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 
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
non-commutative diagrams in SMC categories; our proof of S. Mac Lane
conjecture proves also the correctness of the modified
description
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