Matrix coordinates

Coordinates from express_in_basis(f,Hom(X,Y)) belong to one Hom space. A global matrix realization is stronger: it assigns compatible coordinates to objects as well as morphisms, so that identities and composition are expressed by identity matrices and matrix multiplication. The method matrix(f) is available only when the concrete category model provides such coordinates.

In many implemented models these coordinates come from a faithful $k$-linear functor

\[\label{eq:matrix-realization-functor} U:\mathcal C\longrightarrow\operatorname{Vec}_k\]

that is implicit in the representation. Each object has a vector space and a basis, and matrix(f) represents $U(f)$. The realization can be encoded in the stored objects and morphisms without being represented by a separate Julia functor value. A model can also provide useful matrices by another documented coordinate construction; the meaning of matrix(f) is therefore part of that model's interface.

Row-vector convention

TensorCategories.jl's matrix models use row coordinates: a morphism $f:X\to Y$ has one matrix row per source basis vector and one column per target basis vector. Consequently, for $f:X\to Y$ and $g:Y\to Z$,

\[\label{eq:row-coordinate-composition} M_{g\circ f}=M_fM_g.\]

The following example uses ordinary finite-dimensional vector spaces only to display this convention:

using TensorCategories, Oscar
V = vector_spaces(QQ)
X = VectorSpaceObject(V, 2)
Y = VectorSpaceObject(V, 3)
Z = VectorSpaceObject(V, 1)
f = morphism(X, Y, matrix(QQ, [1 0 2; 0 1 3]))
g = morphism(Y, Z, matrix(QQ, 3, 1, [1, 2, 3]))
@assert matrix(g ∘ f) == matrix(f)*matrix(g)
matrix(compose(f, g))

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Faithfulness does not make every matrix a categorical morphism. In representations it must be an intertwiner; in graded spaces it must preserve degrees. Computing categorical kernels by linear algebra also needs exactness and reconstruction of the categorical object and its maps.

Linear coordinates do not determine tensor structure

A faithful linear realization need not preserve tensor products. Even when matrix(f) and matrix(g) are defined, matrix(f ⊗ g) need not be the ordinary Kronecker product: the source and target coordinates may be built from decomposition bases or other model-specific choices. The category's tensor_product and associator methods supply this additional structure.

A fiber functor is a much stronger realization: it is exact, faithful, and coherently monoidal. The precise distinction is explained under Fiber functors and semisimple coordinates, after the tensor and functor interfaces on which it depends.

Continue with Direct sums, kernels, and decompositions.