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Chapter 5 Bicartesian Categories
CHAPTER 5
BICARTESIAN CATEGORIES
In this chapter, we study the proof-theoretical properties of A , v, T, and 1 that are independent of distributivity. The appropriate class of categorical models for this purpose is the class of small bicartesian categories. 5.1. Definition
A bicartesian category is a Cartesian category C with the following additional structure: (4) A bifunctor (-) v (-) : C X C + C. (5) A distinguished object 1 E ObC. (6) Two adjunctions a. and a&,where a. = {au(A, B, C ) : C(A v B, C)+ C(A, C) X C ( B , C) E ArEns I A, B, C
E ObC},
and
I
a&= {&,(A): C(1, A ) + { * } E ArEns A E ObC}.
5.2. Examples 5.2.1. A category C is bicartesian iff it has finite products and finite
coproducts. Hence, in particular, lattices with smallest and largest elements are bicartesian categories.
5.2.2. COUNTER-EXAMPLE. The set of all functions f : [0, 1]+ R satisfying the condition f ( t x + (1 - t ) y ) > t f ( x )+ (1 - t ) f ( y ) for 0 < t < 1, bounded above by the upper half of the unit circle centred at (f,O), 54
5.31
T H E CATEGORY
Fbc(X)
5.5
partially ordered by f s g iff f ( x ) s g ( x ) for all x E [0, 11, with f A g = {min(f(x), g(x)) 1 x E [0, 11) is a lower semilattice with largest element that is not a lattice. Hence we have a natural example of a Cartesian category which is not bicartesian. 5.3. The category Fbc(X)
Small bicartesian categories are the objects of a category bcCat whose arrows are functors F satisfying the conditions of arrows in cCat and have the additional property that F ( A v B )= F ( A ) v F ( B ) , F(1)= I, and F ( a ; ' ( A ) (*)) = a ; l ( F ( A ) (*)) for all A, B E Obdom(F), and that a,(F(A), F ( B ) , F ( C ) ) ( F ( f )=) ( F ( g ) ,F ( h ) ) for all A, B,C E Obdom(F) and all f, g, h E Ardom(F) for which a,(A, B,C)(f)= (g, h ) . There exists an obvious forgetful functor Ubc : bcCat+ Cat. We now extend the definition of Fc and construct a left adjoint Fbc of Ubc. 5.3.1. DEFINITION. The language of Fbc(X) is the sublanguage bcL(X) of L(X) generated by ObX, T, A , I, v, and ArX. 5.3.2. DEFINITION.The labelled deductive system of Fbc(X) is the
subsystem bc.&(X)of &X) generated by Axioms (AI), (A2), (AlO), (A1I ) , (A12), (A13), (A14), (A15),and Rules ( R I ) , (R3), and (R4).
5.3.3. REMARK.A comparison with 4.3.1 and 4.3.2 shows that bcL(X) and bc&X) result from cL(X) and c&X) by the inclusion of I,v, and (A1 l), (A14), (A15), and (R4), respectively. 5.3.4. DEFINITION. The relation = is the smallest equivalence relation on Der(bc&X)) satisfying the conditions of Definition 4.3.3, and the following additional requirements: (10) If f = g and h = k, then [f, h ] = [g. k l . (1 1 ) comp([f, 81, vn*)= f. (12) comp([f, gl, 6) = g. (13) [comp(k, d), comp(k, v31= k. (14) If d o m u ) = I,then f = 7*.
We now define the category Fbc(X) by modifying and extending the
56
BICARTESIAN CATEGORIES
15.3
definition of Fc(X): (1) ObFbc(X) = bcL(X). (2) ArFbc(X) = Der(bcb(X))/=. (3) As in 4.3.3. (4) As in 4.3.3. (5) As in 4.3.3. (6) As in 4.3.3, with the inclusion of T*, d , and T:, determined by Axioms ( A l l ) , (A14), and (AlS). (7) As in 4.3.3. (8) As in 4.3.3. (9) As in 4.3.3. (10) The image of f : A + B E ArX in Fbc(X) is afn. ( 1 1 ) For all derivable labelled sequents f : A + C and g : B + C, ugni = gin. (12) For all derivable labelled sequents f : A + B and g : C + D, af.1v !dl = U[comp(.rm*(B,D ) ,f), com~(.rr,*(B,D ) ,g)Ib (13) For all A, B E ObFbc(X), &,(A, B, A v B)( l(A v B))= (TA*(A,B ) , &A, U ? ) and &'(A)( * ) = T*(A). This completes the description of Fbc(X). We call the category Fbc(X) the free bicartesian category generated by X. The calculations required to show that it is indeed bicartesian are similar to those in 4.3.3. On the arrows of Cat, the functor Fbc is defined by adapting and extending Definition 4.3.4 thus:
run,
urf,
5.3.5. DEFINITION. Let H : C + D be an arrow of Cat, and Fbc(C) and Fbc(D) be the free bicartesian categories generated by C and D. Then
F%c(H) : Fbc(C)+ Fbc(D) is the functor satisfying the following equations: (1)-(lo) As in 4.3.4, with Fbc in place of Fc. (11) Fbc(H)(A v B ) = Fbc(H)(A) v Fbc(H)(B) for all A, B E ObFbc(C). (12) Fbc(H)(I) = 1. (13) Fbc(H)(lf,gl) = [Fbc(H)Cf), Fbc(H)(g)l for all f , g E ArFbW). (14) Fbc(H)(.r*(A)) = T*(F~c(H)(A)) for all A E ObFbc(C). (15) Fbc(H)(nf(A, B ) ) = d(Fbc(H)(A), Fbc(H)(B)) for all A, B E ObFbc(C). (16) F%c(H)(r,*(A, B)) = r?(Fbc(H)(A), Fbc(H)(B)) for all A, B E ObFbc(C).
5.41
T H E DEDUCTIVE SYSTEM
bcA(X)
57
Once again, the verification that Ubc and Fbc are adjoint functors is routine.
