Monoidal functors

Let $F\colon\mathcal C\to\mathcal D$ be a functor between monoidal categories. A strong monoidal structure on $F$ includes natural isomorphisms

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

and a compatible unit isomorphism, satisfying the associator and unit coherence equations (Etingof et al., 2015; §2.4). The direction in equation \eqref{eq:monoidal-functor-tensorator} is the direction used by TensorCategories.jl.

A tensor functor in the terminology of Etingof et al. (2015), Definition 4.2.5 is an exact faithful $k$-linear monoidal functor between multiring categories. Some literature uses this term for a strong monoidal functor without the exactness and faithfulness hypotheses; the manual uses the EGNO meaning.

Monoidal structure is additional data. A functor that preserves tensor-product classes in a Grothendieck ring does not thereby acquire tensorators, and an objectwise identification $F(X\otimes Y)\cong F(X)\otimes F(Y)$ does not supply a coherent choice of them.

The interface

The implemented solvers assume additive $k$-linear behavior, strict preservation of the represented unit, and normalized unit tensorators. The constructor checks the source and target and the image of the unit, but it does not prove that the supplied functor is additive or linear.

monoidal_structure_candidates(F; check=false) searches for tensorators in supported split fusion categories. The solution scheme can have positive dimension, in which case sampling candidates is not a classification and an empty sample is not a proof of nonexistence. The option check=true re-evaluates the monoidal-functor equation on each returned candidate. monoidal_structures(F) currently restricts its generic complete solver to the normalized case in which the source has one simple object.

For monoidal functors $F,G$, a monoidal natural transformation is an ordinary natural transformation compatible with the two tensorators. The method monoidal_natural_transformations(F,G) solves these additional equations for the supported finite additive models.

Continue with fiber functors.