Pre-Monoidal Categories with Controlled Coherence Defects

Pre-Monoidal Categories with Controlled Coherence Defectsa Qiang Dai [email protected]d.edu (Dated: December 8, 2025) Abstract We construct and analyze a concrete pre-monoidal category in which the associator exhibits a controlled, block-local coherence defect. The resulting structure—the block-cyclic category—has objects given by finite cyclic sets, morphisms given by rotations, and a tensor product defined by block concatenation. Naturality and the triangle axioms uniquely determine the form of the associator, which acts nontrivially only on the terminal tensor block through an integer-valued defect function. We derive a necessary arithmetic condition on this defect for the pentagon identity to hold, and show that non-additive choices such as a linear offset produce a minimal, explicitly computable pentagon anomaly interpretable as a localized categorical curvature. This yields a transparent and tractable model in which coherence fails in a controlled and geometrically meaningful manner, illustrating how local associator anomalies can encode structural effects analogous to curvature or torsion in higher-categorical and physical contexts. aThis manuscript is a preprint submitted to the Journal of Mathematical Physics. It omits a technical appendix of explicit endomorphism maps included in the version submitted for peer review; some of the maps are available upon request. This preprint has not yet undergone peer review. 1 I. INTRODUCTION Monoidal and higher monoidal categories occupy a central place in modern mathematical physics. They provide the structural language for topological field theories, extended operators, tensor network models, and higher gauge theory. In these settings, coherence – in particular the associator and its pentagon identity – encodes geometric or topological invariants, and deviations from coherence correspond to curvature, flux, or torsion. These ideas appear prominently in the foundational work of Joyal-Street [7], Baez-Lauda [8], Schreiber [16], Yetter [9], and many others. Across this literature, defects in associativity are typically modeled abstractly through 3cocycles in group cohomology, reflecting the well-known classification of scalar associators by H3(G, U(1)). In applications to condensed matter physics, lattice gauge theory, and topological phases [10–12], these cocycles govern anomaly inflow, projective representations, and topological actions. In categorical electromagnetism and higher gauge theory [13–15], the curvature of a U(1)-connection appears categorically as the failure of a higher associativity or functoriality condition. A notable feature of this body of work is that the coherent structures (or their failures) are specified at the level of cohomological data: one begins with a 3-cocycle, often written ω∈ Z3(G, U(1)), and treats it as an abstract obstruction to coherence. The underlying monoidal or higher monoidal category, however, is typically not constructed explicitly. Instead it is assumed to exist, after which coherence axioms are imposed or relaxed according to the cocycle. This approach is extremely powerful, but it obscures the fine-grained categorical mechanisms by which these defects arise and propagate, and it provides little intuition for how coherence might fail in a localized or geometrically meaningful way. a. Marginal contribution of this work. The present paper provides, to the best of our knowledge, the first explicit, elementary, and fully computable model of a pre-monoidal category in which: • naturality and the triangle axioms hold identically for all choices of associator deformation, • the block-local form of the associator is uniquely forced by these axioms, • coherence may fail locally and in a controlled manner, • all coherence anomalies are governed by a single integer-valued defect function δ:N→ Zacting on the terminal block, • comparison of the two composite paths in Mac Lane’s pentagon yields a necessary number-theoretic congruence involving δ; failure of this congruence guarantees a pentagon defect, while satisfaction of the congruence does not, in general, ensure coherence. This makes the block-cyclic category a concrete setting in which associator defects can be computed explicitly, rather than posited abstractly, and in which the boundary between monoidal and pre-monoidal behavior can be studied with complete mathematical transparency. This explicit construction stands in contrast with the prevailing paradigm in which one specifies a 3-cocycle first and searches for a monoidal category realizing it. Here, the direction 2 is reversed: naturality and unitor constraints force the associator to act only on the rightmost tensor block, and the entire coherence structure is reduced to a single tunable defect δ(c) local to that block. The pentagon identity then becomes an arithmetic constraint on δ, whose failure we interpret as a discrete curvature. b. Relation to categorical electromagnetism. In categorical