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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?

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    $\begingroup$ Let's see. "Regular skeletal Reedy category" is Cisinski's categorie skeletique reguliere. A categorie skeletique reguliere is a categorie skeletique normale $\mathcal A$ such that every nondegenerate map $A \to X$ in $\hat{\mathcal A}$ with $A$ representable is a monomorphism. A categorie skeletique reguliere in turn is a categorie skeletique with no nonidentity isomorphisms. And a categorie skeletique is what the nlab calls a Cisinski-generalized Reedy category. $\endgroup$
    – Tim Campion
    Commented Jul 23, 2018 at 21:51
  • $\begingroup$ Note that a categorie skeletique normale is the same thing as a Cisinski-generalized Reedy category which happens to be an ordinary Reedy category. So all told, a regular skeletal Reedy category is a Reedy category which satisfies the above condition (every nondegenerate map $A \to X$ in $\hat{\mathcal A}$ with $A$ representable is a monomorphism), as well as the condition from Definition 2.4 from the nlab page linked to above (to ensure that it is a categorie skeletique). $\endgroup$
    – Tim Campion
    Commented Jul 23, 2018 at 21:54
  • $\begingroup$ Drat! There's a typo in my first comment. In the second-to-last sentence, it should say "normale" rather than "reguliere". $\endgroup$
    – Tim Campion
    Commented Jul 23, 2018 at 21:58
  • $\begingroup$ Also, for the category to be (pre)regular, that regularity property only needs to hold for representables, not all presheaves. $\endgroup$
    – Harry Gindi
    Commented Jul 23, 2018 at 22:00
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    $\begingroup$ Good catch. That simplifies things (and makes your question nontrivial!)-- it just says that all the morphisms in $\mathcal A_+$ are monomorphisms. So all told, we have a Reedy category $\mathcal A$ where the morphisms in $\mathcal A_+$ are monomorphisms, the morphisms in $\mathcal A_-$ are split epimorphisms, and a morphism in $\mathcal A_-$ is determined by its sections. $\endgroup$
    – Tim Campion
    Commented Jul 23, 2018 at 22:22

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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.

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