5.4. The deductive system bcA(X) We now expand the deductive system cA(X) to an unlabelled deductive system bcA(X) generating a sequential category which contains an isomorphic copy of Fbc(X) and has the desired syntactic properties. The proof-theoretical interest of bcA(X) lies in the fact that it illustrates the connection between cut eliminability and the distributivity of A and v . We explain this remark by examining the nature of the cut-free representations in A(X) of the distributivity arrows
f : A v ( B A C)+(A v B ) A ( A v C),
h :( A A B ) v ( AA C)+ A
A
( B v C),
g : ( A v B ) A( A v C ) + A v (B
k :A
A
A
c),
( B v C)+ ( A A B ) v ( AA
C),
existing in any sequential category qua distributive lattices as described in 1.1.30. A+A B+B A+A c+ c A-AB BC+AB A-AC BC+AC A+AvB BAC+AVB A+AvC BAC+AVC A v ( B A C )+ A v B A v ( B A C)+Av C A v ( B A C)+(A v B ) A ( Av C)
B+B
(1)
C+C
A+A BC+B BC+C A+A A+A(B A C) BC+BAC A+ A(B A C ) BA+A(BAC) BC+A(BAC) B ( A v C)+ A(B A C) A ( A v C)+ A(B A C) ( A v B ) ( A A C)+ A(B A C ) ( A v B ) A ( A v C)+ A A ( B A C)
(2)
c+ c
A+A A+A B+B AB+A AC+A AB+BC AC+BC AAB+A AAC+A AAB+BvC AAC+BVC (AA B)v (AA B v (AAB)v(AAC)+A ( A A B ) v ( A A C ) + A A ( B v C)
c)+
c
(3)
58
BICARTESIAN CATEGORIES
[5.4
B+B C-PC A+A B+BC C+BC A-A A(B v C ) + A BvC+BC A(B v C ) + A A(B v C ) + BA A(B v C)+ BC A(B v C ) + A ( A A C ) A(B v C ) + B(A A C ) A(B v C)+ ( A A B ) ( A A C ) A A ( B v c)+ ( A A B ) v ( A A
(4)
c)
Derivations (l), (Z), (3), and (4) represent the arrows f, g, h, and k, respectively, with obvious abbreviations. The generality of Rules (R10) and (R12) required in the above derivations may be classified as follows:
The greater generality of (R10) in Derivation (2) can be avoided by replacing the single instance of (R13) towards the end of the derivation be three instances of (R13) higher up in the derivation. We notice that onIy Derivations (2) and (4) require antecedents, respectively succedents, of length greater than 1. Since either one of the inequalities represented by arrows g and k forces a lattice in which it holds to be distributive, it is clear that the length of the succedents of instances of (R10) and the length of the antecedents of instances of (R12) must be restricted to being no greater than 1. On the other hand, the arrows g and k are still representable by these restricted forms of (RIO) and (R12) if the cut rule (Rl) is admitted in full generality: Let a = A A (Bv C ) , P = A ( A A B ) v ( A A C), and let
P
a+
A
( ( AA
4
B)v C), y = A A (C v ( A A B ) ) , 6 = r
P, P+ Y, Y-*
be the following derivations:
6
5.41
T H E DEDUCTIVE SYSTEM
bcA(X)
59
B+B C+C A+A B+B B+BC C+BC AB+A AB+B BvC+BC AB+AAB _- ~~A(B v C)+ B ( A A B ) C A(B v C ) B + ( A A B)C A(B v C)A(B v C)+ ( A A B ) C ( A A B)C ~
~
A+A A(B v C)+ A A A ( B v C)+ A P = A
A
( A A ( B v C))(A A ( B v C))+ ( ( A A B ) v CN(A A B ) v C ) A A (B v C)+(A A B )v C ( B v C)+ A A ((AA B ) v
c)
A+A B+B AB+B AB+A AB+AAB AB+ C ( A A B )
c+c
C + C(AA B ) B+ C v ( A A B ) C + C v ( A A B ) A+A ( AA B ) v C+ C v ( AA B) A ( ( A A B ) v C)+ A A ( ( A A B ) v c ) + C v ( A A B) A A ( ( A A B ) v C)+ A A A ( ( A A B ) v C)+ C v ( A A B ) q= A A ( ( A A B ) v C)+ A A ( C v ( A A B ) ) A
A
A+A B+B AB+A AB+B AB+AAB AAB+AhB C+C C+C A+A C + C ( A A B ) A A B+ C ( A A B ) AC+A AC+C C v ( A A B)+ C ( A A B ) AC+Ar\C A(C v ( A A B))+ C ( A A B ) ( A A C ) A ( C v ( A A B ) ) C + ( A A B ) ( A A C ) A(C v ( A A B ) ) A ( Cv ( A A B ) ) + ( A A B ) ( A A C ) ( A A B ) ( A A C ) ( A A ( C v ( A A B ) ) ) ( AA ( C v ( A A B)))+ ((A A B ) v (A A C))((A A B ) v ( A A C ) ) A A (C v ( A A B ) ) + ( A A B ) v ( A A C)
Then the derivation
represents the arrow k. The arrow g is represented similarly. The cut
60
BICARTESIAN CATEGORIES
p.5
rule (R1) is therefore not admissible in full generality as a rule of inference of bcA(X). Hence we have a non-trivial example of a subsystem of A(X) without a cut elimination theorem: 5.4.1. COUNTER-EXAMPLE. The deductive system consisting of Axioms
(Al), (A3), and (A4), and Rules (RI), (R2), (R3), (RS), (R6), (RIO), and (R12), with the succedents in (RIO) and the antecedents in (R12) restricted to sequences of length 1, does not admit a cut elimination theorem. Fortunately, we can formulate the deductive system bcA(X) by means of two special cases of (Rl) for which a cut elimination theorem holds. The unlabelled deductive system of Fbc(X) is the subsystem bcA(X) of A(X) generated by axioms (Al), (A3), (A4), and the following restrictions of (Rl), (R2), (R3), (RS), (R6), (RlO), (Rll), (R12), and (R13): (R1)
r-+y A ~ A - + Q , ArA+Q,
rjQyq y r+mq
+
~
5.5. The semantics of Der(bcA(X))
We extend the interpretation of Der(cA(X)) in ArFc(X) to an interpretation of Der(bcA(X)) in ArFbc(X) by means of the following canonical arrows of Fbc(X), determined by au: (4) d ( A , B) : A + A v B for all A, B E ObFbc(X), where
a d A , B, A v B)(l(A v B ) ) = (T?(A,B ) , d ( A , B ) ) . ( 5 ) S*(A): A + A v A for all A E ObFbc(X), where
oG'((l(A), I(A))) = S*(A).
