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. 1 Summary of results Thomason’s famous result claims that symmetric monoidal categories model all connective spectra [Tho95]. The discovery of a symmetric monoidal structure on the category of structured spectra [EKMM97] suggests that a similar structure should exist on an adequate category with symmetric monoidal categories as objects. The first aim of this work is to give a reasonable candidate for a tensor product of symmetric monoidal categories. We define such a tensor product for two symmetric monoidal categories by means of a generating graph and relations. It has the following properties. Let SMC denote the 2-category with objects symmetric monoidal categories, with 1-cells symmetric monoidal functors and 2-cells monoidal natural transformations. The tensor product yields a 2-functor SMC × SMC → SMC. This one is part of a 2-categorical structure on SMC that is 2-categorical version of the symmetric monoidal closed categories. Moreover this structure is rather simple since: its “canonical” arrows, i.e those involved for the associativity, the symmetry and the left and right unit cancellation laws, are 2-natural; all coherence axioms for the above arrows are strictly commuting diagrams. Actually this last point was quite unexpected. Eventually from the above structure one can deduce a symmetric monoidal closed structure on the category SMC/∼ quotient of SMC by the congruence ∼ generated by its 2-cells. Here are now, in brief, the technical results in the order in which they occur in the paper. The existence of an internal hom, a tensor and a unit for SMC and the fundamental properties defining and relating those are established first. From this, further properties of the above structure can be derived, such as the existence of associativity, symmetry and unit laws involving 2-natural arrows satisfying coherence axioms. The proofs are rather computational for establishing a few key facts