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.

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.