5.51
THE SEMANTICS OF
Der(bcA(X))
( 6 ) (a*)-I(A,B, C ) : ( A v B ) v C + A v ( B v C) for ObFbc(X), where
61
all
A , B, C E
a,I(,r?(A v B , C)&(A, B ) , (u,'(T?(Av B, C).rr,*(A,B ) , r , * ( AV B, C ) ) ) = ( a * ) - ' ( AB, , C).
5.5.I . DEFINITION. The interpretation of Der(bcA(X)) in Fbc(X) is the function S : Der(bcA(X))+ ArFbc(X) satisfying Conditions (1)-(7) of 4.5.1 and the following additional equations: (8) S(l+) = l(1): I+ 1.
(Ca v 0)v
*
f
As in Fc(X), the equivalence classes of Der(bcA(X)) obtained by defining f = g iff S c f ) = S ( g ) are plentiful enough to classify the arrows of Fbc(X): 5.5.2. THE COMPLETENESSTHEOREM FOR Der(bcA(X)). For every f E Der(bc&X)) there exists a g E Der(bcA(X)) such that S ( g ) = vl E
ArFbc(X).
62
BICARTESIAN CATEGORIES
[5.5
PROOF.We modify and extend the proof of Theorem 4.5.2. (1) As in 4.5.2. (2) As in 4.5.2, with -the following addition: If efn = 1(1), let g be the derivation l+, and if 1Lfl=1(A v B), let g be the derivation
h k B+B A+A A+AB B+AB AvB+AB AvB+AvB (3) As in 4.5.2, with the following addition: I f f quotes Axioms (All), (A14), or (AlS), let g be the derivations
I+ l+A
h A+A A+AB A+AvB
k B+ B B+AB B+AvB
respectively. (4) As in 4.5.2. (5) As in 4.5.2. (6) If the last line of f consists of an application of (R4), i.e., f is a derivation of the form P 4 u:A+C u:B+C [u, u ] :A v B + C and if 1Ipn= S ( h ) and [qn = S ( k ) , let g be the derivation
h k A+C B+C AvB+C
0
5.5.3.REMARK.We repeat Remark 4.5.3 concerning the use of dots in order to display the instances of (R5) and (R6) unambiguously. 5.5.4. COROLLARY. The category Fbc(X) is isomorphic to a subcategory of the sequential category generated by the deductive system bcA(X) and the interpretation S : Der(bcA(X))+ ArFbc(X). 0
5.61
THE S Y N T A X OF
Fbc(X)
63
5.6. The syntax of Fbc(X)
We extend Theorem 4.6.1 to bcA(X) and show that the arrows of Fbc(X) have a composition-free description:
5.6.1. THE CUT ELIMINATION THEOREMFOR bcA(X). Every f € Der(bcA(X)) is equivalent to a cut-free g E Der(bcA(X)). PROOF. By Theorem 4.6.1, with Clause (C.26) generalized to allow non-empty Q and 9,together with Clauses (C.4), (C.7), (C.8), (C.9), (C.12), (C.15), (C.21), (C.22), (C.29), (C.371, (C.38), (C.43), and (C.44) of the cut elimination algorithm, f reduces to a cut-free derivation g . It remains to show that the reduction steps preserve the meaning of f. Most cases can be disposed of by duality: (C.7) is dual to (C.2), (C. 15) is dual to (C.14), (C.21) is dual to (C.34), ((2.22) is dual to (C.35), (C.29) is dual to (C.42), ((2.37) is dual to (C.18), (C.38) is dual to (C.19), (C.43) is dual to (C.26), and (C.44) is dual to ((2.27). Hence we are left with Cases (C.4), (C.8), (C.9), and (C.12), and the task of re-examining Case ((2.26). A brief reflection shows that (C.8) is dual to (C.4), (C.9) is dual to (C.3), and the two subcases arising in (C.12) are analogous, respectively dual, to (C.2.1). In the case of (C.4), we must show that the following reduction preserves equivalence: g
+@
r -+f
Q
~
~
g
f r-+QaB9 p+o
r+Qa@z
P +@
g
+@ a+@
Since g is cut-free by hypothesis, we may take 0 to be T. Hence the above derivations are equivalent provided that the diagram A v ( B v C)(AvB)vC ( I V T ) V T
IVT
A vT
IVS*
64
[5.6
BICARTESIAN CATEGORIES
commutes for all A, B, C E ObFbc(X). By Axiom (Ml) of monoidal categories, Diagram (1) commutes iff Diagram (2) commutes, where (2) results from (1) by the replacement of a*(A, B, C) by (a*)-'(A, B, C). But by the uniqueness of terminal arrows, Diagram (2) commutes iff
- 1 I 1
A v ( B v C)
IV(TVT)
A v (T v T)
IVS*
A
(a*)-'
(3)
(AvB)vC (IVT)VT
iva*
5.61
THE S Y N T A X OF
Fbc(X)
65
in the proof of Theorem 4.6.1. By the naturality of a,', the right-hand side of Case (3) is equivalent to the derivation
r+ayvsq
YVs+'(YAp
r+@aApq
and by the joint naturality of aa and arrthis derivation is equivalent to the left-hand side of Case (3). This proves the cut elimination theorem for bcA(X). 0 Using the completeness theorem for Der(bcA(X)) and arguing as in the proof of Lemma 4.6.2, we obtain an analogous result for bcA(X):
5.6.2. LEMMA.If a = T and p derivable in bcA(X). 0
= 1 in
Fbc(X), then + a and l3+
are
In order to enable us to establish an analogue of Corollary 4.6.3 for bicartesian categories, we require additional preliminaries: 5.6.3. LEMMA.A = A v I= A
A
T for all A E ObFbc(X).
