Pita Nerve in Operadic Categories

• Pita nerve is a categorical construction defined by the unique decomposition of morphisms into order-preserving and quasi-bijection parts in operadic categories.
• It is built as a sequence of categories (Pₙ) connected by face and degeneracy maps, where non-strict simplicial identities are managed via specific coherence 2-cells.
• When quasi-bijections are invertible, the Pita nerve gains the decomposition space property, enabling applications in incidence coalgebras and bialgebra constructions.

The pita nerve is a categorical and simplicial construction associated with strictly factorisable operadic categories, arising from the unique factorisation of morphisms into order-preserving and quasi-bijection parts—referred to as the pita factorisation. It plays a foundational role in the simplicial theory of operadic categories, mediating between the combinatorics of factorisations and the higher coherence properties needed for the operadic nerve in categorical and homotopical contexts. The pita nerve is formally a top-lax (oplax) simplicial object in $\mathbf{Cat}$, with strict simplicial identities except those involving repeated applications of the top face maps, which are instead coherent up to specified 2-cells. In the case where quasi-bijections are invertible, the pita nerve becomes a decomposition space with significant implications for incidence coalgebras and bialgebras (Batanin et al., 28 Dec 2025).

1. Operadic Categories and the Pita Factorisation

An operadic category $\mathcal{O}$ is defined by a category equipped with chosen local terminal objects (one in each connected component), a cardinality functor into the skeletal category of finite ordinals $\mathbb{F}$, and, for each arrow in a slice category and each $i$, a fibre functor

$$( - )^{-1}(i): \mathcal{O}/X \longrightarrow \mathcal{O}.$$

This data, together with strictly functorial fibre maps and subject to axioms (A1)–(A5) of Batanin–Markl (preservation of terminals, compatibility with cardinals, associativity of iterated fibres), defines the structure.

Within $\mathcal{O}$, a morphism is a quasi-bijection if every fibre is a chosen local terminal. A morphism is order-preserving if its image under the cardinality functor is order-preserving in $\Delta$.

A factorisable operadic category admits for every arrow $f$ a factorisation

$f = \eta(f) \circ \pi(f)$

with $\pi(f)$ a quasi-bijection and $\eta(f)$ order-preserving. If, for each composable pair, the filler $\eta(f/g)$ in the associated prism diagram is unique, $\mathcal{O}$ is strictly factorisable. This uniqueness renders the factorisation functorial and well-defined, with arrows written as

$T \xrightarrow{\pi_f} T' \xrightarrow{\eta_f} S,$

with $\pi_f$ a quasi-bijection and $\eta_f$ order-preserving [(Batanin et al., 28 Dec 2025), Lemma 3.14].

2. Construction of the Pita Nerve

Given a strictly factorisable operadic category $\mathcal{O}$, the sequence of categories $(P_n)_{n \geq 0}$, called the pita nerve, is constructed as follows:

• Objects in $P_n$: Chains

$$T_n \xrightarrow{f_n} T_{n-1} \xrightarrow{f_{n-1}} \cdots \xrightarrow{f_2} T_1 \xrightarrow{f_1} T_0$$

such that for each $k$, the composite $T_k \to \cdots \to T_0$ is order-preserving.

• Morphisms in $P_n$: Commutative diagrams where each horizontal map is a quasi-bijection, fibrewise order-preserving with respect to the corresponding vertical arrows.

These $P_n$ assemble into a pre-simplicial object in $\mathbf{Cat}$ via strict face (for $0 \leq i < n$) and degeneracy (for $0 \leq i \leq n$) functors:

• $d_i$ omits the $i$th arrow,
• $s_i$ inserts an identity at level $i$.

Top face maps $d_n$ are defined by a reflection process involving the category $W_n$ of all chains (not all necessarily order-preserving), first applying the genuine top face and then left-adjoint reflection back into $P_{n-1}$ [(Batanin et al., 28 Dec 2025), §2].

