Tensor Categories
Let $k$ be a field. Then a multitensor category is a category $\mathcal C$ which is
- locally finite
- k-linear
- abelian
- rigid monoidal
such that
- $\otimes$ is bilinear on morphisms.
We call a multitensor category
- tensor if additionally $\mathrm{End}(\mathbb 1) = k$,
- multifusion if semisimple and finite and
- fusion if tensor, semisimple and finite.
Thus to implement a one of the above one simply has to implement the interfaces which are part of the definition.
Multifusion Categories
Semisimple k-linear abelian categories have such a structure that allows one to describe them up to equivalence by matrices. Let $\{X_i \mid i \in \mathcal I\}$ be the set of non-isomorphic simple objects in $\mathcal C$. This we can establish an equivalence
$$ \mathcal C \cong \bigoplus\limits_{i ∈ \mathcal I} \mathrm{Vec}_k$$
of abelian categories via $X \mapsto \bigoplus \mathrm{Hom(X_i, X)}$. For the detailed construction we refer to Daisuke Tambara, Shigeru Yamagami (1998).
This basically implies that morphisms are merely matrices allowing us to efficiently compute thins like subobjects etc. Thus we want to choose any faithful functor
\[F \colon \mathcal C \to \mathrm{Vec}_k\]
to provide a method
matrix(f::YourMorphism)::MatElem
.
This will open acces to the following operations which are necessary for many computations with fusion categories like for example the computation of the categorical center.
Missing docstring for eigenvalues
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Missing docstring for simple_subobjects
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Missing docstring for express_in_basis
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Missing docstring for left_inverse
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Missing docstring for right_inverse
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