nLab non-canonical isomorphism

Idea

Often in category theory, requirements that certain things “be isomorphic” are more precisely stated as saying that a certain “canonical” morphism is invertible. However, in certain situations, it happens that the existence of a non-canonical isomorphism automatically entails the “correct” stronger property that the canonical morphism is invertible.

For now, see the references for details.

Examples

• If $\mathcal{C}^{\otimes}$ a symmetric monoidal (∞,1)-category which attains a symmetric monoidal equivalence to a cartesian symmetric monoidal (∞,1)-category, then there is a contractible space of symmetric monoidal equivalences $\mathcal{C}^\otimes \simeq \mathcal{C}^\times$ lifting the identity endofunctor of $\mathcal{C}$.

References

• Stephen Lack, Non-canonical isomorphisms [arXiv:0912.2126]

• Fernando Lucatelli Nunes, Pseudoalgebras and non-canonical isomorphisms [arxiv:1711.02051]