I am trying to figure out an axiomatization for Reedy categories such that the product of representables admits a shuffle decomposition. Cisinski suggested imposing a condition, which we will state after recalling the following definition from [1]:

Recall that a quadruple $(C,C^-,C^+, \lambda:\operatorname{Ob}C\to \mathbb{N})$ is called a *skeletal category* (catégorie squelettique), where $C$ is a small category, $C^-$ and $C^+$ are wide subcategories of $C$, and $\lambda$ is a dimension function such that:

- The isomorphisms of $C$ belong to both $C^-$ and $C^+$.
- If $f:a\to a^\prime$ is a map in $C^+$ (resp. $C^-$) that isn't an isomorphism, we have $\lambda(a) < \lambda(a^\prime)$ (resp. $\lambda(a^\prime) < \lambda(a)$)
- Every morphism of $C$ admits a factorization $\delta\sigma$ where $\delta$ belongs to $C^+$ and $\sigma$ belongs to $C^-$ satisfying a uniqueness property where if $\delta^\prime \sigma^\prime$ is another factorization, there exists a unique $\tau$ such that $\tau \sigma= \sigma^\prime$ and $\delta=\delta^\prime\tau$.
- All arrows $\sigma$ in $C^-$ admit sections, and any two such arrows $\sigma$ and $\sigma^\prime$ are equal if they have the same sections

Recall that a skeletal category $C$ is called *normal* if none of its objects has non-identity automorphisms.

Recall that a presheaf $X$ on a skeletal category $C$ is called *regular* if every nondegenerate section $c\to X$ is monic.

Recall that a skeletal category $C$ is called regular if it is normal and all of its representables are regular.

What Cisinski proposed was a further axiom:

Say that a regular skeletal category $C$ is *Cartesian* if the class of regular presheaves on $C$ is closed under the Cartesian product.

The statement of the shuffle decomposition, then, should be that given any two objects $c_1,c_2$ of $C$, every nondegenerate section $c\to c_1\times c_2$ factors through a nondegenerate section $c^\prime\to c_1\times c_2$ with $\lambda(c^\prime)=\lambda(c_1) + \lambda(c_2)$.

The regularity of products in this case asserts, if this is true, that $c_1\times c_2$ is isomorphic to the union of representables $c$ satisfying the dimension condition.

## Question:

Can the existence of such factorizations for nondegenerate sections of products of representables (factoring through a nondegenerate section of additive dimension) be proven from just these axioms? Cisinski seemed to imply that this was the case, but I can't seem to figure out how to deduce the existence of such factorizations.

[1] D.-C. Cisinski, Les Préfaisceaux comme modèles des types d'homotopie