Equivalence via surjections

Tom Leinster

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arXiv · 2508.20555

Equivalence via surjections

Tom Leinster

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Abstract

Many types of categorical structure obey the following principle: the natural notion of equivalence is generated, as an equivalence relation, by identifying $A$ with $B$ when there exists a strictly structure-preserving map $A \to B$ that is genuinely (not just essentially) surjective in each dimension and faithful in the top dimension. We prove this principle for four types of structure: categories, monoidal categories, bicategories and double categories. The last of these theorems suggests that the right notion of equivalence between double categories is Campbell's gregarious double equivalence, a conclusion also reached for different reasons in recent work of Moser, Sarazola and Verdugo.

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Tom Leinster. 2025-08-28. Equivalence via surjections. https://arxiv.org/abs/2508.20555

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