Pseudo Algebras and Pseudo Double Categories

TL;DR

This paper explores the relationship between pseudo algebras over 2-theories, pseudo double categories, and bicategories, revealing structural equivalences and special cases like crossed modules.

Contribution

It demonstrates that pseudo algebras over the 2-theory of categories can be viewed as pseudo double categories with folding or as 2-functors into bicategories, establishing new categorical equivalences.

Findings

Pseudo algebras over 2-theories relate to pseudo double categories with folding.

Foldings are equivalent to connection pairs and thin structures.

Strict 2-algebras with one object and invertible morphisms are crossed modules.

Abstract

As an example of the categorical apparatus of pseudo algebras over 2-theories, we show that pseudo algebras over the 2-theory of categories can be viewed as pseudo double categories with folding or as appropriate 2-functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2-cells of the horizontal 2-category. As a special case, strict 2-algebras with one object and everything invertible are crossed modules under a group.