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 $\alpha:h\circ f \to k\circ g$. The name arises because a square in the resulting cubical structure is formally a quintet $(f,g,h,k,\alpha)$ (since just knowing the globular cell $\alpha$ does not determine the decomposition of its domain and codomain as composites).
Hence given a 2-category $\mathcal{C}$, it induces a double category $Sq(\mathcal{C})$ whose
objects are the objects of $\mathcal{C}$;
horizontal morphisms are the 1-morphisms of $\mathcal{C}$;
vertical morphisms are also the 1-morphisms of $\mathcal{C}$;
2-morphisms are the 2-morphisms of $\mathcal{C}$ 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
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 September 28, 2022 at 19:26:56. See the history of this page for a list of all contributions to it.