Double category
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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
[edit | edit source]- ^ 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
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Further reading
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- 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
