Pita Factorisation in Operadic Categories

• Pita factorisation is a canonical morphism decomposition technique in strictly factorisable operadic categories that uniquely factors any morphism into a quasi-bijection and an order-preserving map.
• The method constructs a disjoint union of fibres with a canonical reordering, ensuring uniqueness via the universal property of the left adjoint reflection.
• This factorisation underpins the construction of the pita nerve, guaranteeing coherent operadic nerves and enabling incidence bialgebra structures when quasi-bijections are invertible.

Pita factorisation is a canonical and uniquely determined morphism decomposition technique in strictly factorisable operadic categories. Given any morphism $f:T\to S$ in such a category, pita factorisation expresses $f$ uniquely as a composite $f = \eta_f \circ \pi_f$, where $\pi_f$ is a quasi-bijection (whose fibres are category-local terminal objects and is fibrewise order-preserving relative to $\eta_f$) and $\eta_f$ is strictly order-preserving. This structure has substantial implications for the simplicial and algebraic behaviour of operadic categories, yielding the pita nerve (an oplax simplicial object in Cat), guaranteeing coherence properties of the operadic nerve, and, when quasi-bijections are invertible, a decomposition-space structure enabling incidence bialgebra constructions (Batanin et al., 28 Dec 2025).

1. Strictly Factorisable Operadic Categories and the Pita Factorisation Theorem

An operadic category $\mathcal O$, in the sense of Batanin–Markl, consists of a category equipped with designated local terminal objects per component, a cardinality functor $|\!-\!|:\mathcal O \to \mathsf{Fin}$ (where $\mathsf{Fin}$ refers to the skeletal category of finite ordinals), and fibre-objects/fibre-maps compatible with standard operadic axioms.

A morphism $\sigma:T\to S$ in $\mathcal O$ is a quasi-bijection if each fibre is a chosen local terminal object; $\lambda:T\to S$ is order-preserving if $|\lambda|$ is order-preserving on the ordinal level.

The category $\mathcal O$ is strictly factorisable if each morphism $f:T \to S$ admits a unique factorisation $T \xrightarrow{\pi_f} T' \xrightarrow{\eta_f} S$ so that $\pi_f$ is a quasi-bijection, $\eta_f$ is order-preserving, and $\pi_f$ acts fibrewise order-preserving relative to $\eta_f$.

The precise result: In strictly factorisable operadic categories, every morphism $f$ decomposes uniquely as $f = \eta_f \circ \pi_f$ with $\pi_f$ quasi-bijection and fibrewise order-preserving, $\eta_f$ strictly order-preserving. This is designated the pita factorisation.

2. Construction and Uniqueness of $\pi_f$ and $\eta_f$

Given $f:T \to S$ where $|S| = \{1,\dots, n\}$:

• Each fibre $f^{-1}(i)$ has a cardinality and an order-preserving injection $$\epsilon_{f,i}: f^{-1}(i) \longrightarrow \mathrm{dom}(f)$$.
• The sum of fibres $T' = \sum_{i\in |S|} f^{-1}(i)$ is formed as a disjoint union.
• The canonical "re-ordering" bijection $\pi_f:T\to T'$ aligns elements of $T$ blockwise into the order structure of $T'$.
• The induced $\eta_f: T' \to S$ maps each fibre-block entirely to the corresponding $i \in |S|$ monotonically.

By the category axioms, $(\pi_f, \eta_f)$ genuinely factors $f$, and uniqueness follows because any other possible factorisation with the correct properties must coincide on cardinalities and thus on the underlying categorical structure in $\mathcal O$ due to the universal property of the left adjoint reflection $r_S$ to the subcategory of order-preserving arrows over $S$.

3. Key Properties and Supporting Lemmas

The pita factorisation displays several critical properties:

• Identity under order-preservation: If $f$ is order-preserving, then $\pi_f = \mathrm{id}$ and $\eta_f = f$.
• Naturality: In appropriate commutative diagrams arising from factorisations of composites $gf$, both horizontal arrows are quasi-bijections, and the relevant square is fibrewise order-preserving.
• Idempotence and interaction: The assignments $\pi(\!-\!)$, $\eta(\!-\!)$ obey

$$\pi(\pi(f)) = \mathrm{id},\quad \pi(\eta(f)) = \mathrm{id},\quad \eta(\pi(f)) = \mathrm{id},\quad \eta(\eta(f)) = \eta(f),$$

and for composable $f, g$: $\eta(g)\,\eta(f/g) = \eta(gf)$.

