Functors and natural transformations
Functors
All functor types in TensorCategories.jl are subtypes of TensorCategories.AbstractFunctor. A functor must provide its domain and codomain categories and its action on objects and morphisms. The general constructor
F = functor(C, D, object_map, morphism_map)returns a Functor from C to D. It can then be applied directly as F(X) or F(f). The functions domain(F) and codomain(F) return its source and target categories.
The constructor stores the two supplied maps; it does not verify the functor laws. In particular, the user must ensure that
\[\label{eq:implemented-functor-laws} F(\operatorname{id}_X)=\operatorname{id}_{F(X)}, \qquad F(g\circ f)=F(g)\circ F(f).\]
Likewise, linearity, additivity, and exactness are properties of the supplied maps rather than consequences of the constructor. A reusable category-specific implementation can instead define a new subtype of TensorCategories.AbstractFunctor, store the data it needs, and implement its action on objects and morphisms by Julia call methods.
Functor composition follows the same order as morphism composition: compose(F,G) and G ∘ F represent the functor $G\circ F$. The identity functor of a category is id(C).
Example: Defining a functor
Here is the identity functor on finite-dimensional rational vector spaces, entered through the general constructor.
using TensorCategories, Oscar
C = vector_spaces(QQ)
F = functor(C, C, X -> X, f -> f)
X = VectorSpaceObject(C, 2)
f = morphism(X, X, matrix(QQ, [1 1; 0 1]))
@assert domain(F) == C && codomain(F) == C
@assert F(id(X)) == id(F(X))
@assert F(f ∘ f) == F(f) ∘ F(f)
F(f)Vector space morphism with
Domain:Vector space of dimension 2 over Rational field.
Codomain:Vector space of dimension 2 over Rational field.Natural transformations
The abstract type NaturalTransformation is a subtype of Morphism. For additive functors between Krull–Schmidt categories, the concrete type TensorCategories.AdditiveNaturalTransformation represents a natural transformation by its components on a chosen family of indecomposable objects.
Suppose $F,G\colon\mathcal C\to\mathcal D$ are additive functors and $S_1,\ldots,S_r$ are indecomposable representatives from which the objects under consideration can be decomposed. A transformation is entered as
eta = TensorCategories.AdditiveNaturalTransformation(
F, G, indecomposables, components
)where indecomposables is the list $[S_1,\ldots,S_r]$ and components specifies morphisms $\eta_{S_i}\colon F(S_i)\to G(S_i)$. Components may be given in the same order as the objects, or as pairs S => eta_S; omitted pairs are interpreted as zero components.
For a decomposition $X\cong\bigoplus_j S_j$ with inclusions $i_j\colon S_j\to X$ and projections $p_j\colon X\to S_j$, the implementation reconstructs the component on $X$ as
\[\label{eq:additive-natural-transformation-extension} \eta_X = \sum_j G(i_j)\circ\eta_{S_j}\circ F(p_j).\]
Thus eta(X) is available for decomposable objects as soon as the category can compute direct_sum_decomposition(X, indecomposables).
The generic direct_sum_decomposition method uses decomposition into simple objects and therefore applies to semisimple categories. A nonsemisimple Krull–Schmidt model can use the same representation of natural transformations after providing its own direct_sum_decomposition method.
The constructor does not check naturality. The supplied components define a natural transformation only if
\[\label{eq:natural-transformation-naturality-linear} G(f)\circ\eta_S=\eta_T\circ F(f) \qquad \text{for every }f\colon S\to T\]
between the chosen indecomposable objects. This includes all endomorphisms of each indecomposable, not only morphisms between distinct objects.
The function Nat(F,G; indecomposables=objects) computes the vector space of additive natural transformations by solving these linear equations. If the keyword is omitted, it calls indecomposables(domain(F)); the generic fallback can enumerate indecomposables only in the semisimple case. A nonsemisimple Krull–Schmidt model must provide its own method or pass a suitable list explicitly. The solver requires additive functors with the same domain and codomain, finite bases for the relevant Hom spaces, and a common coefficient field. Evaluating the resulting transformations away from the supplied representatives additionally requires effective direct-sum decompositions. The solver returns a NaturalTransformations Hom space; its basis elements can be obtained with basis.
Example: Components on indecomposable objects
Over $\mathbb F_3$, the category $\operatorname{Rep}_{\mathbb F_3}(C_2)$ has two simple, hence indecomposable, objects. Since there are no morphisms between the two simple objects, a natural endomorphism of the identity functor may act on them by independent scalars.
using TensorCategories, Oscar
C = representation_category(GF(3), cyclic_group(2))
S = simples(C)
F = id(C)
eta = TensorCategories.AdditiveNaturalTransformation(
F, F, S, [S[1] => id(S[1]), S[2] => -id(S[2])]
)
X, inclusions, projections = direct_sum(S[1], S[2])
@assert eta(X) ∘ inclusions[1] == inclusions[1] ∘ eta(S[1])
@assert eta(X) ∘ inclusions[2] == inclusions[2] ∘ eta(S[2])
N = Nat(F, F; indecomposables=S)
@assert int_dim(N) == 2
int_dim(N)2The two-dimensional result records the independent scalar components on the two simple objects. In a nonsemisimple Krull–Schmidt category, morphisms between distinct indecomposables and non-scalar endomorphisms impose further equations through \eqref{eq:natural-transformation-naturality-linear}.
Continue with tensor products and associators.