## Proof of a Conjecture of S. Mac Lane (1996)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Soloviev96proofof,

author = {Sergei Soloviev},

title = {Proof of a Conjecture of S. Mac Lane},

year = {1996}

}

### OpenURL

### Abstract

Some sufficient conditions on a Symmetric Monoidal Closed category K are obtained such that a diagram in a free SMC category generated by the set A of atoms commutes if and only if all its interpretations in K are commutative. In particular, the category of vector spaces on any field satisfies these conditions (only this case was considered in the original Mac Lane conjecture). Instead of diagrams, pairs of derivations in Intuitionistic Multiplicative Linear logic can be considered (together with categorical equivalence). Two derivations of the same sequent are equivalent if and only if all their interpretations in K are equal. In fact, the assignment of values (objects of K) to atoms is defined constructively for each pair of derivations. Taking into account a mistake in R. Voreadou's proof of the "abstract coherence theorem" found by the author, it was necessary to modify her description of the class of noncommutative diagrams in SMC categories; our proof of S. Mac Lane conjecture proves also the correctness of the modified description. 1 Preface Since the notion of Symmetric Monoidal Closed (SMC) Category, in its axiomatic formulation, was introduced, the category of vector spaces over a field was considered as one of its principal models (see, e.g., [4]). The structure of an SMC category includes tensor product and internal hom-functor, and corresponding natural transformations. A diagram commutes in the free SMC category, iff it commutes in all its models, including vector spaces. But "how faithfully" does the notion of SMC category capture the categorical properties of a model? For example, Does a diagram commute in a free SMC category (and hence in all SMC categories) iff all instantiations by vector spaces give a commutative diagram? The positive answer would...

### Citations

226 |
Closed categories
- Eilenberg, Kelly
- 1966
(Show Context)
Citation Context ... notion of Symmetric Monoidal Closed (SMC) Category, in its axiomatic formulation, was introduced, the category of vector spaces over a field was considered as one of its principal models (see, e.g., =-=[4]-=-). The structure of an SMC category includes tensor product and internal hom-functor, and corresponding natural transformations. A diagram commutes in the free SMC category, iff it commutes in all its... |

41 |
Linear logic and lazy computation
- Lafont
- 1987
(Show Context)
Citation Context ...ctive system" in [17], though deductive systems for SMC categories can be found in earlier works [11], [13]. The calculus may be considered also as the multiplicative intuitionistic linear logic,=-= see [5]-=-.) Axioms 9 A ! A (identity) ! I (unit) Structural Rules \Gamma ! A A; \Delta ! B \Gamma; \Delta ! B (cut) \Delta \Gamma! I \Sigma \Gamma! A \Delta; \Sigma \Gamma! A (wkn) Logical rules \Gamma ! A \De... |

29 |
Deductive systems and categories i
- Lambek
- 1968
(Show Context)
Citation Context ...oms and rules. (This definition does not differ essentially from the definition of the “unlabelled deductive system” in [17], though deductive systems for SMC categories can be found in earlier works =-=[11]-=-, [13]. The calculus may be considered also as the multiplicative intuitionistic linear logic, see [5].) Axioms 9A → A (identity) → I (unit) Structural Rules Γ → A A,∆ → B (cut) Γ,∆ → B ∆ −→ I Σ −→ A... |

25 |
The category of finite sets and cartesian closed categories
- Soloviev
- 1983
(Show Context)
Citation Context ...all briefly the history of the question.) A similar result concerning free Cartesian Closed (CC) Categories and the category of finite sets was proved in the beginning of 80's (Statman [16], Soloviev =-=[14]-=-), and is known as Statman's finite completeness theorem. This work was supported under the ESPRIT project CLICS-II, and mostly carried out while the author was employed by BRICS, a centre of the Dani... |

16 |
Algebra of proofs
- Szabo
- 1978
(Show Context)
Citation Context ... or less generally known, I am unaware of a published record. M. Barr (e-mail to the author) suggested a similar question (not published) independently about 1970. In "Algebra of Proofs" by =-=M.E.Szabo [17]-=-, chapter 8, one can find an assertion that a diagram is commutative in a free SMC category iff every its interpretation in the SMC category of real Banach spaces is commutative. The following stateme... |

