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.
Definitions
[edit]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
[edit]- ↑ C. Ehresmann. Catégories Structurées. Ann. Sci. Ecole Norm. Sup 80, pp 349-425. 1963.
- ↑ C. Ehresmann. Catégories et Structures, Dunod, Paris, 1965.
- ↑ Morton 2009, § 1. Introduction.
- ↑ Morton 2009, Definition 2.2.1.
References
[edit]- 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
[edit]- 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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