Necessity of shapes for coherence results in category theory

Question

The classic coherence theorems of MacLane (Natural associativity and commutativity, Rice U. studies, 1963) talked about natural transformations between functors. By 1971 (Kelly-MacLane, Coherence in closed categories, JPAA) and 1972 (Kelly, An abstract approach to coherence, LNM 281), functors had been replaced by "shapes"---formal combinations of given functors---with comments to indicate that this replacement is necessary to give correct statements. It is not too hard to build examples that show that the usual coherence statements are false if stated for functors and not shapes. However, I have never seen an example in print. Does anyone know of a reference for an example?

For the curious, here is an example. Let $A_n$ be $n$-tuples of integers regarded as sequences of length $n$, and let $A_m\otimes A_n$ be $A_{m+n}$ obtained by concatenating the sequences. This makes $\otimes$ strictly associative. Build an (idiotic) isomorphism $\alpha$ from $(X\otimes Y)\otimes Z$ to $X\otimes(Y\otimes Z)$ by negating the values contributed by $X$ and $Z$ but not $Y$. Note that the two functors connected by $\alpha$ are identical even though the shapes are different. Now check that the usual pentagonal diagram connecting all five shapes on four variables commutes. Thus our mongrel $\alpha$ is coherent on shapes, but not on the functors themselves. All five functors on four variables are the same, but are connected by 16 different isomorphisms, corresponding to all the elements in $\mathbf (Z_2)^4$, that can be built from $\alpha$.

Answer 1

There is an example in

• Saunders Mac Lane. Topology and Logic as a Source of Algebra (Retiring Presidential Address), Bulletin of the AMS 82:1, January 1976.

See Section 5 ("Coherence and canonical maps"), p.29:

Hence, this theorem really amounts to the statement that, in a monoidal category, every diagram of canonical maps $a, b, c$ must commute. It was proved by Mac Lane [73] and independently by Epstein. Mac Lane's then proof of the theorem was quite correct, but his statement was deficient. He did not discuss words at all, so that he considered not the formal word $A\otimes B$ in two letters $A$ and $B$, but only the functor $M\times M \to M$ which it determines (on any monoidal category $M$). As a result, his formulation, carefully examined, confused the word $A\otimes B$ with the word $B\otimes A$ and his theorem would thus assert that the two central maps $c:A\otimes B \to B\otimes A$ and $1: A\otimes B \to A\otimes B$ are equal. Kelly subsequently set this all straight [50].

([50] is G. M. Kelly. On Mac Lane's conditions for coherence of natural associativities, commutativities, etc., J. Algebra 1 (1964), 397-402. MR 32 #132.)