- Email: [email protected]
Chapter 3 Symmetric Monoidal Categories
CHAPTER 3
SYMMETRIC MONOIDAL CATEGORIES
We now study the effect of the symmetry of the operations A and v on the considerations of Chapter 2. The appropriate class of categorical models for this purpose is the class of small symmetric monoidal categories. 3.1. Definition
A symmetric monoidal category is a monoidal category C with the following additional structure: (4) A natural isomorphism u,where CT
= {cT(A,B ) : Ax
B + B x A E Arc I A, B E ObC}.
The category C satisfies three additional commutativity conditions, for all A, B, C E ObC: LI
(M4) A x ( B x C)----*
( A xB ) x C -
AxB 31
I7
C x (A x B )
32
SYMMETRIC MONOIDAL CATEGORIES
13.3
3.1.1. REMARK.Axioms (Mlt(M6) are known as the MacLane-Kelly coherence conditions for a,A, p, and u.Their independence was proved [1963], they entail that in KELLY[1964]. As was first shown in MACLANE all diagrams whose edges are constructed by means of N, a,A, p, a,and 1, and none of whose vertices contains two occurrences of the same object, with the possible exception of I, commute.
3.2. Examples 3.2.1. The monoidal categories in Examples 2.2.2, 2.2.3, 2.2.4, 2.2.5 and 2.2.8 all have a natural symmetric structure. 3.2.2. For any commutative ring K, the category mod of left K modules with the usual tensor product as bifunctor and the ring K as distinguished object, carries a symmetric monoidal structure, and so does the category ModK of right K-modules. 3.2.3. COUNTER-EX~~MPLES. In addition to the trivial counter-examples provided by the non-commutative monoids in Example 2.2.1, the following construction, suggested by Michael Barr, shows that not every monoidal category of R-R-bimodules admits a symmetric structure: Let RModR be the monoidal category of R-R-bimodules of Example 2.2.6, where R = k [ x ] is a ring of polynomials over a field k, and let RMRand RNR be two R-R-bimodules whose actions are determined by the conditions RM1 RR, N R = RR, & ( m , x) = m for all m E M , and +(x, n) = 0 for all n E N. Then the tensor product M @ R N = 0, since m @ n = & ( m , x ) @ n = m @+(x, n) = m 8 0 = 0, and N @ R M= R. Hence RModR is not symmetric monoidal with respect to the tensor product.
3.3. The category Fsm(X) Small symmetric monoidal categories are the objects of a category smCat whose arrows are the arrows F of mCat satisfying the property that F(u(A,B ) ) = a ( F ( A ) ,F ( B ) ) ,for all A, B E Obdom(F). There exists an obvious forgetful functor Usm : smCat+ Cat, and we now extend the definition of Fm to construct a left adjoint functor Fsm : Cat+ smCat of Usm.
3.41
T H E DEDUCTIVE SYSTEM
smA(X)
33
3.3.1. DEFINITION.The language of Fsm(X) is the sublanguage smL(X) of L(X) generated by ObX, I, I, and ArX. 3.3.2. DEFINITION.The labelled deductive system of Fsm(X) is the subsystem smb(X) of b(X) generated by Axioms (Al), (A2), (A3), (A4), ( A 9 , (A6), (A7), (h), (A9), and Rules (81) and (R2). 3.3.3. REMARK.A comparison with 2.3.1 and 2.3.2 shows that smL(X) = mL(X), and that smb(X) results from m&X) by the inclusion of Axiom (A5). 3.3.4. DEFINITION. The relation = is the smallest equivalence relation on Der(smb(X)) satisfying the conditions of the equivalence relation defined in 2.3.4 and the following additional requirements: (16) compCf I g, a )= comp(a, g I f). (17) comp(comp(a, a),a)= comp(comp(a I 1, a),1 I a). (18) comp(h, a)= p. (19) comp(c+,a)= l(dom(a)). The category Fsm(X) is defined analogously to the category Fm(X) with Clause (7) in the definition of Fm(X) now also mentioning Axiom (AS).We call the category Fsm(X) the free symmetric monoidal category generated by X. The values of Fsm on the arrows of Cat are defined as in 2.3.6, with the following additional clause: (1 1) Fsm(H)(a(A, B)) = a(Fsm(H)(A), Fsm(H)(B)) for all A, B E ObFsm(X). As in Chapter 2, the verification that Usm and Fsm are adjoint functors is routine. We now extend the composition-free description of Fm(X) to a composition-free description of Fsm(X). 3.4. The deductive system smA(X) The unlabelled deductive system of Fsm(X) is the subsystem smA(X) of A(X) generated by Axioms (Al), (A2), and the following restrictions of Rules (Rl), (R2), (R4), (R8), and (R9):
34
SYMMETRIC MONOIDAL CATEGORIES
[3.5
3.4.1. REMARK.The deductive system smA(X) results from mA(X) by the inclusion of Rule (R4) as additional rule of inference.
