In mathematics, especially

category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. 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 In category theory in mathematics, a 2-category is a category (mathematics), category with "morphisms between morphisms", called 2-morphisms. A basic example is the category Cat of all (small) categories, where a 2-morphism is a natural transforma ...

; namely, the notion of a bicategory is obtained by enrichment, while the notion of a double category is obtained by

internalization Internalization may refer to: * Internalization (sociology) *Internalization (psychology) ** Internalizing disorder *Internalization (biology) Endocytosis is a cellular process in which substances are brought into the cell. The material to be ...

. Precisely, a double category is a category

internal Internal may refer to: *Internality as a concept in behavioural economics *Neijia, internal styles of Chinese martial arts *Neigong or "internal skills", a type of exercise in meditation associated with Daoism * ''Internal'' (album) by Safia, 2016 ...

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.

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