Double category

In mathematics, especially category theory, a double category is a generalization of a category where instead of morphisms, we have vertical morphisms, horizontal morphisms and 2-morphisms. Introduced by Ehresmann in 1960s, the notion may be compared with that of a bicategory; namely, the notion of a bicategory is obtained by enrichment, while the notion of a double category is obtained by internalization. Precisely, a double category is a category internal to Cat (roughly meaning a category object).

Just as iterating the process of obtaining the notion of a 2-category leads to that of an n-category, iterating the process for a double category leads to that of an n-fold category.

Definitions

Double categories can be understood as being made from the following data

• A class of objects
• A class of vertical morphisms between objects
• A class of horizontal morphisms between objects
• A class of 2-morphisms between horizontal morphisms filling in squares

In this presentation both the vertical and horizontal morphisms give a category structure to the objects. These categories are called edge categories.

Footnotes

References

• Morton, Jeffrey C. (2009). "Double bicategories and double cospans" (PDF). Journal of Homotopy and Related Structures. 4 (1): 389–428. arXiv:math/0611930.
• Ehresmann, Charles (1963). "Catégories structurées". Annales Scientifiques de l'École Normale Supérieure. 80 (4): 349–426. doi:10.24033/asens.1125.
• Kelly, G. M.; Street, Ross (1974). "Review of the elements of 2-categories". In Kelly, Gregory M. (ed.). Category Seminar: Proceedings of the Sydney Category Theory Seminar, 1972/1973. Lecture Notes in Mathematics. Vol. 420. Springer. pp. 75–103. doi:10.1007/BFb0063101. ISBN 978-3-540-06966-9. MR 0357542.

Further reading

• Palmquist, Paul H. (1971). "The double category of adjoint squares". Reports of the Midwest Category Seminar V. Lecture Notes in Mathematics. Vol. 195. pp. 123–153. doi:10.1007/BFb0072309. ISBN 978-3-540-05442-9.
• https://ncatlab.org/nlab/show/double+category
• https://math.stackexchange.com/questions/1649138/on-the-definition-of-double-categories
• https://math.stackexchange.com/questions/2395428/whats-a-double-category-with-one-object
• http://pantodon.jp/index.rb?body=double_category in Japanese

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