9 |
Autonomous categories, with an appendix by Po Hsiang Chu
- Barr
- 1979
(Show Context)
Citation Context ...n find an assertion that a diagram is commutative in a free SMC category iff every its interpretation in the SMC category of real Banach spaces is commutative. The following statement can be found in =-=[1]-=- (1979): Theorem A diagram commutes in all closed symmetric monoidal categories iff it commutes in the category of real vector spaces. It is formulated in the appendix (by Po-Hsuang Chu, introducing t... |

7 |
The structure of free closed categories
- Jay
- 1990
(Show Context)
Citation Context ..."twisted applications" of \Gamma ,- !. These facts are based mostly on the definition of j for sequent derivations and known facts about F(A). For other methods that can be used to check j s=-=ee, e.g., [6]-=-, [13]. Proposition 5.2 (Cut-elimination theorem, cf. [9], [8].) Every derivation in L is equivalent to a cut-free derivation of the same sequent. Proposition 5.3 (Kelly-Mac Lane coherence theorem for... |

7 |
A generalization of the theorem of
- Herzog, Kelly
(Show Context)
Citation Context ...rences of the subformulas of the form C \Gamma ,- D with D constant and C non-constant. Then / j '. There is also a "cut-elimination" theorem, proved by Kelly and Mac Lane, [9], Theorem 6.5,=-= see also [8]. (It allo-=-ws actually some "controlled" use of cut.) Definition 3.4 Let / : A ! B, ' : A\Omega B ! C, ' 0 : A ! B \Gamma ,- C be arbitrary canonical maps. Then denote ! / ?* ) eBC ffi (id B\Gamma ,- C... |

3 | Topology and Logic as a Source of Algebra - Lane - 1976 |

3 |
Closed categories and Proof Theory
- Mints
- 1981
(Show Context)
Citation Context ...d rules. (This definition does not differ essentially from the definition of the "unlabelled deductive system" in [17], though deductive systems for SMC categories can be found in earlier wo=-=rks [11], [13]-=-. The calculus may be considered also as the multiplicative intuitionistic linear logic, see [5].) Axioms 9 A ! A (identity) ! I (unit) Structural Rules \Gamma ! A A; \Delta ! B \Gamma; \Delta ! B (cu... |

3 |
On the conditions of full coherence in closed categories
- Soloviev
- 1990
(Show Context)
Citation Context ...sive algorithm checking commutativity of a diagram can be extracted.) The following remark should be made. We have used the description of non-commutative diagrams, suggested by Voreadou in our paper =-=[15]-=-. The main results of [15] remain valid in spite of necessary modifications in their proofs (we are going to show it elsewhere). In this paper several computational lemmas from [15] are used. They do ... |

3 |
Coherence and non-commutative diagrams in closed categories
- Voreadou
- 1977
(Show Context)
Citation Context ...with opposite variances. Often only the interpretations (or even only one canonical interpretation) in the category N(K) of functors and natural transformations over a category K were considered [9], =-=[18]. This &qu-=-ot;lifting" is closely connected with the interpretations j \Gamma j J , because j/j J always is a component of natural transformation in N(K)). Meanwhile, in general not all the components of th... |

2 | Coherence in Category Theory and the Church-Rosser property. To appear in: Notre Dame Journal of Formal Logic, also accessible from the WWW-site: linus.socs.uts.edu.au/ cbj
- Jay
(Show Context)
Citation Context ...y never published. However, soon after the appearance of [17] it was realized that its proofs contained flaws, and the "Theorem" published in [1] remained a conjecture. Furthermore, in a rec=-=ent paper [7]-=- C.B. Jay has shown that normalization result in chapter 8 is non-correct. The detailed analysis of the proofs presented in chapters 7 and 8 of [17] (concerning nonsymmetric and symmetric monoidal clo... |

1 | Invariance and -definability - Completeness - 1982 |

1 | Invariance and λ -definability - Completeness - 1982 |