The 2-Categorical View of the Prop-Categorical Semantics

ϕ1, . . . , ϕn⊢ψ[x/x^′] [Γ, x:σ]

=-Adj ϕ1, . . . , ϕn, x=σx^′⊢ψ[Γ, x:σ, x^′ :σ]

Φ⊢ψ[Γ, x:σ]

∀-Adj Φ,⊢ ∀x,σ(ψ) [Γ]

Φ, ψ⊢θ[Γ, x:σ]

∃-Adj Φ,∃x:σ(ψ)⊢θ[Γ]

Figure 3.2: Adjoint rules for∀,∃and=.

the lists of variables inΓσandΓ^′_σrespectively. ThenSJψ[x/x^′][Γ, x:σ]K=ψ^S[¯x/¯x^′][ΓS,Γσ]/∼and

SJϕ1[Γ, x:σ]K⊗. . .⊗SJϕn[Γ, x:σ]K≤SJψ[x/x^′][Γ, x:σ]K

⇐⇒ GJϕ^S₁[Γ_S,Γ_σ]K⊗. . .⊗GJϕ^S_n[Γ_S,Γ_σ]K≤GJψ^S[¯x/¯x^′][Γ_S,Γ_σ]K

⇐⇒ T ⊢Lϕ^S₁, . . . , ϕ^S_n ⊢ψ^S[¯x/¯x^′] [ΓS,Γσ]

⇐⇒ T ⊢_Lϕ^S₁, . . . , ϕ^S_n, x1=σ₁ x^′₁, . . . , xn=σ_nx^′_n ⊢ψ^S [ΓS,Γσ,Γ^′_σ] (=-Adj and 3.4)

⇐⇒ T ⊢Lϕ^S₁, . . . , ϕ^S_n,

n

O

i=1

xi=σi x^′_i⊢ψ^S [ΓS,Γσ,Γ^′_σ] (⊗-Ref ore-Ref)

⇐⇒ SJϕ1[Γ, x:σ, x^′:σ]K⊗. . .⊗SJϕn[Γ, x:σ, x^′:σ]K⊗SJx=σx^′[Γ, x:σ, x^′:σ]K

≤SJψ[Γ, x:σ, x^′:σ]K.

Therefore,Ssatisfies all interpretations of=-Adj.

3. For allc∈Ob(C),

F_c×c^p (Eqc) =Q(aF,c,c)(EqF^oc),

whereaF,b,c:F^o(b×c)→F^ob×F^ocis the change-in-product isomorphism. GivenK: (D, Q)→(E, R), we defineK◦Fby(K◦F)^o:=K^o◦F^oand(K◦F)^p:=K_F^po·F^p. For parallel morphismsF, H: (C, P)→ (D, Q), we define a2-cellη:F ⇒Hto be a natural transformationη:F^o⇒H^osuch thatF^p=Qη·H^p.

C Pos P

D QF

^o

QH

^o

P

H^o F^o

η F^p H^p

Q H^p F^p

Qη

Compositions of2-cells is just as it is inCat. Ifηis a natural isomorphism, we callηa2-isomorphism. For a logicL, we now letFA_Lbe the corresponding full sub-2-category ofFA.

Theorem 5. Let(C, P),(D, Q)∈Ob(FA)andF: (C, P)→(D, Q)be a morphism. Then for each theory T, eachT-modelSin(C, P)gives rise to aT-modelF(S)in(D, Q).

Proof. First we define the structureF(S). For each sort symbolσ,F(S)JσK:=F^o(SJσK), for each function symbolf: σ1, . . . , σn → τ,F(S)JfK :=F^o(SJfK)◦a⁻¹_Γ , and for each relation symbolR ⊆σ1, . . . , σn, F(S)JRK:=Q(a⁻¹_Γ )◦F_S^p

JΓK

(SJRK), whereaΓ:F^o(SJΓK)→F(S)JΓKis the change-of-product morphism andΓ =x1:σ1, . . . , xn:σn. A product preserving functor preserves the satisfaction of equations-in-context since by induction on the complexity of an arbitrary term-in-contextM :τ[Γ],

F(S)JM :τ[Γ]K=F^o(SJM :τ[Γ]K)◦a⁻¹_Γ .