3.5. The semantics of Der(smA(X))
We extend the interpretation of Der(mA(X)) in ArFm(X) to an interpretation S : Der(smA(X))+ ArFsm(X) by means of Clauses (1)-(6) of 2.5.1 and the following additional definition:
Analogously to the identification in 2.5, we put f = g iff Scf) = S ( g ) , and obtain a bijection Der(smA(X))/= = ArFsm(X): 3.5.1. THE COMPLETENESS THEOREMFOR Der(smA(X)). For every f E Der(smb(X)) there exists a g E Der(smA(X)) such that S ( g ) = afJl E ArFsm(X).
3.61
THE SYNTAX OF
Fsrn(X)
35
PROOF. The theorem follows from the proof of Theorem 2.5.2, with Case (3) augmented as follows: If f quotes (h), and S ( h ) = 1(A) and S ( k ) = l ( B ) , let g be the derivation h k B+B A + A BA-BxlA AB+BnA AMB+BIA
0
3.5.2. REMARK.In the proof of Theorem 3.5.1, it was tacitly assumed that the notation reveals the active formulas of instances of (R4). In cases of ambiguity, the active formulas will be highlighted by means of dots enclosing them. Thus
h .AA.A+ B .AA.A + B
and
h A.AA. --* B A.AA.+ B
denote different derivations. Since A ( 1 ) = p ( 1 ) in Fsm(X), such distinctions are unnecessary in the case of (R2). 3.5.3. COROLLARY.The category Fsm(X) is isomorphic to a subcategory of the sequential category generated by the deductive system smA(X) and the interpretation S : Der(smA(X))+ ArFsm(X). 0
3.6. The syntax of Fsm(X)
The cut elimination theorem for mA(X) extends to smA(X) and affords the same syntactic advantages. 3.6.1. THE CUT ELIMINATION THEOREM FOR smA(X). Every f € Der(smA(X)) is equivalent to a cut-free g E Der(smA(X)).
The proof of Theorem 3.6.1 uses the following result:
3.6.1.1. THE COHERENCETHEOREMFOR smCat (MacLane). If X is
36
SYMMETRIC MONOIDAL CATEGORIES
[3.6
discrete, then for all A, B E ObFsm(X) with the property that they contain no object of X more than once, Fsm(X)(A, B ) contains at most one element. 0 PROOF OF THEOREM3.6.1. The theorem follows from the proof of Theorem 2.6.1, together with Clauses (C.20.1-4) and (C.36) of the cut elimination algorithm described in Appendix C, provided that the mentioned clauses preserve equivalence. But an inspection shows that nowhere do they depend on the identity of formulas other than the two instances of the cut formula. Hence the required equivalences are immediate consequences of the coherence theorem for symmetric monoidal categories. This proves the cut elimination theorem for smA(X). 0 The following counter-example shows that the coherence theorem for smCat does not hold unconditionally: 3.6.2. COUNTER-EXAMPLE. Let A = B N B, with B E ObX. Then the set Fsm(X)(A, A) contains two distinct elements: 1(B N B ) and a ( B , B).
Using Clauses (D.l), (D.3), (DS), (D.6), (D.24), (D.26), (D.27), (D.40), and (D.43) of the reducibility relation L defined in Appendix D, together with the coherence theorem for smCat, we can strengthen the cut elimination theorem: 3.6.3. THE NORMALIZATION THEOREM FOR smA(X). Every f E Der(smA(X)) reduces to a unique equivalent normal g E Der(smA(X)). 0
The usefulness of normal derivations derives from the fact that they are effectively calculable by the cut elimination and normalization algorithms, and that this process makes the equality relation in Fsm(X) decidable: 3.6.4. THE CHURCH-ROSSER THEOREMFOR smA(X). If f = g , then there exists a normal h E Der(smA(X)) such thaf f 2 h and g 2 h.
