Catalogue of categories
This chapter is a lookup guide to the models and datasets distributed with the package. It includes complete category implementations, constructors from structural data, and entries providing a specified subset of the categorical structure. The table is ordered alphabetically by family; its detailed pages are independent reference entries rather than a sequence of prerequisites.
Each linked entry gives the mathematical context, a precise literature or dataset reference, and the relation between that presentation and the current implementation. Entries that provide only fusion rules or imported associator data without complete basis metadata are marked accordingly.
Choosing a model
A concrete model retains the objects and morphisms from the mathematical realization, as in vector spaces, graded spaces, representations, and equivariant sheaves. Use such a model when that linear algebra is part of the problem. A SixJCategory retains a split semisimple skeleton together with fusion multiplicities and structural matrices; use it when $F$-symbols are the input or desired output. Some further models store only fusion rules or simple multiplicities. They support only the operations stated in their entries.
Database keys and constructor parameters select stored presentations, rather than tensor-equivalence classes. Each entry supports the structures stated on its detailed page.
Here an artifact is a versioned data archive installed by Julia for a particular package release. Loading an artifact reads the pinned copy shipped with that release; it does not query the current version of an external database.
Available models and data
| Family | Entry point | Representation and scope |
|---|---|---|
| AnyonWiki | anyonwiki(r,m,n,i,a,b,p) | Exact decoding of a pinned artifact record into the package's structural-matrix convention |
| AnyonWiki split centers | anyonwiki_center(r,m,n,i,a,b,p) | Precomputed split skeletal braided centers for available AnyonWiki inputs of rank at most 5 |
| Bicharacter-braided graded spaces | graded_vector_spaces(K,G,χ) | Trivial associator and a supplied bicharacter braiding for finite abelian $G$ |
| Cocycle-twisted graded spaces | graded_vector_spaces(K,G,ω) | Same objects and tensor product; associator, evaluation, and pivotal scalar depend on $\omega$; braiding is unavailable for a nontrivial twist |
| Cocycle twists computed with GAP/HAP | twisted_graded_vector_spaces(K,G,i) | Backend-generated cocycles; class enumeration and tuple/coefficient translations depend on the installed HAP version |
| Convolution categories | convolution_category(K,X::GSet) | Standard convolution bifunctor; the detailed entry states the validation status of associator and duality data |
| Dihedral constructions | I2(m,K), I2subcategory(m,K) | Published fusion interpretation with Kauffman–Lins recoupling; the full model lacks complete rigidity data |
| $E_6$ data | E6subfactor() | Rank 3, fusion multiplicity 2; associators do not satisfy the pentagon |
| Equivariant coherent sheaves | coherent_sheaves(K,X) | Representations of orbit stabilizers |
| Extended Haagerup | TensorCategories.extended_haagerup(K) | $M$–$M$ even-part fusion rules only; no supplied associators |
| Fibonacci categories | fibonacci_category(K,a) | Two algebraic associator choices |
| Finite-dimensional vector spaces | vector_spaces(K) | Basis vectors and matrices |
| Finite-group representations | representation_category(K,G) | Action matrices and intertwiners |
| Finite sets | Sets() | Finite-set model with products and coproducts; current object-equality and dictionary-map validation limitations; no monoidal interface |
| Generic quantum $\mathfrak{sl}_2$ recoupling model | sl2_representations(K,q) | Sparse simple-multiplicity model; not the semisimple root-of-unity quotient |
| Graded vector spaces | graded_vector_spaces(K,G) | Degrees and degree-preserving matrices |
| Haagerup $H_1$ artifact | haagerup_H1() | Published $H_1$ fusion ring with an exact artifact associator; no basis change to a published gauge is recorded |
| Haagerup $H_2$ and $H_3$ | haagerup_H2(), haagerup_H3() | Stored exact AnyonWiki $F$- and pivotal data; no braiding |
| Haagerup exact $H_3$ center | haagerup_H3_center() | Exact split center in a nonunitary gauge, with $F$-, $R$-, and pivotal data |
| Haagerup numerical $H_3$ center | numeric_unitary_center_H3() | Decimal approximations to unitary-gauge $F$- and $R$-data; no stored pivotal data |
| Haagerup $H_3$ formulas | TensorCategories.unitary_haagerup_H3_wolf(K; p1, p2) | Unexported Wolf-formula implementation; Appendix B blocks are transposed into package coordinates |
| Additional Haagerup $H_2$ data | unitary_haagerup_H2() | Exact table whose source does not record a basis change to a published gauge |
| Ising | ising_category(K,s,q) | Tambara–Yamagami for $A=C_2$; optional braiding |
| $\mathrm{SU}(3)_3$ subcategory | TensorCategories.su_3_3_subcategory(K) | Rank 4, multiplicity 2; not exported |
| Symmetric Verlinde categories | symmetric_verlinde_category(p; plus=false) | Semisimplification of $\operatorname{Rep}_{\mathbf F_p}(C_p)$; optionally restrict to $\operatorname{Ver}_p^+$ |
| Tambara–Yamagami | tambara_yamagami(K,A,s,χ) | Explicit bicharacter formulas |
| Trivial fusion category | trivial_fusion_category(K) | Intended rank-one constructor; currently unavailable |
| Vercleyen–Slingerland data | cat_fr_8122(n), cat_fr_9143() | Fusion rings and ancillary sources identified; the imports lack the basis metadata needed for an entrywise comparison |
| Verlinde modular | verlinde_category(K,m,l,t) | Default $l=t=1$ in Kauffman–Lins conventions; other parameters use package-specific normalizations |
General constructions
Categories obtained from these models are documented with their respective constructions: Drinfeld centers, products and scalar extension, internal module categories, and group actions and equivariantization. Their availability is governed by the operations supplied by the input model.