A convenient category of cubes

Abstract

We claim that the cube category whose morphisms are the interval-preserving monotone functions between finite Boolean lattices is a convenient general-purpose site for cubical sets. This category is the largest possible concrete Eilenberg-Zilber variant excluding the reversals and diagonals. The category admits as monoidal generators all functions between the singleton and two-element ordinals and all monotone surjections from finite Boolean lattices to the two-element ordinal. Consequently, morphisms in the minimal symmetric monoidal variant of the cube category containing coconnections of one kind can be characterized as the interval-preserving semilattice homomorphisms between finite Boolean lattices. There exists a model structure on our variant of cubical sets that is at once Quillen equivalent to and left induced from the classical model structure on simplicial sets along triangulation. This model structure is proper and hence its fibrations interpret Martin-Lof dependent types.