PROOF.Since Fsm(X) is free on X and since
L
preserves equivalence, it
3.61
THE S Y N T A X OF
Fsm(X)
37
is sufficient to show that distinct normal derivations f, g : A + a represent distinct arrows in Ens. In the light of Theorem 3.6.1, we may assume that X is discrete. By Clauses (D.6), (D.27), (D.40), and (D.43) of 2 , we may assume that f and g contain no instances of (R9), and by Clauses (D.I), (D.3). (D.5), and (D.6) that f and g contain no instances of (R2). Thus if I is the only atomic subformula of a,then A is empty because of the absence of instances of (R2), and by Theorem 3.6.1, f quotes an axiom iff g quotes an axiom. Under these conditions, neither f nor g contains an instance of (R4). By Clause (D.26), the same is true if both f and g end with an instance of (R8). In these cases, the result therefore follows from Theorem 2.6.3. Two possibilities remain: 1. Derivation f ends with (R8), and therefore contains no instances of (R4),and g ends with (R4). 2. Both f and g end with (R4). It is clear from the nature of (D.24) and (D.26) that the following examples are typical: (1) f and g are the derivatives A+A B+B AB+AnB
g A + A B+B AB+AnB BA+AnB
with A = B E ObX. (2) f and g are among the derivations A-+A B+B AB+AnB C+C ABC -+ ( A I B ) n C BAC + ( A n B ) n C
A+A B+B C+C AB+AxB ABC + ( A n B ) x C ACB + ( A n B ) n C
A+A B+B AB-+AnB C+C ABC+ ( An B ) n C BAC + ( A n B ) x C BCA -+ ( A x B ) n C
A+A B+B AB+AnB C+C ABC -+ ( A n B ) 11: C ACB+ ( A nB)n C CAB -+ ( A n B ) n C
38
SYMMETRIC M O NO I DAL CATEGORIES
A+A B+B AB+AMB C+C A B C + ( A x( B)x( C
[3.6
A+A B+B AB+AnB C+C ABC + (Ax(B ) MC ACB + ( A x( B)x( C CAB + ( A 10: B ) n C CBA + ( A x( B ) XI C
with A = B = C E ObX. Let M be an infinite set, consider Ens as a symmetric monoidal category with respect to Cartesian products, and let FM : Fsm(X) +Ens be the unique functor which preserves the symmetric monoidal structure of Fsm(X) exactly and agrees with the constant functor ConstM : X-Ens on X. Then it follows from the definition of the interpretation S that F M ( S ( f )#) F M ( S ( g ) )Since . M is infinite, the functor FM will separate all similar derivations containing any finite number of instances of (R8). 0 3.6.5. COROLLARY. The word problem f o r the functor Fsm is solvable. 0
It is clear that all normal derivations in smA(X) of a sequent A + B have the same width and are effectively determined by the syntax of Fsm(X) relative to any fixed assignment of axioms to the top nodes of the underlying trees. Hence Theorems 3.5.1, 3.6.1, and 3.6.3 characterize ArFsm(X): 3.6.6. THE COMPUTABILITY THEOREMFOR Fsm(X). Relative to X, the sets Fsm(X)(A, B ) are computable f o r all A , B E ObFsm(X). Cl 3.6.7. COROLLARY. The embedding X + Fsm(X) defined by f + afl is full and faithful.
PROOF.Similar to the proof of Corollary 2.6.6. 0 3.6.8. REMARK.If X is discrete, and Al, . . . , A , are n not necessarily distinct objects of X, then a calculation in Ens, regarded as a symmetric monoidal category with respect to Cartesian products, shows that any arrow f : A1 . A. --* B in the sequential category generated by X deter-
- -
3.61
THE SYNTAX OF
Fsm(X)
39
mines @(n) n ! distinct arrows of Fsm(X). Moreover, Fsm(X)(A, B ) is empty for all A, B E ObFsm(X), if A and B do not contain the same number of occurrences of an object of X. As in the case of Fm(X), the number of objects isomorphic to any given object of Fsm(X) is infinite by virtue of the presence of I.
