Functors and natural transformations

A functor must specify what it does to both objects and morphisms. Giving only a map on isomorphism classes, or only a Grothendieck-ring homomorphism, does not specify a functor on the category.

Additional adjectives refer to additional compatibility:

The last row follows Etingof et al. (2015), Definition 4.2.5; some authors use “tensor functor” without the exactness or faithfulness requirements.

The generic constructor below stores object and morphism maps; it does not infer any of these stronger properties from the Julia functions supplied to it.

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.

functor(C,D,obj_map,mor_map) stores the two maps, with source domain(F) and target codomain(F). A custom subtype of TensorCategories.AbstractFunctor can instead implement call methods. The constructor does not check the identity and composition laws. Functor composition has the same order as morphism composition: compose(F,G) means $G\circ F$.

A natural transformation $\eta:F\Rightarrow G$ has components $\eta_X:F(X)\to G(X)$ satisfying

\[\label{eq:natural-transformation-naturality} G(f)\circ\eta_X=\eta_Y\circ F(f) \qquad(f:X\to Y).\]

An additive natural transformation stores components on specified indecomposables and extends them using direct-sum decompositions. Nat(F,G) solves naturality equations in supported finite additive models whose listed indecomposables generate all objects under finite direct sums. It requires finite Hom bases, effective direct-sum decompositions and coordinates, additive functors, and the same base field for source and target. Semisimplicity is not a formal requirement of this solver. A component on a non-split simple $S$ must satisfy naturality with respect to its whole endomorphism algebra:

\[\label{eq:natural-transformation-simple-test} G(d)\circ\eta_S=\eta_S\circ F(d) \qquad\bigl(d\in\operatorname{End}(S)\bigr).\]

An arbitrary matrix at each simple does not suffice. When $F=G$, this says that the component commutes with the induced endomorphisms.

Tensor structure is extra data

A monoidal functor has tensorators

\[\label{eq:monoidal-functor-tensorator} J_{X,Y}:F(X)\otimes F(Y)\longrightarrow F(X\otimes Y),\]

satisfying the associator compatibility (Etingof et al., 2015; §2.4). The implemented solvers require additive $k$-linear behavior, strictly unit-preserving functors, and normalized unit tensorators. The implementation checks the source and target categories and the image of the unit, but does not verify additivity or $k$-linearity of the supplied functor. These are not solvers for arbitrary unit constraints.

monoidal_structure_candidates(F; check=false) searches for candidates between split fusion categories. This can sample a positive-dimensional solution scheme and is not a complete classification. An empty result does not prove nonexistence. The solved equations already impose coherence; check=true re-evaluates the monoidal-functor axiom on each returned candidate. monoidal_structures(F) currently restricts its generic complete solver to the normalized case in which the source fusion category has exactly one simple object.

Functors between skeletal models

The later skeletal fusion model is implemented by SixJCategory. For two such categories, functor(C,D,images) stores the chosen images of the simple objects of $C$ and uses semisimple block formulas on objects and morphisms. This is a low-level constructor: supply exactly one object of $D$ for each simple of $C$, with compatible parents and functor laws. These conditions are assumed rather than checked. Coherent tensorators remain additional data, and compatibility of the images with the fusion rules is required before such tensorators can exist. The monoidal-structure methods above can search for that data. The returned functor does not declare is_additive, so the generic Nat(F,G) solver does not accept it directly.

Continue with Fiber functors and semisimple coordinates.