Breen Polytope Axiom in 2-Categories

• Breen Polytope Axiom is a higher-coherence law in symmetric strict monoidal 2-categories that generalizes the hexagon axiom to control 2-braidings.
• It employs the CMKZ 2-connection and hexagonator series to bridge analytic iterated integrals with categorical pasting diagrams.
• Verifying the axiom is pivotal for advancing higher representation theory, categorified gauge theory, and constructing extended topological invariants.

The Breen polytope axiom is a higher-coherence law central to the structure of symmetric strict monoidal cochain 2-categories equipped with infinitesimal 2-braidings. It arises as a categorical generalization of the hexagon axiom in braided monoidal categories, controlling the coherence of “hexagonator” 2-morphisms in braided 2-categories. The axiom is realized at the analytic level through 2-holonomy of a specialized 2-connection—the Cirio-Martins-Knizhnik-Zamolodchikov (CMKZ) connection—over configuration spaces, and is crucial for the integration (“Cartier integration”) of infinitesimal 2-braidings. Satisfying the Breen polytope axiom ensures the associativity and coherence of categorified braidings, enabling their use in higher representation theory, topology, and mathematical physics (Kemp, 3 Aug 2025).

1. Foundational Context: Infinitesimal 2-Braidings and Coherence

In the framework of symmetric strict monoidal cochain 2-categories—categories enriched over finitely-generated cochain complexes in degrees $[-1,0]$—an infinitesimal 2-braiding is specified by a pseudonatural endomorphism $t: \otimes \Rightarrow \otimes$ of the monoidal product functor. For strictness, components such as $t_{1(23)} \equiv t_{12} + t_{13}$ and $t_{(12)3} \equiv t_{23} + t_{13}$ are required, as detailed in equations (2.15–2.16) (Kemp, 3 Aug 2025).

The axiom enters when $t$ is totally symmetric and coherent: the higher relationators $L$ and $R$ (encoding the failure of the four-term relations to hold strictly) must satisfy $-R_{213} = L + R = -L_{132}$. The Breen polytope axiom emerges as a constraint ensuring that multiple higher-order compositions of the braiding, associator, and hexagonators yield coherent structures.

2. The Hexagonator Series and 2-Holonomy Construction

The analytic apparatus for realizing the Breen polytope axiom leverages the hexagonator series, constructed as a 2-holonomy with respect to the CMKZ 2-connection, over the configuration space $Y_3$ of three distinct points in $\mathbb{C}$. For $n=2$, the CMKZ 2-connection is given by

$$A_{KZ} = \sum_{1 \leq i < j \leq 3} \omega_{ij} t_{ij}, \quad B_{CM} = 2\sum_{1 \leq i < j < k \leq 3} \left(\omega_{ij}\wedge \omega_{ki} L_{ijk} + \omega_{jk}\wedge \omega_{ki} R_{ijk}\right),$$

where $\omega_{ij} = \frac{dz_i-dz_j}{z_i-z_j}$ encodes the differential geometry of the configuration space (Kemp, 3 Aug 2025).

By pulling back the connection along a specific birational morphism, new coordinates allow the explicit expression of $\nabla$ and $\Delta$ in terms of the objects $L$ and $R$: $$\nabla = (t_{12} z + t_{23}(z-1)) dz + \ldots, \quad \Delta = 2 \left(\frac{L}{zv} + \frac{R}{(z-1)v}\right) dz \wedge dv.$$

The 2-holonomy associated to a composite 2-path—built from “vertically-interpolative” and “horizontally-interpolative” 2-paths—calculates the transformation between two sides of the hexagon diagram. Segment 2-holonomies $W^{(P_V)}$, $W^{(P_{III})}$, and $W^{(P_{IV})}$ are expressed as explicit iterated integrals in the source parameters, yielding analytic series expansions in the deformation parameters.