SYMMETRIC MONOIDAL CATEGORIES
We now study the effect of the symmetry of the operations A and v on the considerations of Chapter 2. The appropriate class of categorical models for this purpose is the class of small symmetric monoidal categories. 3.1. Definition
A symmetric monoidal category is a monoidal category C with the following additional structure: (4) A natural isomorphism u,where CT
= {cT(A,B ) : Ax
B + B x A E Arc I A, B E ObC}.
The category C satisfies three additional commutativity conditions, for all A, B, C E ObC: LI
(M4) A x ( B x C)----*
( A xB ) x C -
AxB 31
I7
C x (A x B )
32
SYMMETRIC MONOIDAL CATEGORIES
13.3
3.1.1. REMARK.Axioms (Mlt(M6) are known as the MacLane-Kelly coherence conditions for a,A, p, and u.Their independence was proved [1963], they entail that in KELLY[1964]. As was first shown in MACLANE all diagrams whose edges are constructed by means of N, a,A, p, a,and 1, and none of whose vertices contains two occurrences of the same object, with the possible exception of I, commute.
3.2. Examples 3.2.1. The monoidal categories in Examples 2.2.2, 2.2.3, 2.2.4, 2.2.5 and 2.2.8 all have a natural symmetric structure. 3.2.2. For any commutative ring K, the category mod of left K modules with the usual tensor product as bifunctor and the ring K as distinguished object, carries a symmetric monoidal structure, and so does the category ModK of right K-modules. 3.2.3. COUNTER-EX~~MPLES. In addition to the trivial counter-examples provided by the non-commutative monoids in Example 2.2.1, the following construction, suggested by Michael Barr, shows that not every monoidal category of R-R-bimodules admits a symmetric structure: Let RModR be the monoidal category of R-R-bimodules of Example 2.2.6, where R = k [ x ] is a ring of polynomials over a field k, and let RMRand RNR be two R-R-bimodules whose actions are determined by the conditions RM1 RR, N R = RR, & ( m , x) = m for all m E M , and +(x, n) = 0 for all n E N. Then the tensor product M @ R N = 0, since m @ n = & ( m , x ) @ n = m @+(x, n) = m 8 0 = 0, and N @ R M= R. Hence RModR is not symmetric monoidal with respect to the tensor product.
3.3. The category Fsm(X) Small symmetric monoidal categories are the objects of a category smCat whose arrows are the arrows F of mCat satisfying the property that F(u(A,B ) ) = a ( F ( A ) ,F ( B ) ) ,for all A, B E Obdom(F). There exists an obvious forgetful functor Usm : smCat+ Cat, and we now extend the definition of Fm to construct a left adjoint functor Fsm : Cat+ smCat of Usm.
3.41
T H E DEDUCTIVE SYSTEM
smA(X)
33
3.3.1. DEFINITION.The language of Fsm(X) is the sublanguage smL(X) of L(X) generated by ObX, I, I, and ArX. 3.3.2. DEFINITION.The labelled deductive system of Fsm(X) is the subsystem smb(X) of b(X) generated by Axioms (Al), (A2), (A3), (A4), ( A 9 , (A6), (A7), (h), (A9), and Rules (81) and (R2). 3.3.3. REMARK.A comparison with 2.3.1 and 2.3.2 shows that smL(X) = mL(X), and that smb(X) results from m&X) by the inclusion of Axiom (A5). 3.3.4. DEFINITION. The relation = is the smallest equivalence relation on Der(smb(X)) satisfying the conditions of the equivalence relation defined in 2.3.4 and the following additional requirements: (16) compCf I g, a )= comp(a, g I f). (17) comp(comp(a, a),a)= comp(comp(a I 1, a),1 I a). (18) comp(h, a)= p. (19) comp(c+,a)= l(dom(a)). The category Fsm(X) is defined analogously to the category Fm(X) with Clause (7) in the definition of Fm(X) now also mentioning Axiom (AS).We call the category Fsm(X) the free symmetric monoidal category generated by X. The values of Fsm on the arrows of Cat are defined as in 2.3.6, with the following additional clause: (1 1) Fsm(H)(a(A, B)) = a(Fsm(H)(A), Fsm(H)(B)) for all A, B E ObFsm(X). As in Chapter 2, the verification that Usm and Fsm are adjoint functors is routine. We now extend the composition-free description of Fm(X) to a composition-free description of Fsm(X). 3.4. The deductive system smA(X) The unlabelled deductive system of Fsm(X) is the subsystem smA(X) of A(X) generated by Axioms (Al), (A2), and the following restrictions of Rules (Rl), (R2), (R4), (R8), and (R9):
34
SYMMETRIC MONOIDAL CATEGORIES
[3.5
3.4.1. REMARK.The deductive system smA(X) results from mA(X) by the inclusion of Rule (R4) as additional rule of inference.
