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, ^{[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. ^[3] Precisely, a double category is a category internal to Cat (roughly meaning a category object). ^[4]

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.

Footnotes

1. C. Ehresmann. Catégories Structurées. Ann. Sci. Ecole Norm. Sup 80, pp 349-425. 1963.
2. C. Ehresmann. Catégories et Structures, Dunod, Paris, 1965.
3. Morton 2009 , § 1. Introduction.
4. Morton 2009 , Definition 2.2.1.

References

• Morton, Jeffrey C. (2009). "Double bicategories and double cospans". Journal of Homotopy and Related Structures. 4 (1): 389–428. arXiv: math/0611930 .
• 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