A category of cubical sets

The base category C is the full subcategory of the category of posets having for objects finite power of the poset 0 6 1. We write [1] the poset 0 6 1. We write I, J,K, . . . the object of C and 1I : I → I the identity map of I. If f : J → I and g : K → J we write fg : K → I their composition. If I is an object of C, we have two constant maps c0 : I → [1] and c1 : I → [1]. We write π1 : I × [1] → I and π2 : I × J → [1] the projection maps and if f : I → J and g : I → [1] we write (f, g) : I → J × [1] the pairing map. For any object I we define e0 = (1I , c0) : I → I × [1] and e1 = (1I , c1) : I → I × [1]. We may write I instead of I × [1] and f : J → I the map f(j, b) = (f j, b). We have the lattice operations ∧,∨ : [1] → [1].