If, for each formula-in-contextϕ[Γ],

F(S)Jϕ[Γ]K=Q(a⁻¹_Γ )(F_S^p

JΓK

(SJϕ[Γ]K)), (3.5)

then ifSsatisfies a sequent-in-contextϕ1, . . . , ϕn⊢ψ[Γ], sinceF_S^p

JΓK

andQ(a⁻¹)are monotoneL-algebra

homomorphisms,

n

O

i=1

SJϕi[Γ]K≤SJψ[Γ]K

=⇒

n

O

i=1

Q(a⁻¹_Γ )◦F_S^p

JΓK

(SJϕ_i[Γ]K)≤Q(a⁻¹_Γ )◦F_S^p

JΓK

(SJψ[Γ]K)

=⇒

n

O

i=1

F(S)Jϕ_i[Γ]K≤F(S)Jψ[Γ]K,

where ifn = 0, thenNn i=1 =e_S

JΓK. And soF(S)satisfiesϕ1, . . . , ϕn ⊢ψ[Γ]. It follows thatF(S)is a T-model and so we only need to prove that Equation 3.5 holds.

Consider a formula-in-contextϕ[Γ]. Supposeϕ[Γ]is atomic. Case 1:ϕ[Γ]is of the formR(M1, . . . , Mn)[Γ], whereR⊆τ₁, . . . , τ_n, and eachM_i:τ_i[Γ]is well-formed. Ifγis the listγ= [M₁[Γ], . . . , M_n[Γ]], we define

SJγK:=⟨SJM₁[Γ]K, . . . , SJM_n[Γ]K⟩.

Note that

F(S)JγK=aΓ^′◦F^o(SJγK)◦a⁻¹_Γ Then,

F(S)JR(M1, . . . , Mn)[Γ]K

=Q(F(S)JγK)(F(S)JRK)

=Q(F(S)JγK)◦Q(a⁻¹_Γ′)(F_S^p

JΓ^′K

(SJRK))

=Q(a⁻¹_Γ )◦Q(F^o(SJγK))◦F_S^p

JΓ^′K

(SJRK)

=Q(a⁻¹_Γ )◦F_S^p

JΓK

◦P(SJγK)(SJRK)

=Q(a⁻¹_Γ )◦F_S^p

JΓK

(SJR(M1, . . . , Mn)[Γ]K).

Otherwise,ϕ[Γ]is of the formM₁ =_τ M₂[Γ]. LetΓ^′ = [x₁ : τ, x₂ : τ]andγ: Γ → Γ^′ be the context

morphism represented by[M1[Γ], M2[Γ]]. Then,

F(S)JM1=τ M2[Γ]K

=Q(F(S)JγK)(Eq_F(S)

JτK)

=Q(a⁻¹_Γ )◦Q(F^o(SJγK))◦Q(a_Γ^′)◦Eq_F(S)

JτK

=Q(a⁻¹_Γ )◦Q(F^o(SJγK))◦F_S^p

JτK×SJτK

(Eq_S

JτK)

=Q(a⁻¹_Γ )◦F_S^p

JΓK

◦P(SJγK)(Eq_S

JτK)

=Q(a⁻¹_Γ )◦F_S^p

JΓK

(SJM1=τM2[Γ]K).

Now supposeϕ[Γ]is of the form3(ϕ₁, . . . , ϕ_n)[Γ], where3is somen-ary operation inLωand for each i∈ {1, . . . , n},F(S)Jϕ_i[Γ]K=Q(a⁻¹_Γ )◦F_S^p

JΓK

(SJϕ_i[Γ]K). Then,

F(S)J3(ϕ1, . . . , ϕn)[Γ]K

=3^Q(F(S)JΓK)(F(S)Jϕ₁[Γ]K, . . . , F(S)Jϕ_n[Γ]K)

=3^Q(F(S)JΓK)(Q(a⁻¹_Γ )◦F_S^p

JΓK

(SJϕ1[Γ]K), . . . , Q(a⁻¹_Γ )◦F_S^p

JΓK

(SJϕn[Γ]K))

=Q(a⁻¹_Γ )◦F_S^p

JΓK

(3^P(SJΓK)(SJϕ₁[Γ]K, . . . , SJϕ_n[Γ]K))

=Q(a⁻¹_Γ )◦F_S^p

JΓK

(SJ3(ϕ1, . . . , ϕn)[Γ]K).

