Fusion and multifusion categories

We now combine finiteness and semisimplicity with the monoidal and rigid structures. Over an algebraically closed field, a multifusion category is a finite semisimple multitensor category, and a fusion category is a finite semisimple tensor category (Etingof et al., 2015; Definition 4.1.1). Here finite has the meaning fixed for finite abelian categories. Equivalently, a fusion category has a simple tensor unit; over an algebraically closed field Schur's lemma then identifies its endomorphism algebra with the coefficient field.

No choice of simple representatives, bases, or matrix coordinates is part of this definition. In particular, a fusion category need not be presented by fusion rules or by structural matrices.

Arbitrary coefficient fields

TensorCategories.jl also works over fields that are not algebraically closed. It uses the terminology of Mäurer and Thiel (2024), §2.1:

• a weak multifusion category is a finite semisimple $k$-linear rigid monoidal category, without requiring the tensor unit to be simple;
• a weak fusion category is a weak multifusion category whose tensor unit is simple;
• a multifusion category is a split weak multifusion category; and
• a fusion category is a split weak fusion category.

Here split means that every simple object $S$ is scalar:

\[\label{eq:fusion-split-simple} \operatorname{End}_{\mathcal C}(S)\cong k.\]

The adjective weak refers only to splitness over the chosen coefficient field. It does not weaken rigidity, semisimplicity, or the monoidal coherence axioms. Over an algebraically closed field the weak and split notions agree.

Separability in positive characteristic

Over an imperfect field, some authors include separability in the definition of a multifusion category. For a finite semisimple category this condition is automatic over a perfect field, including number fields and finite fields, but can be stronger over an imperfect field (Sanford, 2025; Definition 2.9 and pp. 3–4). The package predicates use the semisimple convention above and do not independently test this separability condition.

The interface

Generic implications between these predicates are implemented in FrameworkChecks.jl; individual category types can provide more direct methods. The predicates report properties of the category over its current field. Extending the field can change both the simple objects and which of the four predicates applies.

Example: Group representations

Finite-group representations show separately the effects of semisimplicity and splitness. Let $G=C_3$. Over $\mathbb F_2$, Maschke's theorem gives a weak fusion category, but the irreducible polynomial $x^2+x+1$ produces a two-dimensional simple object with endomorphism field $\mathbb F_4$, so the category is not split. Over $\mathbb F_3$, Maschke's condition fails and the category is not weak fusion at all.

using TensorCategories, Oscar
G = cyclic_group(3)
C2 = representation_category(GF(2), G)
C3 = representation_category(GF(3), G)
@assert is_weak_fusion(C2) && !is_fusion(C2)
@assert !is_weak_fusion(C3)
(is_weak_fusion(C2), is_fusion(C2), is_weak_fusion(C3))

(true, false, false)

This example uses concrete representations and intertwiners throughout.

Continue with Grothendieck rings.