Berger’s Wreath Product Category

• Berger’s wreath product category is a combinatorial framework that defines n-categories via iterative wreath product constructions and rooted tree structures.
• It employs Berger–Joyal duality to establish an equivalence between Θₙᵒᵖ and combinatorial n-disk categories, clarifying coherence in higher category theory.
• Its formalism, based on pullback constructions and Segal functor techniques, extends naturally to cyclic and crossed simplicial group settings.

Berger's wreath product category refers to a categorical and combinatorial framework—stemming from the work of Clemens Berger on higher category theory—where iterative wreath product constructions organize strictly globular structures central to the definitions of higher categories, opetopes, and related objects. This structure is now fundamental in the understanding of the category Θₙ, which provides an explicit combinatorial model of n-categories and is dual to Joyal’s n-disk category, as established by Berger–Joyal duality (Cecil et al., 14 Sep 2025).

1. Foundational Definitions and the Inductive Wreath Product Construction

For each $n\geq 1$, Berger’s wreath product category Θₙ is constructed inductively by

$$\Theta_1 := \Delta,\quad \Theta_{n+1} := \Delta \wr \Theta_n,$$

where $\Delta$ is the simplex category and $\wr$ is Berger's categorical wreath product. The objects of Θₙ can be regarded as rooted trees with n levels, or, equivalently, as n-dimensional pasting diagrams for strictly globular n-categories.

Concretely, Θₙ is defined as follows:

• Base case ($n=1$): $\Theta_1 = \Delta$.
• Inductive step: Each $(n+1)$-cell in Θₙ₊₁ consists of a "tree" whose branches are labeled by n-cells, and whose arity is specified by the objects of $\Delta$. The wreath product encodes the assembly of higher-dimensional cells from lower-dimensional constituents.

A morphism in Θₙ is thus determined by compatible collections of morphisms in $\Delta$ (level 1) and in Θₖ for $1 \leq k < n$, together with the wreathing data that tracks the combinatorics of assembling lower-dimensional shapes into higher ones.

2. Berger–Joyal Duality and the Category of n-Disks

The Berger–Joyal duality establishes that the category Θₙ is (equivalent to) the opposite of Joyal’s combinatorial n-disk category, $\mathcal{D}_n$: $\Theta_n^\mathrm{op} \simeq \mathcal{D}_n$ (Cecil et al., 14 Sep 2025).

The n-disk category $\mathcal{D}_n$ has objects that are combinatorial n-disks: strings of sets and structure maps

$$X_0 \xrightarrow[s_0,\, t_0]{\, } X_1 \xrightarrow[s_1,\, t_1]{\, } \cdots \xrightarrow[s_{n-1},\, t_{n-1}]{\, } X_n$$

equipped with projections $p_k$ satisfying the globularity conditions $$p_k \circ s_{k-1} = p_k \circ t_{k-1} = \mathrm{id}$$. Morphisms are compatible tuples of set maps commuting with these structure maps. The equivalence with $\Theta_n^\mathrm{op}$ is constructed by an inductive argument involving a duality for the generalized wreath product functor and is functorial in $n$.

This duality provides a highly explicit combinatorial description of strict n-categories and organizes many coherence problems in higher category theory by mapping geometrically-defined disks to inductively constructed trees.

3. Categorical Wreath Product Formalism and the Pullback Construction

Berger's categorical wreath product is formalized as a particular pullback in $\mathrm{Cat}$, the category of small categories. Given a functor $T: \mathcal{C} \to \mathcal{C}$ (usually $T = \gamma$, where $\gamma$ is a Segal-type functor sending an object to the set of "edges"), the generalized wreath product is defined by the pullback

$$A \wr^T B = \mathrm{PB}\left( \begin{array}{ccc} A \wr^T B & \rightarrow & T(B) \ \downarrow & & \downarrow\,T(!_B) \ A & \rightarrow & T(*) \end{array} \right)$$

where $!_B: B \to *$ is the unique functor to the terminal category and $T(*), T(B)$ are the images under $T$.