PROOF.The existence of the required isomorphisms follows at once from the Yoneda lemma since
66
BICARTESIAN CATEGORIES
and
[ X ,A v TI
[ X ,A ] x [ X ,TI
[5.6
[ X ,A ]
for all X , Y E ObFbc(X). In contrast to the situation in cA(X), it is no longer true in bcA(X) that we can take all active formulas in instances of (R2) to be atomic. Since Fbc(X) is non-distributive, the derivation A+A A(B v C ) + A with A, B , C E ObX, for example, cannot be replaced by the derivation A+A A+A AB+A AC+A A(B v C )+ A A dual argument shows that the active formulas of instances of (R5) cannot be taken to be atomic. However, a slightly weaker version of Lemma 4.6.4 and its dual still holds for bcA(X):
5.6.4. LEMMA.For every cut-free f EDer(bcA(X)) there exists an equivalent cut-free g E Der(bcA(X)) containing no instances of (R3) and (R6), and no instances o f (R2) and (R5) whose active formulas are o f the form (Y A p and a v p, respectively. . 0
For the purpose of the proof of our final lemma, we regard the category Ens of 2.2.7 as a bicartesian category, with the empty set 0 as an initial and a fixed one-element set { * } as a terminal object. Similarly, we regard the category comRng of commutative rings with 1 and ring homomorphisms as bicartesian, with the ring of integers Z as an initial and the trivial ring 0 as a terminal object. (The product objects of comRng are Cartesian products with coordinate-wise operations (cf. 6.2.2 for the definition of addition, for example), and the coproduct objects are tensor products of rings, taken over Z (cf. LANG[1965]).) 5.6.5. LEMMA.The objects l ~ and l T v T are neither initial nor terminal objects of Fbc(X).
5.61
T H E S Y N T A X OF
Fbc(X)
67
PROOF. Let Const 0 : X + comRng and Const { * } : X + Ens be the constant functors with object values Const O(A) = 0 and Const { * } ( A )= { * } for all A E ObFbc(X),and let FO: Fbc(X) + comRng and F{*): Fbc(X) + Ens be the induced functors of bicartesian categories. Then F o ( l A I)= Z x Z in comRng, and F+,(T v T) = {*} + {*} in Ens. Obviously, Z x Z + 0. Suppose that Z x Z = Z,and let
be a product diagram determined by this isomorphism. Then I @ ) = comp(h, T A )= comp(h, n,,), and therefore T A = np Since ~ A ( ub), = a and n J a , b ) = b, we have a = b for all a, b E Z . A contradiction. A cardinality argument establishes that { * } + { * } is isomorphic to neither 0 nor {*}. 0 5.6.6. COROLLARY. The initial and terminal objects of Fbc(X) are characterized by the following properties: (1) I f a 1T and p = I , then T and 1 are the only atomic subformulas of a and p. (2) a ~ p = T ifa=p=T. (3) a v p = T if a ( @= ) T and P ( a ) = I . (4) ( ~ ~ p = l i f ~ ( ~ ) ~ T u n d p ( co rui f) a= =I p , pI. (5) a r v p = l i f a r = p = I . 0
5.6.7. COROLLARY. If a = T, there exists a unique derivation g of + a consisting at most of instances of (A3), (RS), (RIO), and (R13), and n o instance of (R5) in g has a n actiue formula of the f o r m a v p. 0
5.6.8. COROLLARY. If p =I,there exists a unique derivation h of p + consisting a t most of instances of (A4), (R2), (R1 I), and (R12), and n o instance of (R2) in h has a n active formula of the f o r m a A p. 0 We now extend the definition of the normality of derivations of cA(X) to the derivations of bcA(X) by defining f E Der(bcA(X)) to be normal if it is normal in the sense of Appendix D, and satisfies four additional requirements:
68
BICARTESIAN CATEGORIES
[5.6
(1) f contains no instance of (R3) and (R6). (2) Unless f is a derivation mentioned in Cases (3) and (4) below, f contains no instances of (R2) and (R5) whose active formulas are of the form (Y v p and (Y A p, respectively. (3) If f derives r+@, and if one of the disjunctions of the formulas of @ is isomorphic to T, then f is of the form g
+@
r+ (p(')r where g is the unique derivation of + @ compatible with Corollary 5.6.7, and where (a)consists of the instances of (R2) required to derive r+@ from +@. (4) If f derives r+@, and if one of the conjunctions of the formulas of r is isomorphic to I and no disjunction of the formulas of @ is isomorphic to T, then f is of the form
where h is the unique derivation of + compatible with Corollary 5.6.8, and where ( 7 ) consists of the instances of (R5) required to derive r+@ from r + . By combining Theorem 5.6.1 with the preceding lemmas and corollaries, and extending the reducibility relation 2 of Appendix D in the obvious way beyond the extension required in Chapter 4, we obtain the desired analogue of Theorem 4.6.5 for Der(bcA(X)): 5.6.9. THE NORMALIZATIONTHEOREM
FOR bcA(X). Every f E Der(bcA(X)) reduces to a unique equivalent normal g E Der(bcA(X)).
PROOF.The theorem follows from Theorem 4.6.5 and the normalization algorithm defined in Appendix D, provided the reductions in Conditions (D.4), (D.10), (D.17), (D.21), (D.34), (D.36), (D.37), (D.38), (D.59, (D.591, and (D.62) preserve equivalence. But this is clear: (1) The equivalences required for (D.4), (D.10), (D.17), (D.21), (D.361, and (D.55) are immediate consequences of the associativity of composition.
5.61
THE SYNTAX OF
Fbc(X)
69
(2) The equivalences required for (D.34) are a consequence of the functoriality of v . (3) The equivalences required for (D.37) and (D.59) follow from the naturality of am. (4) The equivalences required for (D.38) follow from the naturality of a*.
(5) The equivalences required for (D.62) are consequences of the coherence of a*,i.e., Theorem 2.6.1.1. This proves the normalization theorem for bcA(X). 0 In view of the duality of A and v in bcA(X), Theorem 4.6.6 therefore generalizes immediately to bcA(X). Hence we have an effective characterization of commutativity in Fbc(X): 5.6.10. THECHURCH-ROSSER THEOREM FOR bcA(X). I f f = g, then there exists a normal h E Der(bcA(X)) such that f 2 h and g L h. 0 5.6.11. COROLLARY. The word problem f o r the functor Fbc is sol-
vable. cl
As in the previous cases, all normal derivations of a sequent A + B in bcA(X) have the same width because of the restriction on the antedents of instances of (R12) and succedents of instances of (R10) to single formulas, and are effectively determined by the syntax of Fbc(X) relative to any fixed assignment of axioms to the top nodes of the underlying trees. Hence Theorems 5.5.2, 5.6.1, and 5.6.9 characterize ArFbc(X): 5.6.12. THE COMPUTABILITY THEOREMFOR Fbc(X). Relative to X, the sets Fbc(X)(A, B ) are computable f o r all A, B E ObFbc(X). 0 5.6.13. COROLLARY. The embedding X + Fbc(X) defined by f full and faithful.