Supposeϕ[Γ]is of the formΩ_x:σ(ψ) [Γ]forΩ∈Lq. LetI=SJΓKandX =SJσK. Forc=c₁× · · · ×c_n, we leta_c:F^o(c)→F^oc₁× · · · ×F^oc_n, anda_c1,c2: F^o(c₁×c₂)→F^oc₁×F^oc₂be the change in product isomorphisms. Then

F(S)JΩx:σ(ψ)[Γ]K

= Ω_F(S)

JΓK,F(S)JσK◦Q(a^F(S)_Γ,x:σ)(F(S)Jψ[Γ, x:σ]K)

=Q(a⁻¹_I )◦Ω_Fo(I),X◦Q(aI ×id_F(X))◦Q(a^F(S)_Γ,x:σ)(F(S)Jψ[Γ, x:σ]K)

=Q(a⁻¹_I )◦Ω_Fo(I),X◦Q(aI ×id_F(X))◦Q(a^F(S)_Γ,x:σ)◦Q(a⁻¹_S

JΓ,x:σK

)◦F_S^p

JΓ,x:σK

(SJψ[Γ, x:σ]K)

=Q(a⁻¹_I )◦Ω_Fo(I),X◦Q(a⁻¹_I,X)◦Q(F^o(a^S_Γ,x:σ))◦F_S^p

JΓ,x:σK

(SJψ[Γ, x:σ]K)

=Q(a⁻¹_I )◦Ω_Fo(I),X◦Q(a⁻¹_I,X)◦F_I×X^p ◦P(a^S_Γ,x:σ)(SJψ[Γ, x:σ]K)

=Q(a⁻¹_I )◦F_I^p◦Ω_I,X◦P(a^S_Γ,x:σ)(SJψ[Γ, x:σ]K)

=Q(a⁻¹_S

JΓK

)◦F_S^p

JΓK

(SJΩx:σ(ψ)[Γ]K.

The next result says we can identifyT-modelsS∈(C, P)with morphismsS: (CT, PT)→(C, P)which we use to develop an “algebraic” view of entailment.

Theorem 6. LetLbe adequate andT anL-theory. For each(C, P) ∈Ob(FA)andT-modelSin(C, P) there is a morphismS: (CT, P_T)→(C, P)inFA, unique up to a2-isomorphism such thatS(G) =S, where Gis the genericT-model in(C_T, P_T).

Proof. We defineS^oon objects byS^o(Γ) :=SJΓK, and on morphisms by

S^o([M1[Γ]/∼, . . . , Mn[Γ]/∼]) :=⟨SJM1[Γ]K, . . . , SJMn[Γ]K⟩.

SinceSis aT-model,S^ois well defined on morphisms and it is straightforward to show thatS^o: CT → C is a product preserving functor such thatS(G) =S, whereGis the genericT-algebra inCT. We extendS^o by definingS^p_Γ(ϕ[Γ]/∼) := SJϕ[Γ]K, andS^p_Γ is well-defined and monotone becauseSis a T-model. By definition,S(G)agrees withSon relation symbols. It is also straightforward to verify thatS^p: P_T →P◦S^o is a natural transformation, and that for each contextΓ,S^P_Γ is anL-homomorphism.

