Double category explained

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,^{[1] [2]} 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.

References

• Morton. Jeffrey C.. Double bicategories and double cospans. Journal of Homotopy and Related Structures. 4. 389–428. 2009. 1 . math/0611930.
• Kelly . G. M. . Street . Ross . Kelly . Gregory M. . Review of the elements of 2-categories . 10.1007/BFb0063101 . 978-3-540-06966-9 . 357542 . 75–103 . Springer . Lecture Notes in Mathematics . Category Seminar: Proceedings of the Sydney Category Theory Seminar, 1972/1973 . 420 . 1974.

Further reading

• Book: 10.1007/BFb0072309 . The double category of adjoint squares . Reports of the Midwest Category Seminar V . Lecture Notes in Mathematics . 1971 . Palmquist . Paul H. . 195 . 123–153 . 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

Notes and References

1. C. Ehresmann. Catégories Structurées. Ann. Sci. Ecole Norm. Sup 80, pp 349-425. 1963.
2. C. Ehresmann. Catégories et Structures, Dunod, Paris, 1965.