@MISC{Soloviev96proofof, author = {Sergei Soloviev}, title = {Proof of a Conjecture of S. Mac Lane}, year = {1996} }

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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...