Tensor structure and skeletal fusion data
TensorCategories.jl exposes tensor-categorical structure through generic operations. A category may implement only part of this structure. The manual explains the required hypotheses before using dimensions, traces, modular data, or coherence tests.
| Role | Principal public names | Manual |
|---|---|---|
| Tensor product and unit | tensor_product, ⊗, one, associator, inv_associator, distribute_left, distribute_right | Tensor products and duality |
| Duality, traces, and dimensions | dual, left_dual, right_dual, ev, coev, right_ev, right_coev, left_trace, right_trace, tr, left_dim, right_dim, dim, TensorCategories.squared_norm | Tensor products and duality |
| Pivotal and spherical structure | pivotal, spherical, pivotal_structures, set_pivotal!, set_spherical!, is_pivotal, is_spherical | Tensor products and duality |
| Braiding and ribbon data | braiding, reverse_braiding, twist, twists, smatrix, normalized_smatrix, tmatrix | Tensor products and duality, reversing a braiding, fusion and splitting |
| Fusion-category data and predicates | fusion_coefficient, fpdim, is_ring, is_tensor, is_fusion, is_multifusion, is_unitary, is_modular | Fusion and splitting, numerical fusion categories |
| Coherence checks | pentagon_axiom, randomized_pentagon_axiom, hexagon_axiom, monoidal_functor_axiom | Tensor products and duality, skeletal models |
| Skeletal model and structure setters | SixJCategory, six_j_category, skeletonize, set_tensor_product!, set_one!, set_associator!, set_braiding!, set_pivotal! | Skeletal models, working with fusion data |
| Symbol-dictionary extraction | F_symbols, R_symbols, P_symbols | Precise conventions, data exchange |
| Associator-block generation | six_j_symbols | Skeletal models, matrix coordinates |
| Numerical conversion and symbols | numeric, numeric_F_symbols, numeric_R_symbols, numeric_P_symbols, numeric_smatrix, numeric_twists | Numerical fusion categories, numerical computations |
| Tensor powers and generated additive closures | tensor_power, tensor_power_category | Products and related constructions |
| Skeletal fusion subcategories | fusion_subcategory, simple_fusion_subcategories | Products and related constructions, skeletal models |
The structure setters describe a skeletal presentation; a collection of arrays becomes valid fusion-category data only when the relevant axioms and nondegeneracy conditions hold. See the linked chapters before constructing or interpreting $F$- and $R$-symbols.