Hom spaces and linear algebra

TensorCategories.jl records linear enrichment and additive structure separately. Thus is_linear(C) says that the Hom spaces are $k$-modules and composition is $k$-bilinear, while is_additive(C) records finite biproducts and a zero object. When $k$ is a field, the Hom spaces are $k$-vector spaces. This differs slightly from the terminology of Etingof et al. (2015), Definition 1.2.2, where a $k$-linear category is additive by definition. The package uses the same base_ring(C) interface over fields and more general coefficient rings, but algorithms that divide by scalars or use vector-space dimension require a field and finite-dimensional Hom spaces. This chapter describes that field-linear setting.

Operation	Result
`Hom(X,Y)`	An `AbstractHomSpace` for maps $X\to Y$
`End(X)`	`Hom(X,X)`
`basis(H)`	A vector of basis morphisms
`int_dim(H)`	Dimension as a Julia integer
`zero_morphism(X,Y)`	The zero map
`af + bg`	A linear combination of parallel maps
`express_in_basis(f,H)`	Coordinates in `basis(H)`
`endomorphism_ring(X)`	An associative algebra representing `End(X)`

The last operation includes multiplication. Dimension alone does not determine the algebra structure.

using TensorCategories, Oscar
V = vector_spaces(QQ)
X, Y = VectorSpaceObject(V, 2), VectorSpaceObject(V, 3)
H = Hom(X, Y)
@assert int_dim(H) == 6
B = basis(H)
f = 2*B[1] - B[end]
c = express_in_basis(f, H)
@assert sum(c[i]*B[i] for i in eachindex(B)) == f
c

6-element Vector{QQFieldElem}:
  2
  0
  0
  0
  0
 -1

A Hom basis is a choice, not an invariant. Coordinates of structural morphisms must therefore be compared in compatible bases. In the later skeletal fusion model, this dependence becomes the gauge dependence of $F$- and $R$-symbols; their bases and matrix directions are fixed only after the model is introduced.

Over a non-splitting field, a simple object $S$ can have a division algebra $D=\operatorname{End}(S)$ larger than $k$. In a semisimple category, $\operatorname{Hom}(S,X)$ is a right $D$-module by precomposition, and the multiplicity of $S$ in $X$ is its dimension over $D$, not generally its dimension over $k$. See Fusion categories and splitting.

Coordinates in each Hom space do not by themselves specify a functor on objects to vector spaces. The next page explains when morphisms also have a compatible global matrix realization.

Continue with Matrix coordinates.