Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
A quintet construction is an operation that takes a “globular” categorical structure and produces a “cubical” one (see geometric shape for higher structures) in which the squares
are morphisms between composites . The name arises because a square in the resulting cubical structure is formally a quintet (since just knowing the globular cell does not determine the decomposition of its domain and codomain as composites).
Hence given a 2-category , it induces a double category whose
objects are the objects of ;
horizontal morphisms are the 1-morphisms of ;
vertical morphisms are also the 1-morphisms of ;
2-morphisms are the 2-morphisms of between the compositions of the 1-morphisms.
Quintets in a strict 2-category form a strict double category.
Quintets in a bicategory form a double bicategory.
Quintets in a Gray-category form an intercategory.
By a certain stretch of terminology, the singular cubical set of a topological space might be called a quintet construction.
A quintet construction is a left adjoint to a functor that picks out the companion pairs in a cubical structure (see Theorem 1.7 in Grandis and Paré). Thus, functors out of quintet constructions are nothing new; but some interesting applications involve functors into quintet constructions from other cubical structures. For instance:
The concept is due to
It appears spelled out also in
A. Bastiani, Charles Ehresmann, pages 272-273 of Multiple functors. I. Limits relative to double categories, Cah. Top. Géom. Différ. Catég. 15 (1974) 215–292
Andrée Ehresmann and Charles Ehresmann. Multiple functors IV. Monoidal closed structures on , Cahiers de topologie et géométrie différentielle, Volume 20 (1979) no. 1, pp. 59-104. (link)
Grandis, Marco, and Robert Paré. “Adjoint for double categories.” Cahiers de topologie et géométrie différentielle catégoriques 45.3 (2004): 193-240. link
Last revised on March 31, 2023 at 12:21:18. See the history of this page for a list of all contributions to it.