# Axiomatization of the shuffle decomposition

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:

1. The isomorphisms of $$C$$ belong to both $$C^-$$ and $$C^+$$.
2. 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. $$\lambda(a^\prime) < \lambda(a)$$)
3. 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 $$\delta=\delta^\prime\tau$$.
4. 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

• Explicit examples admitting shuffle decompositions as stated are the categories $\Theta_n$ for $0\leq n\leq \omega$. – Harry Gindi Oct 10 '18 at 3:23