PROOF. Similar to the proof of Corollary 2.6.6. 0
+uf]
is
BICARTESIAN CATEGORIES
In this chapter, we study the proof-theoretical properties of A , v, T, and 1 that are independent of distributivity. The appropriate class of categorical models for this purpose is the class of small bicartesian categories. 5.1. Definition
A bicartesian category is a Cartesian category C with the following additional structure: (4) A bifunctor (-) v (-) : C X C + C. (5) A distinguished object 1 E ObC. (6) Two adjunctions a. and a&,where a. = {au(A, B, C ) : C(A v B, C)+ C(A, C) X C ( B , C) E ArEns I A, B, C
E ObC},
and
I
a&= {&,(A): C(1, A ) + { * } E ArEns A E ObC}.
5.2. Examples 5.2.1. A category C is bicartesian iff it has finite products and finite
coproducts. Hence, in particular, lattices with smallest and largest elements are bicartesian categories.
5.2.2. COUNTER-EXAMPLE. The set of all functions f : [0, 1]+ R satisfying the condition f ( t x + (1 - t ) y ) > t f ( x )+ (1 - t ) f ( y ) for 0 < t < 1, bounded above by the upper half of the unit circle centred at (f,O), 54
5.31
T H E CATEGORY
Fbc(X)
5.5
partially ordered by f s g iff f ( x ) s g ( x ) for all x E [0, 11, with f A g = {min(f(x), g(x)) 1 x E [0, 11) is a lower semilattice with largest element that is not a lattice. Hence we have a natural example of a Cartesian category which is not bicartesian. 5.3. The category Fbc(X)
Small bicartesian categories are the objects of a category bcCat whose arrows are functors F satisfying the conditions of arrows in cCat and have the additional property that F ( A v B )= F ( A ) v F ( B ) , F(1)= I, and F ( a ; ' ( A ) (*)) = a ; l ( F ( A ) (*)) for all A, B E Obdom(F), and that a,(F(A), F ( B ) , F ( C ) ) ( F ( f )=) ( F ( g ) ,F ( h ) ) for all A, B,C E Obdom(F) and all f, g, h E Ardom(F) for which a,(A, B,C)(f)= (g, h ) . There exists an obvious forgetful functor Ubc : bcCat+ Cat. We now extend the definition of Fc and construct a left adjoint Fbc of Ubc. 5.3.1. DEFINITION. The language of Fbc(X) is the sublanguage bcL(X) of L(X) generated by ObX, T, A , I, v, and ArX. 5.3.2. DEFINITION.The labelled deductive system of Fbc(X) is the
subsystem bc.&(X)of &X) generated by Axioms (AI), (A2), (AlO), (A1I ) , (A12), (A13), (A14), (A15),and Rules ( R I ) , (R3), and (R4).
5.3.3. REMARK.A comparison with 4.3.1 and 4.3.2 shows that bcL(X) and bc&X) result from cL(X) and c&X) by the inclusion of I,v, and (A1 l), (A14), (A15), and (R4), respectively. 5.3.4. DEFINITION. The relation = is the smallest equivalence relation on Der(bc&X)) satisfying the conditions of Definition 4.3.3, and the following additional requirements: (10) If f = g and h = k, then [f, h ] = [g. k l . (1 1 ) comp([f, 81, vn*)= f. (12) comp([f, gl, 6) = g. (13) [comp(k, d), comp(k, v31= k. (14) If d o m u ) = I,then f = 7*.
We now define the category Fbc(X) by modifying and extending the
56
BICARTESIAN CATEGORIES
15.3
definition of Fc(X): (1) ObFbc(X) = bcL(X). (2) ArFbc(X) = Der(bcb(X))/=. (3) As in 4.3.3. (4) As in 4.3.3. (5) As in 4.3.3. (6) As in 4.3.3, with the inclusion of T*, d , and T:, determined by Axioms ( A l l ) , (A14), and (AlS). (7) As in 4.3.3. (8) As in 4.3.3. (9) As in 4.3.3. (10) The image of f : A + B E ArX in Fbc(X) is afn. ( 1 1 ) For all derivable labelled sequents f : A + C and g : B + C, ugni = gin. (12) For all derivable labelled sequents f : A + B and g : C + D, af.1v !dl = U[comp(.rm*(B,D ) ,f), com~(.rr,*(B,D ) ,g)Ib (13) For all A, B E ObFbc(X), &,(A, B, A v B)( l(A v B))= (TA*(A,B ) , &A, U ? ) and &'(A)( * ) = T*(A). This completes the description of Fbc(X). We call the category Fbc(X) the free bicartesian category generated by X. The calculations required to show that it is indeed bicartesian are similar to those in 4.3.3. On the arrows of Cat, the functor Fbc is defined by adapting and extending Definition 4.3.4 thus:
run,
urf,
5.3.5. DEFINITION. Let H : C + D be an arrow of Cat, and Fbc(C) and Fbc(D) be the free bicartesian categories generated by C and D. Then
F%c(H) : Fbc(C)+ Fbc(D) is the functor satisfying the following equations: (1)-(lo) As in 4.3.4, with Fbc in place of Fc. (11) Fbc(H)(A v B ) = Fbc(H)(A) v Fbc(H)(B) for all A, B E ObFbc(C). (12) Fbc(H)(I) = 1. (13) Fbc(H)(lf,gl) = [Fbc(H)Cf), Fbc(H)(g)l for all f , g E ArFbW). (14) Fbc(H)(.r*(A)) = T*(F~c(H)(A)) for all A E ObFbc(C). (15) Fbc(H)(nf(A, B ) ) = d(Fbc(H)(A), Fbc(H)(B)) for all A, B E ObFbc(C). (16) F%c(H)(r,*(A, B)) = r?(Fbc(H)(A), Fbc(H)(B)) for all A, B E ObFbc(C).
5.41
T H E DEDUCTIVE SYSTEM
bcA(X)
57
Once again, the verification that Ubc and Fbc are adjoint functors is routine.
