On an extension of the notion of Reedy category

We extend the classical notion of a Reedy category so as to allow non-trivial automorphisms. Our extension includes many important examples occuring in topology such as Segal’s category Gamma, or the total category of a crossed simplicial group such as Connes’ cyclic category Lambda. For any generalized Reedy category R and any cofibrantly generated model category E, the functor category E^R is shown to carry a canonical model structure of Reedy type.

💡 Research Summary

The paper proposes a substantial generalization of the classical notion of a Reedy category, allowing non‑trivial automorphism groups at each object. In the traditional setting a Reedy category R is equipped with two wide subcategories R⁺ (the “degree‑increasing” maps) and R⁻ (the “degree‑decreasing” maps) together with a degree function d:Ob(R)→ℕ. The axioms force every non‑identity morphism to lie uniquely in either R⁺ or R⁻ and require that the only automorphisms are identities. This restriction excludes many naturally occurring indexing categories, most notably Segal’s Γ and the total categories of crossed simplicial groups such as Connes’ cyclic category Λ.

The authors introduce the concept of a generalized Reedy category. The definition retains the decomposition of any morphism f:x→y as f = f⁺ ∘ α ∘ f⁻ where f⁺∈R⁺, f⁻∈R⁻ and α∈Aut_R(y). The degree function still increases along R⁺ and decreases along R⁻, but now each object may have a non‑trivial automorphism group G_x = Aut_R(x). The axioms also require that R⁺ and R⁻ are closed under the action of these automorphisms and that any morphism between objects of the same degree is necessarily an automorphism. In this way the classical Reedy structure is recovered when all G_x are trivial, while the new framework embraces categories with rich symmetry.

With this structure in place the paper reconstructs the familiar Reedy machinery—latching and matching objects—by taking equivariant colimits and limits over the over‑ and under‑categories R⁺/r and R⁻\r. Because of the presence of automorphisms, these colimits/limits must be formed in the category of G_r‑objects (i.e., fixed‑point diagrams), which the authors call “weighted” or “fixed‑point” latching and matching objects. For a diagram X∈E^R (E a cofibrantly generated model category) the latching map L_r X→X(r) and the matching map X(r)→M_r X are defined exactly as in the classical case, but now they are G_r‑equivariant morphisms.

The central theorem states that E^R carries a canonical Reedy‑type model structure:

Weak equivalences are objectwise weak equivalences in E.
Reedy cofibrations are those maps f:X→Y such that each latching map L_r Y∪_{L_r X}X(r)→Y(r) is a cofibration in E (equivalently, each L_r X→X(r) is a cofibration).
Reedy fibrations are those maps f:X→Y for which each matching map X(r)→Y(r)×_{M_r Y}M_r X is a fibration in E.

The proof follows the standard small‑object argument but requires a careful choice of generating (trivial) cofibrations that respect the G_r‑actions. For each object r the authors select a set I_r (resp. J_r) of G_r‑equivariant generating cofibrations (resp. trivial cofibrations) in E and then take the union over all r. The resulting sets I_R and J_R satisfy the required lifting and factorization properties, yielding a cofibrantly generated model structure on E^R. The construction shows that the presence of automorphisms does not obstruct the existence of a Reedy model structure; rather, it enriches it by incorporating equivariant homotopical data.

A substantial part of the paper is devoted to examples. The category Γ, which indexes Segal’s Γ‑spaces, acquires a generalized Reedy structure where the automorphism groups are the symmetric groups acting on finite sets. The cyclic category Λ, the total category of the crossed simplicial group underlying Connes’ cyclic homology, also fits the framework; its automorphisms are the cyclic groups C_n. More generally, any total category of a crossed simplicial group (dihedral, quaternionic, etc.) becomes a generalized Reedy category. In each case the Reedy model structure on E^R recovers known homotopy theories (e.g., the model structure on Γ‑spaces or cyclic objects) while providing a uniform, equivariant description.

Finally, the authors discuss future directions. They suggest that the notion can be iterated to define “multi‑Reedy” or “higher‑Reedy” structures, potentially useful for modeling ∞‑categories with internal symmetries. They also point out applications to operad theory, where many operadic indexing categories possess non‑trivial automorphisms, and to algebraic topology contexts such as cyclic (co)homology, equivariant stable homotopy, and modular stacks. By extending the Reedy paradigm to accommodate symmetry, the paper opens a pathway for systematic homotopical analysis of a broad class of diagrammatic objects that were previously inaccessible to classical Reedy techniques.