Fiber functors and semisimple coordinates

The presence of matrices in a category model does not imply a fiber functor. In the split semisimple models discussed below, the matrices are coordinates of a faithful linear realization, but that realization need not preserve tensor products. The distinction matters because every finite split semisimple category has this linear realization, whereas many fusion categories admit no fiber functor.

The split semisimple linear realization

Choose split simple representatives $S_1,\ldots,S_r$. The functor

\[\label{eq:canonical-vector-space-realization} U(X)=\bigoplus_{i=1}^r\operatorname{Hom}_{\mathcal C}(S_i,X)\]

is exact and faithful. After choosing bases of the Hom spaces, a morphism is represented by one matrix block for each simple object. The package's later skeletal $F$-symbol model, implemented by SixJCategory, uses precisely these coordinates: matrices(f) returns the blocks and matrix(f) places them on the diagonal. The dimension of $U(X)$ counts simple multiplicities; it is neither the categorical dimension nor, in general, the Frobenius–Perron dimension.

For the supplied Fibonacci category:

using TensorCategories, Oscar
C = fibonacci_category()
t = simples(C)[2]
@assert size(matrix(id(t))) == (1, 1)
@assert size(matrix(id(t ⊗ t))) == (2, 2)
size(matrix(id(t) ⊗ id(t)))

(2, 2)

The functor $U$ is not generally monoidal. In the example, an isomorphism $U(t)\otimes U(t)\cong U(t\otimes t)$ is already impossible because the two spaces have dimensions $1$ and $2$. Thus matrix(f ⊗ g) is not obtained by forming the Kronecker product of matrix(f) and matrix(g); the tensor product and associator use the category's fusion and decomposition data.

Over a non-splitting field, put $D_i=\operatorname{End}(S_i)$. Precomposition makes $\operatorname{Hom}(S_i,X)$ a right $D_i$-module. Forgetting these modules to the base field still gives a faithful linear realization in the finite semisimple setting, but one copy of $S_i$ contributes $\dim_kD_i$ coordinates. The scalar-block representation of SixJCategory implements only the split case.

When the realization is a fiber functor

A fiber functor on a ring category $\mathcal C$ over $k$ is an exact faithful $k$-linear functor $F:\mathcal C\to\operatorname{Vec}_k$ equipped with a compatible unit isomorphism and coherent tensor isomorphisms

\[\label{eq:fiber-functor-tensorator} J_{X,Y}:F(X)\otimes F(Y)\xrightarrow{\sim}F(X\otimes Y).\]

This is the definition used by Etingof et al. (2015), Definition 5.1.1, p. 91. The usual forgetful functors for $\operatorname{Rep}_k(G)$ and untwisted $\operatorname{Vec}_G$ are fiber functors (Etingof et al., 2015; Example 5.1.2). For $\operatorname{Vec}_G^\omega$, a cohomologically nontrivial class $[\omega]\in H^3(G,k^\times)$ prevents a fiber functor: coherent tensorators would give a $2$-cochain whose coboundary trivializes $\omega$ (Etingof et al., 2015; Example 5.1.3).

The Fibonacci fusion rule $t\otimes t\cong\mathbb 1\oplus t$ gives a simple obstruction. A fiber functor would assign a positive integer $n=\dim_kF(t)$ satisfying $n^2=1+n$, which is impossible. Equivalently, the Grothendieck ring has no dimension homomorphism that takes positive integer values on its distinguished basis.

Coordinate extraction requires faithful linear coordinates and compatibility with composition. A fiber functor supplies more: it identifies tensor products coherently. TensorCategories.jl uses matrix coordinates without assuming this extra structure.

Continue with the implementation checklist, which separates the operations required at each categorical level before the complete matrix example implements them.