Cubical sets with connection form a presheaf category on some category $C$. Is $C$ just the full subcategory of the category of posets whose objects are products of the interval $\Delta[1]$?



Is the site for cubical sets with connections equivalent to a full subcategory of posets?
No. Cubical sets with or without connections do not have diagonals. That is, there is no map $\square^1 \to \square^2$ which maps $0$ to $(0,0)$ and $1$ to $(1,1)$.