3.5. The semantics of Der(smA(X))
We extend the interpretation of Der(mA(X)) in ArFm(X) to an interpretation S : Der(smA(X))+ ArFsm(X) by means of Clauses (1)-(6) of 2.5.1 and the following additional definition:
Analogously to the identification in 2.5, we put f = g iff Scf) = S ( g ) , and obtain a bijection Der(smA(X))/= = ArFsm(X): 3.5.1. THE COMPLETENESS THEOREMFOR Der(smA(X)). For every f E Der(smb(X)) there exists a g E Der(smA(X)) such that S ( g ) = afJl E ArFsm(X).
3.61
THE SYNTAX OF
Fsrn(X)
35
PROOF. The theorem follows from the proof of Theorem 2.5.2, with Case (3) augmented as follows: If f quotes (h), and S ( h ) = 1(A) and S ( k ) = l ( B ) , let g be the derivation h k B+B A + A BA-BxlA AB+BnA AMB+BIA
0
3.5.2. REMARK.In the proof of Theorem 3.5.1, it was tacitly assumed that the notation reveals the active formulas of instances of (R4). In cases of ambiguity, the active formulas will be highlighted by means of dots enclosing them. Thus
h .AA.A+ B .AA.A + B
and
h A.AA. --* B A.AA.+ B
denote different derivations. Since A ( 1 ) = p ( 1 ) in Fsm(X), such distinctions are unnecessary in the case of (R2). 3.5.3. COROLLARY.The category Fsm(X) is isomorphic to a subcategory of the sequential category generated by the deductive system smA(X) and the interpretation S : Der(smA(X))+ ArFsm(X). 0
3.6. The syntax of Fsm(X)
The cut elimination theorem for mA(X) extends to smA(X) and affords the same syntactic advantages. 3.6.1. THE CUT ELIMINATION THEOREM FOR smA(X). Every f € Der(smA(X)) is equivalent to a cut-free g E Der(smA(X)).
The proof of Theorem 3.6.1 uses the following result:
3.6.1.1. THE COHERENCETHEOREMFOR smCat (MacLane). If X is
36
SYMMETRIC MONOIDAL CATEGORIES
[3.6
discrete, then for all A, B E ObFsm(X) with the property that they contain no object of X more than once, Fsm(X)(A, B ) contains at most one element. 0 PROOF OF THEOREM3.6.1. The theorem follows from the proof of Theorem 2.6.1, together with Clauses (C.20.1-4) and (C.36) of the cut elimination algorithm described in Appendix C, provided that the mentioned clauses preserve equivalence. But an inspection shows that nowhere do they depend on the identity of formulas other than the two instances of the cut formula. Hence the required equivalences are immediate consequences of the coherence theorem for symmetric monoidal categories. This proves the cut elimination theorem for smA(X). 0 The following counter-example shows that the coherence theorem for smCat does not hold unconditionally: 3.6.2. COUNTER-EXAMPLE. Let A = B N B, with B E ObX. Then the set Fsm(X)(A, A) contains two distinct elements: 1(B N B ) and a ( B , B).
Using Clauses (D.l), (D.3), (DS), (D.6), (D.24), (D.26), (D.27), (D.40), and (D.43) of the reducibility relation L defined in Appendix D, together with the coherence theorem for smCat, we can strengthen the cut elimination theorem: 3.6.3. THE NORMALIZATION THEOREM FOR smA(X). Every f E Der(smA(X)) reduces to a unique equivalent normal g E Der(smA(X)). 0
The usefulness of normal derivations derives from the fact that they are effectively calculable by the cut elimination and normalization algorithms, and that this process makes the equality relation in Fsm(X) decidable: 3.6.4. THE CHURCH-ROSSER THEOREMFOR smA(X). If f = g , then there exists a normal h E Der(smA(X)) such thaf f 2 h and g 2 h.
