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 at least morally a left adjoint to a functor that picks out the companion pairs in a cubical structure. 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
Last revised on July 13, 2018 at 07:21:46. See the history of this page for a list of all contributions to it.