We now verify Condition 2. Form >0, letΓ =x₁ :σ₁, . . . , x_n :σ_n,Γ^′ =x_n+1 :σ_n+1, . . . , x_n+m : σn+m, and for eachi∈ {1, . . . , n+m}, letπi: SJΓ,Γ^′K→SJσiKbe theith projection map. LetSJΓK=I, SJΓ^′K=I^′and fori∈ {1, . . . , n+m}, letSJσiK=Xi. Leta:SJΓ,Γ^′K→SJΓK×SJΓ^′Kbe the change- of-product isomorphism. Then for eachΩ∈Lq, ifm >0,

S^p_Γ◦ΩΓ,Γ^′(ϕ[Γ,Γ^′]/∼)

=S^p_Γ(Ωx_n+1:σ_n+1· · ·Ωx_n+m:σ_n+m(ϕ)[Γ]/∼)

= Ω_I,Xn+1◦. . .◦Ω_{I×Xn+1...×Xn+m−1,Xn+m}(SJϕ[Γ,Γ^′]K)

= Ω_S

JΓK,SJΓ^′K◦a^−1∗(SJϕ[Γ,Γ^′]K) (by 5)

= Ω_S^o_(Γ),S^o_(Γ′)◦a^−1∗◦S^p_Γ×Γ′(ϕ[Γ,Γ^′]/∼).

Otherwise,Γ^′ = [ ]and

S^p_Γ◦Ω_{Γ,[ ]}(ϕ[Γ]/∼) =S^p_Γ(ϕ[Γ]/∼) =SJϕ[Γ]K

= Ω_S

JΓK,SJ K◦P(π₁^{SJΓK,SJ K})(SJϕ[Γ]K) (by 5)

= Ω_S^o_Γ,S^o_{[ ]}◦P(a⁻¹

S,Γ,[ ])◦S_{Γ×[ ]}^p (ϕ[Γ]/∼).

LetΓ =x1:σ1, . . . , xn:σn,Γ^′=xn+1:σ1, . . . , x2n :σn. Ifn >0,

S^p_Γ×Γ′(EqΓ)

=

n

O

i=1

SJx_i=_σi x_n+i[Γ,Γ^′]K

=

n

O

i=1

P(⟨π^S_i^JΓ,Γ′K, π_i+n^SJΓ,Γ′K⟩)Eq_S

JσiK

=

n

O

i=1

P(⟨π^S_i^JΓKπ^SJΓK,SJΓ

′

K 1 , π^SJΓ

′

K

i π^SJΓK,SJΓ

′

K

2 ⟩ ◦a)Eq_S

JσiK

=P(a)◦

n

O

i=1

P(⟨π_i^SJΓKπ^SJΓK,SJΓ

′

K 1 , π^SJΓ

′

K

i π^SJΓK,SJΓ

′

K

2 ⟩)EqSJσ_iK

=P(a)◦Eq_S^o_(Γ). (by 6)

And ifn= 0,

S^p_{[ ]}(Eq_{[ ]}) =S^p_{[ ]}(e[ ]/∼) =SJe[ ]K=e₁ ( by 6)

=P(a_{S,[ ],[ ]})(e1×1) =P(a_{S,[ ],[ ]})(Eq_S^o_{[ ]}).

Now all that remains to show is that ifF: (CT, P_T) → (C, P)is another morphism such thatF(G) = S(G) =Sthen they are 2-isomorphic.

Leta_F,Γ:F^o(Γ)→F(G)^o(Γ) =S^o(Γ)be the change-of-product isomorphism. Thena_F:F^o⇒S^ois a natural isomorphism, andP a_F·S^p=P a_F·F(G)^p=F^p, from Equation 3.5. Thusa_Fis a2-isomorphism fromFtoS.

Definition 3.3.1. LetF: (C, P)→(D, Q)be a morphism inFA. We define thekernel ofF, denotedkerF to consist of the following data:

1. A relation onOb(C), such thatc1∼c2iffF^oc1=F^oc2. 2. A relation onMor(C)such thatf₁∼f₂iffF^of₁=F^of₂. 3. A relation on⊔_c∈Ob(C)P(c)such thatr₁≺r₂iffF_c^p

1(r₁)≤F_c^p

2(r₂), wherer_i∈P(c_i).

IfK: (C, P)→(E, R)is another morphism, we saykerK≤kerF, if and only if the relations ofkerKare contained in the corresponding relations ofkerF.

