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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Coherence in closed categories, JPAA) and 1972 (Kelly, An abstract approach to coherence, LNM 281), functors had been replaced by "shapes"---formal combinations of given functors---with comments to indicate that this replacement is necessary to give correct statements.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 $\alpha$.

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There is an example in

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.)

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  • $\begingroup$ Thank you, but ... There are similar such statements in the 1971 and 1972 papers I refer to in the question. All these statements describe the difference, but do not argue that the difference makes a difference. Further, the quote above gives the impression that problems only occur in the commutativity part of symmetric monoidal categories. The example in the question shows a problem already in the associativity part of a monoidal category. I was wondering if the literature had a demonstration of a real problem rather than just an awareness of a potential problem. More ... $\endgroup$ – Matt Brin Jun 14 '16 at 21:13
  • $\begingroup$ And ... . I have looked at [50] above and I don't see a demonstration of the problem in question. In fact I don't even see a reference to the problem. I do see mechanics similar to the example in the question that could have been put to use to build a similar example. My guess is that MacLane's statements above of 1976 conflated the contents of the paper [50] with later communications or conversations with G. M. Kelly. The resemblance is strong. $\endgroup$ – Matt Brin Jun 14 '16 at 21:16

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