A key duality is established (Proposition 2.8 in (Cecil et al., 14 Sep 2025)): for any automorphism $\tau: \mathcal{C} \to \mathcal{C}$ (in this context, the opposite category functor),

$\tau(A \wr^T B) \cong \tau(A) \wr^{T^\tau} \tau(B)$

which is crucial in the inductive step of the duality between Θₙ and $\mathcal{D}_n$.

4. Generalizations: Cyclic Categories and Crossed Simplicial Groups

The categorical and duality techniques for Θₙ extend to other base categories beyond the simplex category $\Delta$. For example, if $\Lambda$ denotes Connes’ cyclic category (encoding cyclic orderings), one defines

$$\Theta_n(\Lambda) := \underbrace{\Lambda \wr \cdots \wr \Lambda}_{n \text{ factors}}$$

and a duality

$$\Theta_n(\Lambda)^\mathrm{op} \simeq \mathcal{D}_n(\Lambda)$$

where $\mathcal{D}_n(\Lambda)$ is a suitable "cyclic" n-disk category. This methodology applies to any crossed simplicial group with an appropriate Segal functor $\gamma: C \to \Gamma$ encoding the combinatorial data (e.g., the edge set of a cyclically ordered set) (Cecil et al., 14 Sep 2025).

The classification of Segal functors by sieves, as in Theorem 4.17 and Corollary 4.19, shows that Berger's choice for the simplex category is extremal, and formal properties of the wreath product and compatibility with (op) functors allow for the same inductive arguments.

5. Applications: Higher Category Theory and Structure of Θₙ

Berger's wreath product category Θₙ serves as a universal indexing category for strict n-categorical pasting diagrams and the nerve of strict globular n-categories. Presheaf categories on Θₙ encode models of weak n-categories, and several comparison results (see (Cecil et al., 14 Sep 2025) and the references therein) exploit Θₙ's inductive and combinatorial structure.

Furthermore, the formalism underpins various comparison functors central to higher category theory:

• Realization and strictification functors for higher operads.
• Universal operations (such as traces or coherences) for n-categories.
• The bridge to cubical, simplicial, and cyclic combinatorial models for higher categorical structures.

6. Categorified and Operadic Perspectives, and Further Directions

The explicit construction of Θₙ by wreath products enables extension to colored or cyclic settings, and—by the associated duality—organizes further development such as:

• Definition and analysis of cyclic or modular operads via variants of Θₙ based on the cyclic category.
• Investigation of presheaf or sheaf categories on Θₙ for approaches to (∞,n)-categories.
• Generalizations to crossed simplicial group contexts, classifying all Segal functor–based wreath product categories using sieve-theoretic tools.

A plausible implication is that the wreath product–disk duality continues to underpin new models of higher categorical structures as more exotic base categories (with richer automorphism or symmetry data) are incorporated.

7. Technical Summary Table

Category	Construction	Duality Relation
Θ₁	$\Delta$	$\mathcal{D}_1 \simeq \Delta^{op}$
Θₙ	$\Delta \wr \Theta_{n-1}$	$\Theta_n^{op} \simeq \mathcal{D}_n$
Θₙ(Λ) (cyclic version)	$\Lambda \wr \cdots \wr \Lambda$	$$\Theta_n(\Lambda)^{op} \simeq \mathcal{D}_n(\Lambda)$$
General crossed simplicial G	$C \wr \cdots \wr C$ with Segal functor $\gamma$	$(C \wr \cdots \wr C)^{op} \simeq \mathcal{D}_n(C)$

The Berger–Joyal duality provides an explicit combinatorial and functorial bridge between inductively defined wreath product categories and globular disk categories, governing the foundations of strict higher category theory and serving as a prototype for broad generalizations across combinatorial and categorical frameworks.