Proof of a conjecture of S. Mac Lane Original Research Article

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 (this is the only case 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. Voreadouʹs proof of the “abstract coherence theorem” found by the author, it was necessary to modify her description of the class of non-commutative diagrams in SMC categories; our proof of S. Mac Lane conjecture also proves the correctness of the modified description.