PROOF.Since Fsm(X) is free on X and since
L
preserves equivalence, it
3.61
THE S Y N T A X OF
Fsm(X)
37
is sufficient to show that distinct normal derivations f, g : A + a represent distinct arrows in Ens. In the light of Theorem 3.6.1, we may assume that X is discrete. By Clauses (D.6), (D.27), (D.40), and (D.43) of 2 , we may assume that f and g contain no instances of (R9), and by Clauses (D.I), (D.3). (D.5), and (D.6) that f and g contain no instances of (R2). Thus if I is the only atomic subformula of a,then A is empty because of the absence of instances of (R2), and by Theorem 3.6.1, f quotes an axiom iff g quotes an axiom. Under these conditions, neither f nor g contains an instance of (R4). By Clause (D.26), the same is true if both f and g end with an instance of (R8). In these cases, the result therefore follows from Theorem 2.6.3. Two possibilities remain: 1. Derivation f ends with (R8), and therefore contains no instances of (R4),and g ends with (R4). 2. Both f and g end with (R4). It is clear from the nature of (D.24) and (D.26) that the following examples are typical: (1) f and g are the derivatives A+A B+B AB+AnB
g A + A B+B AB+AnB BA+AnB
with A = B E ObX. (2) f and g are among the derivations A-+A B+B AB+AnB C+C ABC -+ ( A I B ) n C BAC + ( A n B ) n C
A+A B+B C+C AB+AxB ABC + ( A n B ) x C ACB + ( A n B ) n C
A+A B+B AB-+AnB C+C ABC+ ( An B ) n C BAC + ( A n B ) x C BCA -+ ( A x B ) n C
A+A B+B AB+AnB C+C ABC -+ ( A n B ) 11: C ACB+ ( A nB)n C CAB -+ ( A n B ) n C
38
SYMMETRIC M O NO I DAL CATEGORIES
A+A B+B AB+AMB C+C A B C + ( A x( B)x( C
[3.6
A+A B+B AB+AnB C+C ABC + (Ax(B ) MC ACB + ( A x( B)x( C CAB + ( A 10: B ) n C CBA + ( A x( B ) XI C
with A = B = C E ObX. Let M be an infinite set, consider Ens as a symmetric monoidal category with respect to Cartesian products, and let FM : Fsm(X) +Ens be the unique functor which preserves the symmetric monoidal structure of Fsm(X) exactly and agrees with the constant functor ConstM : X-Ens on X. Then it follows from the definition of the interpretation S that F M ( S ( f )#) F M ( S ( g ) )Since . M is infinite, the functor FM will separate all similar derivations containing any finite number of instances of (R8). 0 3.6.5. COROLLARY. The word problem f o r the functor Fsm is solvable. 0
It is clear that all normal derivations in smA(X) of a sequent A + B have the same width and are effectively determined by the syntax of Fsm(X) relative to any fixed assignment of axioms to the top nodes of the underlying trees. Hence Theorems 3.5.1, 3.6.1, and 3.6.3 characterize ArFsm(X): 3.6.6. THE COMPUTABILITY THEOREMFOR Fsm(X). Relative to X, the sets Fsm(X)(A, B ) are computable f o r all A , B E ObFsm(X). Cl 3.6.7. COROLLARY. The embedding X + Fsm(X) defined by f + afl is full and faithful.
PROOF.Similar to the proof of Corollary 2.6.6. 0 3.6.8. REMARK.If X is discrete, and Al, . . . , A , are n not necessarily distinct objects of X, then a calculation in Ens, regarded as a symmetric monoidal category with respect to Cartesian products, shows that any arrow f : A1 . A. --* B in the sequential category generated by X deter-
- -
3.61
THE SYNTAX OF
Fsm(X)
39
mines @(n) n ! distinct arrows of Fsm(X). Moreover, Fsm(X)(A, B ) is empty for all A, B E ObFsm(X), if A and B do not contain the same number of occurrences of an object of X. As in the case of Fm(X), the number of objects isomorphic to any given object of Fsm(X) is infinite by virtue of the presence of I.