Models and the category interface
TensorCategories.jl represents categories, objects, and morphisms as Julia values. A categorical construction calls operations on those values, such as Hom, direct_sum, and associator. Different categories can implement those operations in entirely different ways.
A representation can be stored by group-action matrices. A graded vector space can be stored by a basis and its degrees. A split fusion category can instead be described by simple multiplicities and $F$-symbols. $F$-symbols are one input model; they are not required by the general category interface.
| Value or operation | Meaning |
|---|---|
C::Category | A particular category, including choices such as its field |
X::Object | An object in parent(X) |
f::Morphism | A map from domain(f) to codomain(f) |
Hom(X,Y), End(X) | Morphism spaces, where supported |
compose(f,g) | $g\circ f$ |
X ⊕ Y, X ⊗ Y | Direct sum and tensor product objects |
associator(X,Y,Z) | The specified rebracketing map |
Independent structures
The abstract Julia types do not form a hierarchy of linear, abelian, and monoidal categories. These are independent axes of structure, supplied by methods and recorded by predicates:
| Structure | Mathematical data or property | Predicate |
|---|---|---|
| linear | Hom spaces are $k$-modules (vector spaces when $k$ is a field) and composition is bilinear | is_linear(C) |
| additive | finite biproducts and a zero object | is_additive(C) |
| abelian | additive structure, kernels, and cokernels with the abelian axioms | is_abelian(C) |
| monoidal | tensor product, unit, and coherent associator and unit constraints | is_monoidal(C) |
| rigid | chosen left and right duality data | TensorCategories.is_rigid(C) |
Here linear is the package's enrichment predicate. In the terminology of Etingof et al. (2015), Definition 1.2.2, a $k$-linear category is additive as well as enriched in $k$-vector spaces. TensorCategories.jl records these two requirements separately through is_linear(C) and is_additive(C). Likewise, linearity and monoidality are independent. The operations X ⊕ Y, X ⊗ Y, and associator(X,Y,Z) are available only when the corresponding structure has been implemented.
The package also names standard combinations of these axes. A multiring category is a locally finite $k$-linear abelian monoidal category whose tensor product is $k$-bilinear and exact in each variable. It is a ring category when the canonical map $k\to\operatorname{End}(\mathbb 1)$ is an isomorphism. A multitensor category is a rigid multiring category, and it is a tensor category when it is also a ring category; exactness of tensor product in a multitensor category follows from rigidity (Etingof et al., 2015; Definition 4.1.1, Proposition 4.2.1, and Definition 4.2.3). The corresponding package predicates are is_multiring, is_ring, is_multitensor, and is_tensor. Finite semisimple versions, including the distinction between weak fusion and split fusion categories over a general field, are defined later under Fusion categories and splitting.
Predicates record declarations or backend-specific information about the category; generic code cannot prove all categorical axioms merely from the existence of methods. Algorithms rely on both the declared structure and the operations needed for the computation. For example, semisimple decomposition uses finite-dimensional Hom spaces and linear algebra over the coefficient field, while the scalar $F$-symbol model additionally requires split simple objects.
Reading the interface chapters
The order of the following pages is chosen to introduce dependencies needed by later examples; it is not a chain of mathematical implications. Objects and morphisms come first, followed by linear Hom spaces and optional matrix coordinates. Additive and abelian operations and monoidal operations are then introduced as separate structures before the manual combines them in tensor and fusion categories. Functors follow the structural interfaces, and the last page distinguishes a faithful linear realization from the stronger notion of a fiber functor.
Our terminology and conventions follow Etingof et al. (2015). For non-split categories, we use the extensions described under Fusion categories and splitting. The generic framework and the center, skeletonization, and module algorithms implemented by the package are developed in Mäurer (2026), Introduction and Chapters 2–4.
Continue with Objects, morphisms, and composition.