Tensor product for symmetric monoidal categories

We introduce a tensor product for symmetric monoidal categories with the following properties. Let SMC denote the 2-category with objects small symmetric monoidal categories, arrows symmetric monoidal functors and 2-cells monoidal natural transformations. Our tensor product together with a suitable unit is part of a structure on SMC that is a 2-categorical version of the symmetric monoidal closed categories. This structure is surprisingly simple. In particular the arrows involved in the associativity and symmetry laws for the tensor and in the unit cancellation laws are 2-natural and satisfy coherence axioms which are strictly commuting diagrams. We also show that the category quotient of SMC by the congruence generated by its 2-cells admits a symmetric monoidal closed structure.

💡 Research Summary

The paper presents a novel tensor product for the 2‑category SMC whose objects are small symmetric monoidal categories, whose 1‑morphisms are symmetric monoidal functors, and whose 2‑morphisms are monoidal natural transformations. The authors construct a bifunctor ⊗ : SMC × SMC → SMC together with a unit object 𝟙, and they show that this data equips SMC with the structure of a symmetric monoidal closed 2‑category.

A key feature of the construction is that the associator α, the braiding σ, and the left and right unitors λ, ρ are all 2‑natural transformations. Consequently, the usual coherence diagrams (the pentagon, triangle, and hexagon) commute strictly, not merely up to isomorphism. This strictness is unusual at the 2‑categorical level and simplifies many higher‑dimensional calculations.

The authors define an internal hom