A common misinterpretation of Isbell's obstruction to monoidal strictification

TL;DR

This paper clarifies a common misunderstanding of Isbell's obstruction to strictifying the associator in monoidal categories, showing that naturality is not the true obstacle but rather a hidden assumption about product cones.

Contribution

The paper identifies and corrects a misinterpretation of Isbell's argument, providing a new proof that isolates the actual source of the obstruction to strictification.

Findings

Isbell's original argument relies on a hidden hypothesis, not naturality.

Natural isomorphisms can be strictly realized without obstruction.

No general obstruction exists to strictifying the associator component in cartesian monoidal categories.

Abstract

A monoidal category has a natural isomorphism $\alpha_{A,B,C}\colon(A\otimes B)\otimes C\to A\times (B\otimes C)$ called the associator. In the case where the objects $(A\otimes B)\otimes C$ and $A\otimes(B\otimes C)$ are equal, it is natural to ask whether this map may be taken to be the identity. Isbell gave an argument for an obstruction to strictifying the component of the associator of a cartesian monoidal category at an object $C=C\times C$. This argument has been widely reproduced and is commonly misunderstood as demonstrating that naturality of the associator is the true obstruction to strictification. We consider the hidden hypothesis in this argument, give a new argument not dependent on naturality but on the hidden hypothesis, and finally show that naturality alone is not the issue -- rather the crux of Isbell's argument involves a hidden assumption concerning the product cones. Through this analysis we also resolve that there can be no general obstruction to strictifying a component of the of the associator, even at such an object $C$, in a cartesian monoidal category.