Pseudo Algebras and Pseudo Double Categories

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.

60 pages. Exposition improved, Section 5 rewritten to include the 2-equivalence between crossed modules under groups and double groups with folding. This final version will appear in the Journal of Homotopy and Related Structures.

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