TL;DR
This paper explores specific categorical axioms where the invertibility of a canonical natural transformation is equivalent to the invertibility of any natural transformation, focusing on distributive and (semi-)additive categories.
Contribution
It provides two examples of such axioms and demonstrates that these follow from a general result about monoidal functors.
Findings
Distributive categories satisfy these axioms.
(Semi-)additive categories satisfy these axioms.
The results are derived from a general theorem on monoidal functors.
Abstract
We give two examples of categorical axioms asserting that a canonically defined natural transformation is invertible where the invertibility of any natural transformation implies that the canonical one is invertible. The first example is distributive categories, the second (semi-)additive ones. We show that each follows from a general result about monoidal functors.
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