These relations emanate from the reflection functor formalism and the universal property of the canonical factorisation.

4. The Pita Nerve as an Oplax (Top-Lax) Simplicial Category

Consider the category $W_n$ of chains $$T_n \xrightarrow{f_n} T_{n-1} \xrightarrow{f_{n-1}} \dots \xrightarrow{f_1} T_0$$ and morphisms are fibrewise order-preserving quasi-bijection diagrams. The subcategory $P_n \subset W_n$ consists of chains where each composite $T_k \to T_0$ is order-preserving.

The inclusions $i_n: P_n \hookrightarrow W_n$ admit left adjoints $r_n$, which iteratively apply the pita factorisation at the chain's final map; thus, $P_n$ is a reflective subcategory of $W_n$.

The simplicial structure respects $P_n$ except for the “top” face map, which instead requires passage via the monad $t_{n-1} = i_{n-1} r_{n-1}$ and the composite $d_n = r_{n-1} d_n i_n$.

A top-lax simplicial object in $\mathsf{Cat}$, in this context, has categories $X_n$ with standard face and degeneracy maps preserving all simplicial identities except those involving consecutive top-faces; replacement is via non-invertible comparison 2-cells $\beta_n$ subject to coherence axioms.

For a strictly factorisable operadic category $\mathcal O$, the sequence $P(\mathcal O)_\bullet = (P_0, P_1,\dots)$ with face-degeneracy maps and the $\beta_n$ comparison 2-cells forms a top-lax simplicial category

$$P(\mathcal O): \Delta^{\mathrm{op}} \to \mathsf{Cat}$$

5. Coherence of the Operadic Nerve via Pita Factorisation

The operadic nerve $N_{\mathrm{op}}(\mathcal O)$ is constructed as a simplicial object in the Kleisli category for the free symmetric strict monoidal category monad. Its pattern of non-strictness aligns precisely with the top-lax pattern of the pita nerve $P(\mathcal O)$. Pita factorisation supplies the coherence data that lifts the top-lax structure to a coherent pseudo-simplicial groupoid—a normal oplax simplicial object in $\mathsf{Gpd}$.

Hence, all higher simplicial identities in the operadic nerve are controlled by invertible 2-cells dictated by the factorisation axioms, ensuring coherence necessary for applications such as the theory of decomposition spaces.

6. Invertible Quasi-Bijections and Decomposition Spaces

When all quasi-bijections in $\mathcal O$ are invertible:

• Each $P_n(\mathcal O)$ is a groupoid.
• The top-lax simplicial category $P(\mathcal O):\Delta^{\mathrm{op}}\to\mathsf{Gpd}$ is a pseudo-simplicial groupoid with invertible comparison cells (Jardine’s supercoherence theorem applies).
• $P(\mathcal O)$ then becomes a decomposition space (aka 2-Segal space), characterized by strict Segal groupoid structure in the upper decalage $\operatorname{Dec}^u(P(\mathcal O))$ and pullback-square conditions rooted in the uniqueness of pita factorisation.

Such decomposition spaces possess well-behaved incidence coalgebras and bialgebras, enabling combinatorial enumerative and algebraic applications.

7. Worked Examples

Skeletal Finite Sets $\mathsf{Fin}$

For $\mathcal O = \mathsf{Fin}$, morphisms $f: \underline m \to \underline n$ admit pita factorisation as blockwise reorderings—the “cut-and-reorder” paradigm. $\pi_f$ establishes the block ordering, and $\eta_f$ is the monotone assignment by cardinalities.

Surjections $\mathsf{Surj}$

For the category of surjective maps between finite sets, pita factorisation produces a bijection $\pi_f$ and a monotone surjection $\eta_f$. All quasi-bijections are invertible, so the pita nerve is a decomposition space; the associated incidence bialgebra relates to factorial-weighted Faà di Bruno structures.

Connected Graphs

In the category of connected graphs, morphisms combine edge contractions and indexing. Pita factorisation splits a morphism into a permutation-like reindexing (quasi-bijection) followed by edge-monotone contraction (order-preserving). The category supports strict factorisability and invertible quasi-bijections.

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For further details, see Batanin–Kock–Weber “Pita factorisation in operadic categories” (Batanin et al., 28 Dec 2025), Batanin–Markl “Operadic categories and Koszul duality,” Garner–Kock–Weber “Operadic categories and decalage,” Galvez‐Carrillo–Kock–Tonks “Decomposition spaces, incidence algebras,” and Jardine “Supercoherence.”