The definition of a (pre)regular skeletal Reedy category in the sense of Cisinski generalizes the intuition behind the Eilenberg-Zilber factorization for simplicial sets, specifically the property that the image of any section of a representable is again representable, or equivalently that all nondegenerate sections of representables are monic. 


Of course, the classical Eilenberg-Zilber property for simplicial sets is stronger, namely that this regularity property holds not only for representables but also for products of representables (which yields the shuffle (i.e. Eilenberg-Zilber) decomposition of simplicial prisms that we all know and love). 


Can the regularity of products of representables be deduced from just the axioms of regular skeletal Reedy categories, or is it an additional axiom that must be included? If it is an additional property, are there any easier-to-check equivalent properties?



Products of representables are regular on a regular skeletal Reedy category?
I asked Cisinski by e-mail if this was true, and he said that it isn't. A counterexample is the category of planar trees ($\Omega\_{\mathbf{pl}}$, not $\Omega$, which is not even normal) used in the theory of Dendroidal sets. He said that this property can be axiomatized by asking that the class of regular presheaves is closed under the Cartesian product.


