We provide a number of new construction techniques for cubical complexes and cubical polytopes, and thus for cubifications (hexahedral mesh generation). As an application we obtain an instance of a cubical 4-polytope that has a non-orientable dual manifold (a Klein bottle). This confirms an existence conjecture of Hetyei (1995). More systematically, we prove that every normal crossing codimensi...

We construct a family of cubical polytypes which shows that the upper bound on the number of facets of a cubical polytope (given a fixed number of vertices) is higher than previously suspected. We also formulate a lower bound conjecture for cubical polytopes.

Extended cubical sets (with connections and interchanges) are presheaves on a ground category, the extended cubical site K, corresponding to the (augmented) simplicial site, the category of finite ordinals. We prove here that K has characterisations similar to the classical ones for the simplicial analogue, by generators and relations, or by the existence of a universal symmetric cubical monoid...

We prove a range of $$L^p$$ bounds for singular Brascamp–Lieb forms with cubical structure. pass through sparse and local bounds, the latter proved by an iteration Fourier expansion, telescoping, Cauchy–Schwarz inequality. allow $$2^{m-1}<p\le \infty $$ m dimension cube, extending earlier result that required $$p=2^m$$ . The threshold $$2^{m-1}$$ is sharp in our theorems.

In homotopy type theory (HoTT), higher inductive types provide a means of defining and reasoning about higher-dimensional objects such as circles and tori. The formulation of a schema for such types remains a matter of current research. We investigate the question in the context of cubical type theory, where the homotopical structure implicit in HoTT is made explicit in the judgmental apparatus...

We associate to any (strict) multiple category C three homology theories : the rst one is called the globular homology and it contains the oriented loops of C ; both other ones are called corner homology, the negative one and the positive one, which contain the corners included in C. We make explicit the link between these homology theories and the homotopy of paths in multiple category. We con...

We introduce a deformed product construction for simple polytopes in terms of lowertriangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the k-faces) are “strictly preserved” under projection. Thus, starting from an arbitrary neighborly simplicial (d−2)-polytope...

Several recent papers have addressed the problem of characterizing the f -vectors of cubical polytopes. This is largely motivated by the complete characterization of the f -vectors of simplicial polytopes given by Stanley, Billera, and Lee in 1980 [19] [4]. Along these lines Blind and Blind [5] have shown that unlike in the simplicial case, there are parity restrictions on the f -vectors of cub...

We provide a number of new construction techniques for cubical complexes and cubical polytopes, and thus for cubifications (hexahedral mesh generation). As an application we obtain an instance of a cubical 4-polytope that has a non-orientable dual manifold (a Klein bottle). This confirms an existence conjecture of Hetyei (1995). More systematically, we prove that every normal crossing codimensi...