LetSbe anSg-structure in(C, P)∈Ob(FA)and leta∈ASg. Ifa=ϕ1, . . . , ϕn⊢ϕ[Γ], define

SJaK:= (

n

O

i=1

SJϕi[Γ]K, SJϕ[Γ]K),

and ifais and equationM₁=M₂:τ[Γ], define

SJaK:= (SJM₁:τ[Γ]K, SJM₂:τ[Γ]K).

For eachSg-theoryT, we defineSJTK:={SJaK:a∈A(T)}and forF: (CT, PT)→(C, P)whereTis an L^m-theory, letTh(F)be theSg(T)-theory such that

A(Th(F)) :={a∈ASg:GJaK∈kerF}.

It is straightforward to verify thatTh(F) = Th(F(G)). Thus, for each structureS ∈ (C, P),Th(S) = Th(S(G)) = Th(S). Moreover, wheneverFis2-isomorphic to a parallel2-cellK, thenTh(F) = Th(K).

Let T, and T^′ beSg-theories and(C, P) ∈ Ob(FA). From our prior observations the following are equivalent:

1. T ⊨(C,P)T^′.

2. ∀F: (CSg, PSg)→(C, P)such thatT≤Th(F), thenT^′ ≤Th(F).

3. ∀F: (CSg, PSg)→(C, P)such thatGJTK⊆kerF, thenGJT^′K⊆kerF.

Thus in the sequel, we take Sg-structures in (C, P)to be morphisms F: (CSg, PSg) → (C, P). In the remainder of the section, we show how the prop-categorical semantics provide a natural notion of structural action on theories. This material is not necessary for the development of subsequent sections.

LetH: (CSg, PSg)→(CSg^′, PSg^′)be a morphism and letT1, T2beSg-theories. We can define an action ofHonSgequations and sequents in context which can be extended toT1by taking the union: that isH·T1

is theSg^′theory whose assertions areS

a∈A(T1)H·a. For equations-in-contextM¹=M²:τ[Γ], H·(M¹=M²:τ[Γ]) :={M_i¹=M_i²:τ_i[Γ_H]}ⁿ_i=1,

whereH^o(Mⁱ : τ [Γ]) = [M₁ⁱ : τ₁[Γ_H], . . . , M_nⁱ : τ_n[Γ_H]]fori ∈ {1,2}. And for sequents-in-context ϕ1, . . . , ϕn⊢ϕn+1[Γ],

H·(ϕ1, . . . , ϕn ⊢ϕn+1[Γ]) =ϕ^H₁ , . . . , ϕ^H_n ⊢ϕ^H_n+1[ΓH],

whereH_Γ^p(ϕi[Γ]) = ϕ^H_i [ΓH], for alli ∈ {1, . . . , n+ 1}. Since products in(CSg, PSg)are unique up to permutation of the list of their variable sort pairs, ifK: (CSg^′, PSg^′)→(CSg^′′, PSg^′′), then(K◦F)·T1= K·(F·T1)and clearly,id_(CSg,P_Sg)·T1=T1. Moreover, supposeV ⊆Ob(FA_L)and thatT1⊨^V T2. Let (C, P) ∈ VandF: (CSg^′, PSg^′) → (C, P)such thatGJH·T1K ⊆kerF. ThenGJT1K ⊆ kerF ◦H and sinceT₁⊨VT₂,GJT₂K⊆kerF◦H. It follows thatGJH·T₂K⊆kerFand soH·T₁⊨VH·T₂. Therefore, all logics defined semantically by subcollections of Ob(FA_L) are structural with respect the actions by morphisms.

This allows one to define a (first-order) logic as a closure operator⊢^Sg_L onThSg, for each signatureSg that is structural with respect to the action of some subcollection of morphismsGofFA_L between formula prop-categories. Particularly interesting candidates are when G includes all morphisms between formula prop-categories and whenGconsists of all morphismsH: (CSg, PSg)→(CSg^′, PSg^′)which map single sort contexts to single sort contexts.

Chapter 4

Fibered Universal Algebra