nLab quintet construction

The Quintet Construction

The Quintet Construction


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

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

are morphisms between composites α:hfkg\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,α)(f,g,h,k,\alpha) (since just knowing the globular cell α\alpha does not determine the decomposition of its domain and codomain as composites).

Hence given a 2-category 𝒞\mathcal{C}, it induces a double category Sq(𝒞)Sq(\mathcal{C}) whose



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

  • 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 nCat_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

Last revised on March 31, 2023 at 12:21:18. See the history of this page for a list of all contributions to it.