Fiber functors and matrix realizations

A fiber functor on a ring category $\mathcal C$ over $k$ is a tensor functor

\[\label{eq:fiber-functor} F\colon\mathcal C\longrightarrow\operatorname{Vec}_k,\]

hence an exact faithful $k$-linear functor together with coherent tensor and unit isomorphisms (Etingof et al., 2015; Definition 5.1.1). The monoidal data are essential. A faithful realization of morphisms by matrices need not be a fiber functor.

For $\operatorname{Rep}_k(G)$, forgetting the group action gives the standard fiber functor. The package stores its underlying vector spaces and matrices inside each representation and intertwiner, so computations use this realization even though it is not necessarily constructed as a separate Julia functor value.

By contrast, suppose $\mathcal C$ is a finite split semisimple $k$-linear category with chosen simple representatives $S_1,\ldots,S_r$. The assignment

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

is a faithful exact linear realization after bases are chosen. It gives block matrices for morphisms, but it has no automatic coherent identification of $U(X)\otimes U(Y)$ with $U(X\otimes Y)$. It is therefore not generally a fiber functor. Its vector-space dimension is the total multiplicity of simple summands of $X$, rather than the categorical or Frobenius–Perron dimension.

In a non-split semisimple category, $\operatorname{Hom}(S_i,X)$ is naturally a module over the division algebra $\operatorname{End}(S_i)$. Forgetting that module structure can still give vector-space coordinates, but it must not identify one copy of $S_i$ with a single scalar coordinate.

TensorCategories.jl consequently treats matrix(f), tensor_product, and the associator as separate parts of the interface. A category can support all three without possessing a fiber functor to ordinary vector spaces.

Continue with skeletal fusion categories.