5.4. The deductive system bcA(X) We now expand the deductive system cA(X) to an unlabelled deductive system bcA(X) generating a sequential category which contains an isomorphic copy of Fbc(X) and has the desired syntactic properties. The proof-theoretical interest of bcA(X) lies in the fact that it illustrates the connection between cut eliminability and the distributivity of A and v . We explain this remark by examining the nature of the cut-free representations in A(X) of the distributivity arrows
f : A v ( B A C)+(A v B ) A ( A v C),
h :( A A B ) v ( AA C)+ A
A
( B v C),
g : ( A v B ) A( A v C ) + A v (B
k :A
A
A
c),
( B v C)+ ( A A B ) v ( AA
C),
existing in any sequential category qua distributive lattices as described in 1.1.30. A+A B+B A+A c+ c A-AB BC+AB A-AC BC+AC A+AvB BAC+AVB A+AvC BAC+AVC A v ( B A C )+ A v B A v ( B A C)+Av C A v ( B A C)+(A v B ) A ( Av C)
B+B
(1)
C+C
A+A BC+B BC+C A+A A+A(B A C) BC+BAC A+ A(B A C ) BA+A(BAC) BC+A(BAC) B ( A v C)+ A(B A C) A ( A v C)+ A(B A C) ( A v B ) ( A A C)+ A(B A C ) ( A v B ) A ( A v C)+ A A ( B A C)
(2)
c+ c
A+A A+A B+B AB+A AC+A AB+BC AC+BC AAB+A AAC+A AAB+BvC AAC+BVC (AA B)v (AA B v (AAB)v(AAC)+A ( A A B ) v ( A A C ) + A A ( B v C)
c)+
c
(3)
58
BICARTESIAN CATEGORIES
[5.4
B+B C-PC A+A B+BC C+BC A-A A(B v C ) + A BvC+BC A(B v C ) + A A(B v C ) + BA A(B v C)+ BC A(B v C ) + A ( A A C ) A(B v C ) + B(A A C ) A(B v C)+ ( A A B ) ( A A C ) A A ( B v c)+ ( A A B ) v ( A A
(4)
c)
Derivations (l), (Z), (3), and (4) represent the arrows f, g, h, and k, respectively, with obvious abbreviations. The generality of Rules (R10) and (R12) required in the above derivations may be classified as follows:
The greater generality of (R10) in Derivation (2) can be avoided by replacing the single instance of (R13) towards the end of the derivation be three instances of (R13) higher up in the derivation. We notice that onIy Derivations (2) and (4) require antecedents, respectively succedents, of length greater than 1. Since either one of the inequalities represented by arrows g and k forces a lattice in which it holds to be distributive, it is clear that the length of the succedents of instances of (R10) and the length of the antecedents of instances of (R12) must be restricted to being no greater than 1. On the other hand, the arrows g and k are still representable by these restricted forms of (RIO) and (R12) if the cut rule (Rl) is admitted in full generality: Let a = A A (Bv C ) , P = A ( A A B ) v ( A A C), and let
P
a+
A
( ( AA
4
B)v C), y = A A (C v ( A A B ) ) , 6 = r
P, P+ Y, Y-*
be the following derivations:
6
5.41
T H E DEDUCTIVE SYSTEM
bcA(X)
59
B+B C+C A+A B+B B+BC C+BC AB+A AB+B BvC+BC AB+AAB _- ~~A(B v C)+ B ( A A B ) C A(B v C ) B + ( A A B)C A(B v C)A(B v C)+ ( A A B ) C ( A A B)C ~
~
A+A A(B v C)+ A A A ( B v C)+ A P = A
A
( A A ( B v C))(A A ( B v C))+ ( ( A A B ) v CN(A A B ) v C ) A A (B v C)+(A A B )v C ( B v C)+ A A ((AA B ) v
c)
A+A B+B AB+B AB+A AB+AAB AB+ C ( A A B )
c+c
C + C(AA B ) B+ C v ( A A B ) C + C v ( A A B ) A+A ( AA B ) v C+ C v ( AA B) A ( ( A A B ) v C)+ A A ( ( A A B ) v c ) + C v ( A A B) A A ( ( A A B ) v C)+ A A A ( ( A A B ) v C)+ C v ( A A B ) q= A A ( ( A A B ) v C)+ A A ( C v ( A A B ) ) A
A
A+A B+B AB+A AB+B AB+AAB AAB+AhB C+C C+C A+A C + C ( A A B ) A A B+ C ( A A B ) AC+A AC+C C v ( A A B)+ C ( A A B ) AC+Ar\C A(C v ( A A B))+ C ( A A B ) ( A A C ) A ( C v ( A A B ) ) C + ( A A B ) ( A A C ) A(C v ( A A B ) ) A ( Cv ( A A B ) ) + ( A A B ) ( A A C ) ( A A B ) ( A A C ) ( A A ( C v ( A A B ) ) ) ( AA ( C v ( A A B)))+ ((A A B ) v (A A C))((A A B ) v ( A A C ) ) A A (C v ( A A B ) ) + ( A A B ) v ( A A C)
Then the derivation
represents the arrow k. The arrow g is represented similarly. The cut
60
BICARTESIAN CATEGORIES
p.5
rule (R1) is therefore not admissible in full generality as a rule of inference of bcA(X). Hence we have a non-trivial example of a subsystem of A(X) without a cut elimination theorem: 5.4.1. COUNTER-EXAMPLE. The deductive system consisting of Axioms
(Al), (A3), and (A4), and Rules (RI), (R2), (R3), (RS), (R6), (RIO), and (R12), with the succedents in (RIO) and the antecedents in (R12) restricted to sequences of length 1, does not admit a cut elimination theorem. Fortunately, we can formulate the deductive system bcA(X) by means of two special cases of (Rl) for which a cut elimination theorem holds. The unlabelled deductive system of Fbc(X) is the subsystem bcA(X) of A(X) generated by axioms (Al), (A3), (A4), and the following restrictions of (Rl), (R2), (R3), (RS), (R6), (RlO), (Rll), (R12), and (R13): (R1)
r-+y A ~ A - + Q , ArA+Q,
rjQyq y r+mq
+
~
5.5. The semantics of Der(bcA(X))
We extend the interpretation of Der(cA(X)) in ArFc(X) to an interpretation of Der(bcA(X)) in ArFbc(X) by means of the following canonical arrows of Fbc(X), determined by au: (4) d ( A , B) : A + A v B for all A, B E ObFbc(X), where
a d A , B, A v B)(l(A v B ) ) = (T?(A,B ) , d ( A , B ) ) . ( 5 ) S*(A): A + A v A for all A E ObFbc(X), where
oG'((l(A), I(A))) = S*(A).
