Concrete categories
Most models on this page retain the objects and morphisms of a mathematical realization. They are the natural entry point when those objects and their linear maps matter, rather than only a finite skeletal table of fusion data. Principal signatures, defaults, and parameter choices are described on the linked catalogue pages; use methods(name) for the complete installed method table.
| Family | Principal constructors and types | Catalogue entry |
|---|---|---|
| Finite sets and maps | Sets, SetObject, SetMorphism, SetHomSet | Finite sets |
| Finite-dimensional vector spaces | vector_spaces, VectorSpaces, VectorSpaceObject, VectorSpaceMorphism, VSObject, VSHomSpace | Vector spaces and gradings |
| Group-graded vector spaces | graded_vector_spaces, twisted_graded_vector_spaces, GradedVectorSpaces, GVSObject, GVSMorphism, GVSHomSpace, Cocycle, cyclic_group_3cocycle, unitary_cocycle | Vector spaces and gradings |
| Finite-group representations | representation_category, GroupRepresentationCategory, GroupRepresentation, GroupRepresentationMorphism, GRHomSpace, Representation | Group representations |
| Equivariant coherent sheaves | coherent_sheaves, convolution_category | Equivariant sheaves and convolution |
| Generic quantum $\mathfrak{sl}_2$ model | sl2_representations | $\mathfrak{sl}_2$, Verlinde, and dihedral models |
The generic quantum $\mathfrak{sl}_2$ entry is the exception: it is a skeletal recoupling model with infinitely many simple labels, whose objects are finite direct sums encoded by sparse multiplicity vectors. It does not represent the action of $U_q(\mathfrak{sl}_2)$ by matrices.
Constructors for supplied skeletal fusion categories are listed under fusion data and databases. General constructions such as scalar extension, centers, and internal modules are listed separately because their availability depends on the input category.