3. Second-Order Expansion and Infinitesimal Hexagonators

Critical to the categorical applications is the extraction and comparison of the second-order term in the 2-holonomy series, corresponding to quadratic terms in the deformation parameter (often associated with $\hbar$ absorbed into $t$, $L$, and $R$):

• For $P_V$, the second-order holonomy is

$$(W^{(P_V)})_2 \to -\frac{\pi^2}{6} L - \frac{(\ln \epsilon)^2}{2} (L + R)$$

• For $P_{III}$,

$$(W^{(P_{III})})_2 \to -\frac{\pi^2}{6} R - \frac{(\ln \epsilon)^2}{2} (L + R)$$

• For $P_{IV}$,

$(W^{(P_{IV})})_2 \to i\pi (\ln \epsilon)(L + R)$

Summing the contributions from all relevant 2-paths and canceling divergent terms in $\ln(\epsilon)$, the total second-order term for the right hexagonator is

$(R)_2 = -\frac{\pi^2}{6} (L + 2R)$

This exactly recovers the “infinitesimal hexagonator” result from previous work and confirms the consistency of the analytic and algebraic approaches to categorified coherence (Kemp, 3 Aug 2025).

4. The Categorical Formulation and Proof of the Breen Polytope Axiom

The Breen polytope axiom is encapsulated, in its categorical (index) notation, as a specific relation between pasting diagrams of braidings, associators, and hexagonators (see equation (6.1) in (Kemp, 3 Aug 2025)). After using symmetry, naturality, and algebraic identities, the axiom is rewritten in terms of the infinitesimal 2-braiding $t$ and the four-term relationators $L$, $R$.

To prove the axiom, a contractible 2-loop (a closed 2-path) in $Y_3$ is explicitly constructed, formed by composing six 2-paths corresponding to the “faces” of the Breen polytope. The 2-holonomy of each face computes the relevant modification or congruence (e.g., the series for $e^{i\pi t}$ relative to $e^{i\pi t_{12}}$). By the 2-flatness of the CMKZ connection and the total symmetry, the 2-holonomy of the closed loop must vanish; thus, the two candidate composites of modifications defined by the axiom coincide. This is formalized in equation (6.4) (Kemp, 3 Aug 2025), where the cancellation of all nontrivial contributions demonstrates that the Breen polytope axiom holds under the stated conditions of strictness, total symmetry, and coherence for $t$.

5. Structural and Mathematical Implications

The satisfaction of the Breen polytope axiom under these analytic and algebraic constructions is essential for the correct integration of infinitesimal 2-braidings. In the classical theory (Drinfeld, Cartier), the integration of infinitesimal braidings yields solutions to the Yang–Baxter equation and enables quantum group construction. Analogously, the Breen polytope axiom enables, in the 2-categorical setting, the assembly of a “Cartier integration” of 2-braidings into a fully braided monoidal 2-category.

The analytic realization—by matching the 2-holonomy’s expansion with the algebraic infinitesimal hexagonators—grounds this higher-coherence requirement in the geometry of configuration spaces and iterated integrals. The result ensures the associativity and intercompatibility of all higher-morphism compositions, enabling higher-categorical structures suitable for applications in higher representation theory, categorified gauge theory, and the definition of extended topological invariants.

6. Broader Context and Applications

The Breen polytope axiom’s satisfaction via the CMKZ 2-connection 2-holonomy bridges the analytic (iterated integral and connection theory) and categorical (coherence law) perspectives. Applications include:

• Categorification of quantum groups, via integration of 2-braidings in monoidal 2-categories.
• Development of higher gauge theories, where 2-connections and their holonomy classes organize topological field theories.
• Construction of categorified knot invariants, employing 2-versions of the Knizhnik-Zamolodchikov connection.
• Potential foundation for “higher” deformation–quantization and its role in mathematical physics.

The analytic apparatus suggests further research directions, such as a corresponding treatment of pentagonators (higher associator coherences) and a systematic analysis of categorified monodromy representations.

7. Summary Table: Key Components and Relationships

Component	Analytic Realization	Categorical Meaning
Infinitesimal 2-braiding $t$	Appears in CMKZ 2-connection	Generates braiding 2-morphisms
Hexagonator series	2-holonomy along composite 2-paths	Coherence 2-morphisms (hexagonators)
Four-term relationators $L$, $R$	Second-order terms in 2-holonomy expansion	Measure deviation from four-term relations
Breen polytope axiom	Vanishing of 2-holonomy along closed 2-loop	Equivalence of higher pasting diagrams

Satisfying the Breen polytope axiom is thus both a highly structured analytic requirement (via vanishing 2-holonomy in $Y_3$) and a categorical necessity for constructing braided monoidal structures at the 2-level, with broad implications for modern areas of higher category theory and quantum algebra.