5.51
THE SEMANTICS OF
Der(bcA(X))
( 6 ) (a*)-I(A,B, C ) : ( A v B ) v C + A v ( B v C) for ObFbc(X), where
61
all
A , B, C E
a,I(,r?(A v B , C)&(A, B ) , (u,'(T?(Av B, C).rr,*(A,B ) , r , * ( AV B, C ) ) ) = ( a * ) - ' ( AB, , C).
5.5.I . DEFINITION. The interpretation of Der(bcA(X)) in Fbc(X) is the function S : Der(bcA(X))+ ArFbc(X) satisfying Conditions (1)-(7) of 4.5.1 and the following additional equations: (8) S(l+) = l(1): I+ 1.
(Ca v 0)v
*
f
As in Fc(X), the equivalence classes of Der(bcA(X)) obtained by defining f = g iff S c f ) = S ( g ) are plentiful enough to classify the arrows of Fbc(X): 5.5.2. THE COMPLETENESSTHEOREM FOR Der(bcA(X)). For every f E Der(bc&X)) there exists a g E Der(bcA(X)) such that S ( g ) = vl E
ArFbc(X).
62
BICARTESIAN CATEGORIES
[5.5
PROOF.We modify and extend the proof of Theorem 4.5.2. (1) As in 4.5.2. (2) As in 4.5.2, with -the following addition: If efn = 1(1), let g be the derivation l+, and if 1Lfl=1(A v B), let g be the derivation
h k B+B A+A A+AB B+AB AvB+AB AvB+AvB (3) As in 4.5.2, with the following addition: I f f quotes Axioms (All), (A14), or (AlS), let g be the derivations
I+ l+A
h A+A A+AB A+AvB
k B+ B B+AB B+AvB
respectively. (4) As in 4.5.2. (5) As in 4.5.2. (6) If the last line of f consists of an application of (R4), i.e., f is a derivation of the form P 4 u:A+C u:B+C [u, u ] :A v B + C and if 1Ipn= S ( h ) and [qn = S ( k ) , let g be the derivation
h k A+C B+C AvB+C
0
5.5.3.REMARK.We repeat Remark 4.5.3 concerning the use of dots in order to display the instances of (R5) and (R6) unambiguously. 5.5.4. COROLLARY. The category Fbc(X) is isomorphic to a subcategory of the sequential category generated by the deductive system bcA(X) and the interpretation S : Der(bcA(X))+ ArFbc(X). 0
5.61
THE S Y N T A X OF
Fbc(X)
63
5.6. The syntax of Fbc(X)
We extend Theorem 4.6.1 to bcA(X) and show that the arrows of Fbc(X) have a composition-free description:
5.6.1. THE CUT ELIMINATION THEOREMFOR bcA(X). Every f € Der(bcA(X)) is equivalent to a cut-free g E Der(bcA(X)). PROOF. By Theorem 4.6.1, with Clause (C.26) generalized to allow non-empty Q and 9,together with Clauses (C.4), (C.7), (C.8), (C.9), (C.12), (C.15), (C.21), (C.22), (C.29), (C.371, (C.38), (C.43), and (C.44) of the cut elimination algorithm, f reduces to a cut-free derivation g . It remains to show that the reduction steps preserve the meaning of f. Most cases can be disposed of by duality: (C.7) is dual to (C.2), (C. 15) is dual to (C.14), (C.21) is dual to (C.34), ((2.22) is dual to (C.35), (C.29) is dual to (C.42), ((2.37) is dual to (C.18), (C.38) is dual to (C.19), (C.43) is dual to (C.26), and (C.44) is dual to ((2.27). Hence we are left with Cases (C.4), (C.8), (C.9), and (C.12), and the task of re-examining Case ((2.26). A brief reflection shows that (C.8) is dual to (C.4), (C.9) is dual to (C.3), and the two subcases arising in (C.12) are analogous, respectively dual, to (C.2.1). In the case of (C.4), we must show that the following reduction preserves equivalence: g
+@
r -+f
Q
~
~
g
f r-+QaB9 p+o
r+Qa@z
P +@
g
+@ a+@
Since g is cut-free by hypothesis, we may take 0 to be T. Hence the above derivations are equivalent provided that the diagram A v ( B v C)(AvB)vC ( I V T ) V T
IVT
A vT
IVS*
64
[5.6
BICARTESIAN CATEGORIES
commutes for all A, B, C E ObFbc(X). By Axiom (Ml) of monoidal categories, Diagram (1) commutes iff Diagram (2) commutes, where (2) results from (1) by the replacement of a*(A, B, C) by (a*)-'(A, B, C). But by the uniqueness of terminal arrows, Diagram (2) commutes iff
- 1 I 1
A v ( B v C)
IV(TVT)
A v (T v T)
IVS*
A
(a*)-'
(3)
(AvB)vC (IVT)VT
iva*
5.61
THE S Y N T A X OF
Fbc(X)
65
in the proof of Theorem 4.6.1. By the naturality of a,', the right-hand side of Case (3) is equivalent to the derivation
r+ayvsq
YVs+'(YAp
r+@aApq
and by the joint naturality of aa and arrthis derivation is equivalent to the left-hand side of Case (3). This proves the cut elimination theorem for bcA(X). 0 Using the completeness theorem for Der(bcA(X)) and arguing as in the proof of Lemma 4.6.2, we obtain an analogous result for bcA(X):
5.6.2. LEMMA.If a = T and p derivable in bcA(X). 0
= 1 in
Fbc(X), then + a and l3+
are
In order to enable us to establish an analogue of Corollary 4.6.3 for bicartesian categories, we require additional preliminaries: 5.6.3. LEMMA.A = A v I= A
A
T for all A E ObFbc(X).
