Linear Categories

Let $k$ be any field. A category is called $k$-linear whenever it is enriched over the category of $k$-vector spaces, i.e. all Hom-spaces are $k$-vector spaces and composition is $k$-linear. We need the following method:

• *(λ, f::YourMorphism)::YourMorphism returning the multiplication if a scalar λ.

Rational Forms

In the literature most categories are defined over an algebraically closed field of characteristic 0 or even over $\mathbb C$. This is technically possible to implement utilizing the implementation of algebraic numbers in Nemo.jl.

In general it is very interesting to work with categories not defined over algebraically closed fields. Especially it might be of interest to implement a category that is usually defined over $\mathbb C$ over a finite extension of $\mathbb Q$.

Let $\mathcal C$ be a $K$-linear category and $k \subset K$ a field extension. Then a category $\tilde \mathcal C$ is called a rational form for $\mathcal C$ if the karoubian envelope of $\tilde \mathcal C \otimes K$is equivalent to $\mathcal C$. We call the rational form complete if already $\tilde \mathcal C \otimes K$ is equivalent to $\mathcal C$.

Unfortunately the notion of a complete rational form violates the principle of equivalence. For example the center construction does not preserve the completeness of the rational form.