nLab quintet construction

Idea

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

$$\array{ A & \overset{f}{\to} & B \\ ^g\downarrow & \swArrow & \downarrow^h\\ C & \underset{k}{\to} & D }$$

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.

Examples

• 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.

Applications

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 assignment of homotopy categories and derived functors to model categories can be made into a double pseudofunctor from the double category of model categories to the double category of quintets in Cat (this Prop.). This implies that mates are preserved even when they relate composites of left and right Quillen functors.

Properties

• The quintet construction admits a left adjoint (see Proposition 8 in Multiple Functors IV).

References

The concept is due to

• Charles Ehresmann, Catégorie double des quintettes; applications covariantes, C. R. A. S. Paris 256 (1963), 1891-1894. and in volume III in his collected works.

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 $Cat_n$, 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