PROOF.The existence of the required isomorphisms follows at once from the Yoneda lemma since
66
BICARTESIAN CATEGORIES
and
[ X ,A v TI
[ X ,A ] x [ X ,TI
[5.6
[ X ,A ]
for all X , Y E ObFbc(X). In contrast to the situation in cA(X), it is no longer true in bcA(X) that we can take all active formulas in instances of (R2) to be atomic. Since Fbc(X) is non-distributive, the derivation A+A A(B v C ) + A with A, B , C E ObX, for example, cannot be replaced by the derivation A+A A+A AB+A AC+A A(B v C )+ A A dual argument shows that the active formulas of instances of (R5) cannot be taken to be atomic. However, a slightly weaker version of Lemma 4.6.4 and its dual still holds for bcA(X):
5.6.4. LEMMA.For every cut-free f EDer(bcA(X)) there exists an equivalent cut-free g E Der(bcA(X)) containing no instances of (R3) and (R6), and no instances o f (R2) and (R5) whose active formulas are o f the form (Y A p and a v p, respectively. . 0
For the purpose of the proof of our final lemma, we regard the category Ens of 2.2.7 as a bicartesian category, with the empty set 0 as an initial and a fixed one-element set { * } as a terminal object. Similarly, we regard the category comRng of commutative rings with 1 and ring homomorphisms as bicartesian, with the ring of integers Z as an initial and the trivial ring 0 as a terminal object. (The product objects of comRng are Cartesian products with coordinate-wise operations (cf. 6.2.2 for the definition of addition, for example), and the coproduct objects are tensor products of rings, taken over Z (cf. LANG[1965]).) 5.6.5. LEMMA.The objects l ~ and l T v T are neither initial nor terminal objects of Fbc(X).
5.61
T H E S Y N T A X OF
Fbc(X)
67
PROOF. Let Const 0 : X + comRng and Const { * } : X + Ens be the constant functors with object values Const O(A) = 0 and Const { * } ( A )= { * } for all A E ObFbc(X),and let FO: Fbc(X) + comRng and F{*): Fbc(X) + Ens be the induced functors of bicartesian categories. Then F o ( l A I)= Z x Z in comRng, and F+,(T v T) = {*} + {*} in Ens. Obviously, Z x Z + 0. Suppose that Z x Z = Z,and let
be a product diagram determined by this isomorphism. Then I @ ) = comp(h, T A )= comp(h, n,,), and therefore T A = np Since ~ A ( ub), = a and n J a , b ) = b, we have a = b for all a, b E Z . A contradiction. A cardinality argument establishes that { * } + { * } is isomorphic to neither 0 nor {*}. 0 5.6.6. COROLLARY. The initial and terminal objects of Fbc(X) are characterized by the following properties: (1) I f a 1T and p = I , then T and 1 are the only atomic subformulas of a and p. (2) a ~ p = T ifa=p=T. (3) a v p = T if a ( @= ) T and P ( a ) = I . (4) ( ~ ~ p = l i f ~ ( ~ ) ~ T u n d p ( co rui f) a= =I p , pI. (5) a r v p = l i f a r = p = I . 0
5.6.7. COROLLARY. If a = T, there exists a unique derivation g of + a consisting at most of instances of (A3), (RS), (RIO), and (R13), and n o instance of (R5) in g has a n actiue formula of the f o r m a v p. 0
5.6.8. COROLLARY. If p =I,there exists a unique derivation h of p + consisting a t most of instances of (A4), (R2), (R1 I), and (R12), and n o instance of (R2) in h has a n active formula of the f o r m a A p. 0 We now extend the definition of the normality of derivations of cA(X) to the derivations of bcA(X) by defining f E Der(bcA(X)) to be normal if it is normal in the sense of Appendix D, and satisfies four additional requirements:
68
BICARTESIAN CATEGORIES
[5.6
(1) f contains no instance of (R3) and (R6). (2) Unless f is a derivation mentioned in Cases (3) and (4) below, f contains no instances of (R2) and (R5) whose active formulas are of the form (Y v p and (Y A p, respectively. (3) If f derives r+@, and if one of the disjunctions of the formulas of @ is isomorphic to T, then f is of the form g
+@
r+ (p(')r where g is the unique derivation of + @ compatible with Corollary 5.6.7, and where (a)consists of the instances of (R2) required to derive r+@ from +@. (4) If f derives r+@, and if one of the conjunctions of the formulas of r is isomorphic to I and no disjunction of the formulas of @ is isomorphic to T, then f is of the form
where h is the unique derivation of + compatible with Corollary 5.6.8, and where ( 7 ) consists of the instances of (R5) required to derive r+@ from r + . By combining Theorem 5.6.1 with the preceding lemmas and corollaries, and extending the reducibility relation 2 of Appendix D in the obvious way beyond the extension required in Chapter 4, we obtain the desired analogue of Theorem 4.6.5 for Der(bcA(X)): 5.6.9. THE NORMALIZATIONTHEOREM
FOR bcA(X). Every f E Der(bcA(X)) reduces to a unique equivalent normal g E Der(bcA(X)).
PROOF.The theorem follows from Theorem 4.6.5 and the normalization algorithm defined in Appendix D, provided the reductions in Conditions (D.4), (D.10), (D.17), (D.21), (D.34), (D.36), (D.37), (D.38), (D.59, (D.591, and (D.62) preserve equivalence. But this is clear: (1) The equivalences required for (D.4), (D.10), (D.17), (D.21), (D.361, and (D.55) are immediate consequences of the associativity of composition.
5.61
THE SYNTAX OF
Fbc(X)
69
(2) The equivalences required for (D.34) are a consequence of the functoriality of v . (3) The equivalences required for (D.37) and (D.59) follow from the naturality of am. (4) The equivalences required for (D.38) follow from the naturality of a*.
(5) The equivalences required for (D.62) are consequences of the coherence of a*,i.e., Theorem 2.6.1.1. This proves the normalization theorem for bcA(X). 0 In view of the duality of A and v in bcA(X), Theorem 4.6.6 therefore generalizes immediately to bcA(X). Hence we have an effective characterization of commutativity in Fbc(X): 5.6.10. THECHURCH-ROSSER THEOREM FOR bcA(X). I f f = g, then there exists a normal h E Der(bcA(X)) such that f 2 h and g L h. 0 5.6.11. COROLLARY. The word problem f o r the functor Fbc is sol-
vable. cl
As in the previous cases, all normal derivations of a sequent A + B in bcA(X) have the same width because of the restriction on the antedents of instances of (R12) and succedents of instances of (R10) to single formulas, and are effectively determined by the syntax of Fbc(X) relative to any fixed assignment of axioms to the top nodes of the underlying trees. Hence Theorems 5.5.2, 5.6.1, and 5.6.9 characterize ArFbc(X): 5.6.12. THE COMPUTABILITY THEOREMFOR Fbc(X). Relative to X, the sets Fbc(X)(A, B ) are computable f o r all A, B E ObFbc(X). 0 5.6.13. COROLLARY. The embedding X + Fbc(X) defined by f full and faithful.
PROOF. Similar to the proof of Corollary 2.6.6. 0
+uf]
is