Double Categories

A double category is a generalized 2-category. A 2-category has objects, arrows, and natural transformations; arrows have one type of composition, but natural transformations have two types of composition, vertical and horizontal composition. A double category has two types of arrows, vertical and horizontal arrows, as well as natural transformations.

Every 2-category can be regarded as a double category with only one type of arrow. Similarly, every category can be regarded as a double category with only one type of arrow, using commuting squares of arrows for natural transformations. Every category's slice and coslice categories can be paired to form a double category.

By the Macrocosm Principle, there is a category DblCat whose objects are double categories and arrows are double functors; this is the category of internal categories in Cat. There are two functors which designate the 2-categories resulting from restricting the vertical and horizontal arrows respectively to identities; they are known as the horizontal and vertical 2-categories.

There are several common constructions for double categories which are included from other tables:

• Every restriction category gives a double category.

2 rows where 2cells_hr = "natural transformations"

✎ View and edit SQL

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CREATE VIEW "all_double_categories" AS select vertical_2cat, horizontal_2cat, `2cells_hr`, 'plain' as kind from double_categories
UNION
select total_subcategory, partial_category, 'equivalences of restricted arrows', 'total restriction' from restriction_categories;