formulations of electromagnetism and higher U(1)-gauge theory (notably [13, 14, 16]), curvature is encoded as a failure of certain higher functoriality or coherence laws. In these approaches one typically begins with smooth data (differential forms, Deligne cocycles, bundle gerbes) and promotes them to categorical objects; the underlying monoidal structures are implicitly assumed to behave coherently. Our construction complements this perspective by demonstrating that even at the purely discrete, combinatorial level, coherence defects may be modeled explicitly in an elementary category. More importantly, this explicit model allows one to import the substantial machinery of the categorical electromagnetism literature without reconstructing its foundations. Once the associator defect is realized concretely as a function δ(c), the interpretation of its pentagon obstruction as curvature becomes immediate: the condition that the pentagon commute corresponds precisely to vanishing curvature, and the defect corresponds to a discrete 3cocycle. This allows our block-cyclic category to serve as a “foundational substrate” upon which the constructions of [13, 14] can operate directly. c. Summary of results. The main achievements of this paper are: 1. We construct the block-cyclic category, a pre-monoidal category with objects Cn, morphisms given by rotations, and a tensor product defined by block concatenation. 2. We prove that naturality and the unit axioms uniquely determine the associator, leaving a single free parameter: a defect function δ:N→Zthat controls the rotation of the rightmost tensor block. 3. We derive a necessary number-theoretic criterion for the pentagon diagram to commute. Its failure for generic δillustrates how coherence defects arise locally and propagate through rebracketing operations. 4. We exhibit the canonical defect δ(c) = c+ 1, which produces a minimal and uniform pentagon anomaly interpretable as a discrete, block-local curvature. This provides an explicit categorical model of curvature in the sense of higher gauge theory. 5. We discuss how such a construction can be used as the categorical foundation for U(1) electromagnetism, where the pentagon defect plays the role of the curvature 2-form and its arithmetic structure parallels the cohomological classification of anomalies. d. Outlook. The block-cyclic category thus serves as a testbed for understanding how coherence can fail in controlled ways and how such failures encode geometric or physical information. Because the category is explicit, elementary, and computationally tractable, it provides a foundation on which more sophisticated categorical field theories may be built. In a companion work, we apply similar principles to construct pre-monoidal and monoidal categories with richer grading structures, including examples related to Z6symmetry and categorical models of particle interactions. 3 II. THE BLOCK-CYCLIC CATEGORY We now introduce the block-cyclic category, a concrete example of a pre-monoidal category constructed from finite cyclic sets. Throughout this section, we refer to the technical definitions and proofs in the appendix rather than reproducing them. A. Objects and morphisms Throughout, N={0,1,2, . . . }denotes the non-negative integers and Zdenotes the set of all integers. Definition II.1. The block-cyclic category Chas: • objects Cnfor each n∈N, where Cn={0,1, . . . , n −1}is the cyclic set of size n; • morphisms Cn→Cngiven by cyclic rotations, all of which are invertible. Thus each Cnmay be regarded as an information register with ndistinguishable states, and the isomorphisms are precisely the rotations of this register. B. Tensor product The tensor product of objects is defined additively: Cm⊗Cn=Cm+n. Elements of Cm+nare interpreted as blocks: Cm+n=AtB, |A|=m, |B|=n, and tensor products of morphisms act blockwise. This ensures that naturality of the tensor product is automatic. C. Associator The associator is uniquely determined by naturality and the triangle axioms. As stated in Theorem A 3 and proved in Appendix A 3, the only possible form is a morphism that acts as the identity on the Aand B-blocks and as a rotation on the C-block. Definition II.2 (Block-local associator).For a, b, c ∈N, write Ca+b+c=AtBtC, |A|=a, |B|=b, |C|=c. Given a function δ:N>→Z, define αa,b,c(i) = {i, i < a +b, (a+b) + ((i−(a+b)) + δ(c))mod c, i ≥a+b. This is precisely the form imposed by functoriality, naturality, and the triangle axioms. The only freedom left is the choice of the defect function δ. See Appendix A 3 for a proof of uniqueness. 