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Pseudo algebras and pseudo double categories. (English) Zbl 1184.18003

Summary: 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.

MSC:

18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18C10 Theories (e.g., algebraic theories), structure, and semantics
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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