3. Oplax Simplicial Structure and Coherence

The pita nerve is not a strict simplicial object. The simplicial identities fail to hold strictly only for composites of two top face maps; specifically,

$d_{n+1} d_{n+1} = d_{n+1} d_{n+2}$

is replaced by a specified coherence 2-cell

$$\beta_n : d_{n+1} d_{n+1} \Longrightarrow d_{n+1} d_{n+2},$$

given by a fibrewise order-preserving quasi-bijection of the form $\pi\bigl(\pi(f_{n})\circ\eta(f_{n+1})\bigr)$ (see (Batanin et al., 28 Dec 2025), display (6.10)). All other simplicial identities are satisfied strictly (Lemma 6.6).

The coherence data $\beta_n$ is subject to further pentagon and unit-triangle coherence conditions, ensuring the object's consistency as a normal oplax algebra for the decalage monad on $\mathbf{Cat}$—that is, a top-lax simplicial category. These higher coherences are described by commutative diagrams (equations (6.11) and (6.12) in (Batanin et al., 28 Dec 2025)), whose satisfaction relies on the uniqueness property of the pita factorisation.

The core result (Theorem 6.13 of (Batanin et al., 28 Dec 2025)) establishes that for strictly factorisable $\mathcal{O}$, the assignments $[n] \mapsto P_n$ together with $d_i$, $s_i$, $\beta_n$ constitute a coherent top-lax simplicial object in $\mathbf{Cat}$.

4. Relationship to the Operadic Nerve

The operadic nerve $N_{\mathrm{op}}$ of an operadic category, as developed in [Batanin-Kock-Weber:mainpaper], has a structure governed by a decalage-undecalage formula: $\mathrm{Dec}(N_{\mathrm{op}}) = S P,$ where $S$ is the symmetric-monoidal Kleisli decalage and $P$ is the pita nerve. The coherence of the pita nerve as a top-lax simplicial object, combined with the strictness of the Kleisli decalage on lower faces, guarantees that $N_{\mathrm{op}}$ is a coherent pseudo-simplicial groupoid: all simplicial identities hold up to coherent natural isomorphism (Theorem 4.5 in the main paper).

The locus of non-strictness in the operadic nerve is thus fully reflected in the coherence structure $\beta_n$ of the pita nerve (Batanin et al., 28 Dec 2025).

5. Decomposition Space Property for Invertible Quasi-Bijections

When every quasi-bijection in $\mathcal{O}$ is an isomorphism, the pita nerve $P$ exhibits enhanced properties:

• Each $P_n$ is a groupoid, and all face and degeneracy maps and coherence 2-cells are groupoid-valued.
• The 2-cells $\beta_n$ are invertible, making $P$ a genuine pseudo-simplicial groupoid according to Jardine's supercoherence ((Batanin et al., 28 Dec 2025), Prop. 7.2).
• $P$ becomes a decomposition space: for all $n$, the square

$\begin{tikzcd} P_{n+3} \ar[r,"d_1"] \ar[d,"d_{n+3}"'] & P_{n+2}\ar[d,"d_{n+2}"]\ P_{n+2} \ar[r,"d_1"'] & P_{n+1} \end{tikzcd}$

is a pullback in groupoids; dually, the dual "outer" squares are pullbacks whenever one omits the first or last face (Theorem 7.6).

This property enables the application of the incidence-coalgebra construction to $P$, producing an incidence bialgebra whose comultiplication on a basis element (an order-preserving morphism in $\mathcal{O}$) is

$$\Delta(f) = \sum_{f = e \circ h\,\text{order-preserving}} \eta(h) \otimes e.$$

For the operadic category of surjections, this yields a factorial-weighted version of the Faà di Bruno bialgebra (Batanin et al., 28 Dec 2025).

6. Technical Summary and Significance

The pita nerve encapsulates the combinatorics of factorisations in strictly factorisable operadic categories at the simplicial-categorical level, providing both a framework for higher coherences and a direct bridge to decomposition spaces in the presence of invertible quasi-bijections. Its construction and coherence conditions clarify where strict simplicial identities break down and how this can be controlled using categorical 2-cells.

The pita nerve is a central technical ingredient in the simplicial approach to operadic categories, and its decomposition space property underlies incidence coalgebra and bialgebra constructions in categorified settings, securing connections to classical combinatorial bialgebras via the structure of operads and factorisation combinatorics (Batanin et al., 28 Dec 2025).