according to this formula. The composition of cliques inLaxNomLinPolfollows from the one of lax nom- inal linear polarised relations. That is, it is simply their relational composition (while forgetting polarities in the middle), and it hence follows thatLaxHypCohis simply a subcategory of the category of lax nominal linear polarised relation. Notably, the identity morphism is simply the identity relation.
In order to prove that the composition of separated relations is compatible with the definition of cliques, we proceed by steps and start with this lemma.
Lemma 3.28. • Let w ⊆∗fin|A|NomHypCoh, such that w ∈ [Γ∗(A). Then w ∈Γ(A).
• , Γ(A) ⊆Γ(A).d
Proof. The proof is done by induction on the structure of the formula A. For the first point, the base cases X, I, 0, >, ⊥ are immediate, as they are equal, and the inductive cases are automatic.
The first point is needed to prove the second. Just as in the first point, the proof is done by induction on the structure of the formulas, the base case and the inductive cases for ⊗, M, ⊕ being immediate. Remaining is the case for &, setting A = A1& A2. If w A1 ∈ inl( [Γ(A1)),
then as w ⊆∗fin (A1), it entails, as proven in the first part of the proof, that w A1 ∈inl(Γ(A1)).
Therefore w A2∈inl(Γ(A2)) and by induction w A2 ∈ [Γ(A2). The second case is dealt with
on an equal footing, and thereforeΓ(A1& A2) ⊆Γ(A1& A2) V
. To conclude,Γ(A) ⊆Γ(A).d Lemma 3.29. Let R be a clique of A. Then bR is a clique of ˆA.
In other terms, this expresses that b(.) acts seemingly as a functor from hypercoherent linear nominal polarised relations to hypercoherent lax nominal polarised relations, assuming they indeed form a category. The main property to prove is that for every finite subset w ⊆fin bR , then wis lax coherent: w ∈Γ(A)
V
. For the proof we will rely on the following property, whose proof is immediate.
Proposition 3.30. Let w, w0 ⊆∗
fin(|A|), such that, for any occurrence of atomic formula X within
A we have: • w X , ∅ ⇔ w0 X , ∅ • w X ∈ Γ(X) ⇔ w0 X ∈ Γ(X). • w X ∈ Γ∗(X) ⇔ w0 X ∈ Γ∗(X). • w X ∈ Γ⊥ (X) ⇔ w0 X ∈ Γ⊥(X). • w X ∈ Γ⊥,∗(X) ⇔ w0 X ∈ Γ⊥,∗(X).
Then w ∈ Γ(A) (respectively Γ∗(A),Γ⊥(A),Γ∗,⊥(A)) ⇔ w0 ∈ Γ(A) (respectively Γ∗(A),Γ⊥(A),Γ∗,⊥(A)).
The proof of the proposition is a simple induction. The proposition is true in any category with hypercoherence, such asNomHypCohorLaxHypCoh.
Proof of lemma 3.29. Let R be a clique of ~A. Let w ⊆fin bR and w = {e1· x1, ..., en · xn}
where {x1, ..., xn} ⊆ R and e1, .., en∈Ξ. We do the proof by induction on the sum of the lengths
of the substitutions |e1|+ ... + |en|. If the sum is 0, then w = {x1, ..., xn} ∈ R and therefore
w ∈Γ(A) ⊆Γ(A).d
So suppose the sum is equal to n+1, and we proved the property up to n. Writing x0ifor ei· xi,
we know that w = {x01, ..., x0n} ∈ dΓ(A) ∩ P( bR ) and we want to prove that {[a/b] · x01, ..., x0n} ∈ d
Γ(A). The first case to tackle is the case where b < ν(x0
1). Then [a/b] · x 0
1 = (a, b) · x 0 1, and
the set {(a, b) · x1, ..., xn} is a subset of R . Furthermore, w is now equal to {((a, b) · e1) · ((a, b) ·
x1), e2· x2, ..., en· xn}. The length of the substitutions applied on this set being n, we conclude
by induction hypothesis that w is lax-hypercoherent: {[a/b] · x01, ...., x0n} ∈ dΓ(A).
The harder case is when b ∈ ν(x01). Then let us consider a permutation (b, c), such that c#{x01, ..., x0n, a}. We know that {(b, c)· x01, ..., x0n} ∈ dΓ(A), as proven above. Finally, let us apply the
substitution [a → c] to x01. Applying it changes the lax coherence as much as a permutation (a, d) when d is fresh. This follows from noticing that {[a/c] · (b, c) · x01, ..., x0n} X has same coherence
as {(a, d).(b, c) · x10, ..., x0n} X for every occurrence X of atomic formula in A (since both a, d
are fresh for x02, .., x0n). As a result, if {(a, d).(b, c) · x01, ..., x0n} belongs in dΓ(A), then so does
{[a/d].(b, c)· x01, ..., xn}. As the first member of the previous sentence indeed is lax hypercoherent
(for the same reasons as explained above), then {[a/c]·(b, c)· x01, ..., xn} ∈ dΓ(A). Finally, we apply
the permutation (b, c) back , noticing that (b, c).[a → c].(b, c) · x01 = [a → b] · x01, and therefore as {[a/c].(b, c) · x01, ..., x0n} ∈ dΓ(A) ∩ P( bR ), this entails {(b, c).[a/c].(b, c) · x01, ..., x0n} ∈ dΓ(A) ∩ bR , or, equivalently, w= {[a/b] · x01, ..., x0n} ∈ dΓ(A) ∩ P( bR )
Lemma 3.31. Let w ⊆finA[( B such that w ∈ Γ(A ( B) V
. If w ⊆fin A( B then w ∈ Γ(A ( B).
The proof is straightforward.
Proposition 3.32. Let R : A ( B and Q : B ( C be two linear cliques. Then R ; Q is a linear clique of A( C.
Proof. The linearity and nominal closure follows from the composition of nominal linear po- larised relations. Therefore, in rest of this proof, we forget about the local polarities of the lists. Let w ⊆∗fin R ; Q , and let v ⊆∗fin A × bb B × bC a witness of interaction; that is, for all x in w there exists a unique y ∈ v such that y ∈ bR ; bQ and y A ( C = x. Then suppose w A ∈ Γ(A), then v A ∈ dΓ(A). As v Ab ( bB ⊆∗fin bR , then v bA ( bB ∈ Γ(A ( B),[ entailing v bA ∈ dΓ(A) ⇒ v Γ(B). Hence v d bB ∈ dΓ(B). Doing the same for v bB ( bC, we conclude that v bC ∈ dΓ(C). Now as w C = v Cbthen w C ∈ Γ(C). To sum up, ∀w ⊆∗fin R ; Q .w A ∈ Γ(A) ⇒ w C ∈ Γ(C). A similar reasoning holds for proving ∀w ⊆∗
fin R ; Q.w A ∈ Γ
∗(A) ⇒ w C ∈ Γ∗(C).That is, ∀w ⊆∗
fin R ; Q .w ∈Γ(A ( C).
ThereforeNomHypCohforms a category, and the operation b(.) defines a star-autonomous functor fromNomHypCohtoLaxHypCoh. This is a direct consequence of b(.) being a functor of compact closed categories between their underlying categoriesNomLinPolandLaxNomLinPol.