4 D. Naturality and the triangle axioms The blockwise structure of morphisms and associators implies that naturality holds identically for all choices of δ. Moreover, the triangle identity is satisfied because the left and right unitors affect only the A-block, which is fixed by the associator. A complete proof appears in Appendix A 3. III. PENTAGON COHERENCE AND THE DEFECT FUNCTION We now examine the pentagon identity. Since the associator is block-local, only the Cand D-blocks are affected when comparing the two sides of the pentagon diagram. Let a, b, c, d ∈N. The two composite associators appearing in Mac Lane’s pentagon act identically on the Aand B-blocks, but differ on the Cand D-blocks. Comparing the two paths yields the necessary condition recorded in the next theorem. Theorem III.1 (Necessary condition for pentagon coherence).For the block-local associator of Definition II.2, a necessary condition for the pentagon identity to hold is δ(c+d)≡δ(c) + δ(d) (mod gcd(c, d)). If this congruence fails for some pair (c, d), then the pentagon identity does not commute and a coherence defect is present. A detailed computation is provided in Appendix A 6. We emphasize that the congruence above is necessary but not sufficient; satisfying it does not guarantee that the pentagon commutes, because the two paths act differently on the C-Dboundary. A. Canonical defect: δ(c) = c+ 1 Most choices of δviolate the congruence in Theorem III.1. Among all non-additive functions, the shift δ(c) = c+ 1 plays a distinguished role. It preserves the natural block-local structure, but fails the pentagon test in a uniform, size-independent way: δ(c+d)−δ(c)−δ(d) = −1. Thus the pentagon defect is constant. This yields a model of uniform curvature concentrated at the point where the rebracketing moves the C-block across the D-block. As shown by explicit computation, naturality and the triangle axioms remain intact, while the pentagon identity fails at a single index in a predictable manner. This makes δ(c) = c+ 1 the canonical choice for a minimally curved pre-monoidal category. A noteworthy feature of the canonical deformation δ(c) = c+ 1 is that the induced coherence defect is not uniformly distributed across all tensor products. Instead, coherence is restored exactly when the sizes cand dof the terminal blocks are coprime. In other words, even though δis globally non-additive, the pentagon anomaly becomes arithmetically invisible whenever the obstruction is measured modulo gcd(c, d) = 1. This produces a 5 natural number-theoretic “coherence stratification” inside the pre-monoidal category: tensorings along coprime directions behave monoidally, while those sharing common divisors accumulate curvature. Such arithmetic restoration of coherence is reminiscent of topological anomalies that disappear under reduction modulo a trivial stabilizer. See Appendix A 7 for technical details and mathematical proofs. IV. INTERPRETATION AND OUTLOOK The block-cyclic category provides a toy model of a categorical universe in which: • objects behave as finite information registers, • morphisms are reversible state permutations, • naturality holds identically, • the triangle axioms hold canonically, • the pentagon identity may fail locally in a controlled manner. Associator defects of the form δ(c) = c+ 1 may be interpreted as primordial curvature, torsion, or “twisting” inherent to the categorical substrate. The fact that coherence fails only when four tensor factors are compared suggests a deep analogy with the emergence of curvature from higher-categorical or obstruction-theoretic data. In future work we will analyze how this pre-monoidal structure interacts with braiding operations, centers, and higher-dimensional coherence conditions, as well as its relationship to classical gauge-theoretic structures. The block-cyclic example constitutes the first step in a wider program of modeling physical and geometric phenomena through controlled coherence defects. V. CATEGORICAL CURVATURE AND THE U(1) 3-COCYCLE Given the block-local associator αa,b,c :Ca+b+c→Ca+b+c, defined by αa,b,c(i) = {i, 0≤i < a +b, (a+b) +((i−(a+b)) + δ(c))(mod c), a +b≤i < a +b+c, the pentagon identity compares the defect values δ(c), δ(d), δ(c+d). The pentagon fails exactly when the quantity ∆(c, d) = δ(c) + δ(d)−δ(c+d) is nonzero modulo the appropriate block identifications. We interpret ∆(c, d)as a categorical curvature. 6 A. Normalization and the choice of N To pass from the integer-valued defect δ(c)to a U(1)-valued 3-cochain, we fix a global integer N≥1and define ω(a, b, c) = exp(2πi Nδ(c)). This normalization has three motivations: •δ(c)is an integer rotation, so the natural map into U(1) = R/Zmust factor through asingle global modulus, not a size-dependent modulus such as c,d, or c+d. • The constant Nplays the role of a coupling parameter or quantization unit. Larger Ngives finer phase resolution, and the limit N→ ∞ recovers a continuous gaugetheoretic regime. • The coboundary of ωreproduces the pentagon defect: (dω)(a, b, c, d) = exp(2πi N∆(c, d)). Thus the pentagon holds if and only if ∆(c, d)≡0 (mod N). B. Relation to categorical gauge curvature With this normalization, the curvature of the categorical gauge field is represented by the U(1)-valued 3-coboundary (dω)(a, b, c, d) = exp(2πi N∆(c, d)). The dictionary with abelian gauge theory is: δ(c)←→ discrete gauge potential, ∆(c, d)←→ curvature of the connection, (dω)(a, b, c, d)←→ U(1) holonomy over a 