Reedy categories and the $Theta$-construction

We use the notion of multi-Reedy category to prove that, if $\mathcal C$ is a Reedy category, then $\Theta \mathcal C$ is also a Reedy category. This result gives a new proof that the categories $\Theta_n$ are Reedy categories. We then define elegant Reedy categories, for which we prove that the Reedy and injective model structures coincide.

💡 Research Summary

The paper tackles two intertwined problems in the theory of Reedy categories. First, it introduces the notion of a multi‑Reedy category—a categorical structure that allows morphisms with a single source but possibly many targets—and uses this to prove that the Θ‑construction preserves the Reedy property. Second, it defines a class of “elegant Reedy categories” and shows that for such categories the Reedy model structure coincides with the injective model structure on the associated presheaf category.

The authors begin by recalling the classical definition of a Reedy category: a small category C equipped with two wide subcategories C⁺ (direct) and C⁻ (inverse) together with a degree function deg: ob(C) → ℕ satisfying the usual factorisation and degree‑monotonicity axioms. They then enrich this picture by considering the symmetric multicategory C(∗) whose multimorphisms are tuples of ordinary morphisms with a common source. A multi‑Reedy category consists of a sub‑multicategory C⁺(∗) ⊂ C(∗) and a subcategory C⁻ ⊂ C together with a degree function such that every multimorphism α admits a unique factorisation α = α⁺α⁻ with α⁻ ∈ C⁻ and α⁺ ∈ C⁺(∗), and the degree constraints are appropriately generalized (deg(source) ≤ sum of degrees of targets for α⁺, and deg(source) ≥ deg(target) for α⁻). Proposition 2.5 shows that any multi‑Reedy category yields an ordinary Reedy category by intersecting C⁺(∗) with the underlying category.

With this machinery in place, the Θ‑construction is defined for any small category C. Objects of Θ C are formal strings