# Equivalence via surjections

## TL;DR

This paper establishes a principle for defining equivalences in various categorical structures based on strictly structure-preserving surjective maps, and applies it to categories, monoidal categories, bicategories, and double categories.

## Contribution

It proves a unifying principle for equivalences via surjections across multiple categorical structures, including double categories.

## Key findings

- The principle holds for categories, monoidal categories, bicategories, and double categories.
- Double categories' equivalence aligns with Campbell's gregarious double equivalence.
- Supports recent work on double category equivalences by Moser, Sarazola, and Verdugo.

## 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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Source: https://tomesphere.com/paper/2508.20555