2-surface. This mirrors the classical relations F=dA, dF = 0, with Fthe electromagnetic field tensor. In contrast to approaches that begin by postulating an abstract 3-cocycle, the block-cyclic category provides an explicit, computable source for such curvature: the failure of the associator to satisfy the pentagon law. 7 VI. CONCLUSION We have constructed an explicit and fully computable pre-monoidal category whose associator exhibits a controlled and localized coherence defect. Naturality and the triangle axioms uniquely determine the form of the associator, reducing all nontrivial behavior to a single integer-valued defect function. Rather than enforcing global coherence, this framework isolates the precise mechanism by which the pentagon can fail, thereby providing a concrete model in which associativity is locally obstructed in a quantifiable and geometrically interpretable way. By introducing a U(1)-valued 3-cochain derived from this defect, we obtain a categorical analogue of curvature familiar from abelian gauge theory. In contrast with approaches that begin by postulating a cocycle or curvature form, the block-cyclic category exhibits such curvature intrinsically: it arises directly from the local twist implemented by the associator. The resulting structure supplies a transparent bridge between coherence anomalies in tensor categories and geometric concepts such as holonomy and field strength. The construction is deliberately minimal yet surprisingly versatile. It admits both coherent and non-coherent regimes, makes the source of curvature fully explicit, and offers a discrete setting in which the transition from strict monoidality to controlled pre-monoidality can be studied in detail. This opens several avenues for further development, including braided extensions, higher-categorical generalizations, and the analysis of physical or combinatorial models in which local associator defects play a structural role. In summary, the block-cyclic category provides a foundational example of how coherence can fail locally in a precise and computable manner, and how such failures naturally give rise to geometrically meaningful curvature data. It thereby offers a concrete platform on which to explore the interaction between categorical algebra and gauge-theoretic structures. An unexpected feature of the canonical defect δ(c) = c+ 1 is the appearance of a coherence stratification: the pentagon identity is restored precisely when the terminal block sizes are coprime. This arithmetic locus is dense in N2but forms only a thin subset of the tensor product geometry, suggesting a subtle interaction between categorical curvature and elementary number theory. While phenomena of this kind are intriguing — coprime directions such as the Fibonacci ray exhibiting a persistent restoration of coherence — their deeper implications lie beyond the scope of the present work. Our primary aim here has been to demonstrate that a nontrivial 3-cocycle obstruction can arise naturally and explicitly in a concrete pre-monoidal category, and to show how such coherence defects integrate with a categorical interpretation of electromagnetism. Appendix A: Proofs 1. Category Structure Definition A.1 (Block-Cyclic Pre-Monoidal Category).The block-cyclic pre-monoidal category Cis defined as follows: •Objects: For each positive integer n≥1, an object Cn={0,1, . . . , n −1}(cyclic group of order n). •Morphisms: For each n≥1and k∈Z/nZ, a rotation morphism fk:Cn→Cn defined by fk(i) = (i+k) mod n. All morphisms are isomorphisms. 8 •Tensor Product of Objects:Cm⊗Cn=Cm+n. •Tensor Product of Morphisms: For fk:Cm→Cmand gℓ:Cn→Cn, the tensor product (fk⊗gℓ) : Cm+n→Cm+nis defined blockwise: (fk⊗gℓ)(i) = {(i+k) mod mif 0≤i < m m+ ((i−m) + ℓ) mod nif m≤i < m +n •Associator: For objects a, b, c ≥1, the associator αa,b,c :Ca+b+c→Ca+b+cis blockaware: αa,b,c(i) =      iif 0≤i < a (A block) iif a≤i < a +b(B block) (a+b) + ((i−(a+b)) + δ(c)) mod cif a+b≤i < a +b+c(C block) where δ:N→Zis a defect function depending only on c. 2. Naturality Theorem A.2 (Naturality Always Holds).For any morphisms f:Ca→Ca′,g:Cb→Cb′, h:Cc→Cc′in the block-cyclic category, the naturality square for the associator commutes: αa′,b′,c′◦((f⊗g)⊗h) = (f⊗(g⊗h)) ◦αa,b,c Proof. Since all morphisms are rotations, we can write f=fk:Ca→Ca′,g=gℓ:Cb→Cb′, h=hm:Cc→Cc′for some k, ℓ, m. The naturality square requires: αa′,b′,c′◦((fk⊗gℓ)⊗hm) = (fk⊗(gℓ⊗hm)) ◦αa,b,c Both sides are functions Ca+b+c→Ca′+b′+c′. Case 1: i < a (A block). • LHS: ((fk⊗gℓ)⊗hm)(i) = fk(i) = (i+k) mod a′(since i < a). Then αa′,b′,c′acts as identity on the A’ block, so result is (i+k) mod a′. • RHS: αa,b,c(i) = i(identity on A block). Then (fk⊗(gℓ⊗hm))(i) = fk(i)=(i+ k) mod a′. • Both sides agree. Case 2: a≤i < a +b(B block). • LHS: ((fk⊗gℓ)⊗hm)(i)maps ito the B’ block with rotation. Then αa′,b′,c′acts as identity on B’ block. • RHS: αa,b,c(i) = i(identity on B block). Then (fk⊗(gℓ⊗hm))(i)applies the appropriate rotation to the B block. • Both sides agree by blockwise structure. 9