English

Noun

comma category ‎(plural comma categories)

1. (category theory) A category built on the basis of two other categories with a functor between them, whose objects are pairs object, morphism (morphism, object) where the object belongs to the first (second) category and the morphism to the second (first) category and the domain (codomain) of the morphism is the application of the functor to the object, and whose morphisms are pairs morphism, commuting square (commuting square, morphism) formed by connecting two objects (as defined above) by means of three morphisms, one of them belonging to the first (second) category and the other two to the second (first) category, and the middle morphism being the application of the functor to the first (third) morphism.
2. (category theory) Most generally: A category built on the basis of three other categories with a pair of functors from the flanking categories to the middle category; whose objects are triples object, morphism, object where the first and second objects belong to the first and third categories, respectively, and the morphism to the second category, and the domain of the morphism is the first functor applied to the first object and the codomain of the morphism is the second functor applied to the second object; and whose morphisms are triples morphism, commuting square, morphism formed by connecting two objects (as defined above) by means of four morphisms, one belonging to the first category, a pair to the second category, and one to the third category, where the second morphism is the application of the first functor to the first morphism and the third morphism is the application of the second functor to the fourth morphism.

Hyponyms

• slice category
• coslice category