A framed polytope is the convex closure of a finite set of points in $\mathbb{R}^n$ together with a basis. An $n$-category is a category that is enriched in $(n-1)$-categories. Although these concepts may initially appear to be distant peaks in the mathematical landscape, there exists a trail connecting them, blazed in the 90's by Kapranov and Voevodsky. We will traverse this path, widening and improving it as we address some of their conjectures along the way.