Tensor and unit for symmetric monoidal categories

Let SMC denote the 2-category with objects small symmetric monoidal categories, 1cells symmetric monoidal functors and 2-cells monoidal natural transformations. It is shown that the category quotient of SMC by the congruence generated by its 2-cells is symmetric monoidal closed. 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 of symmetric monoidal categories. Some structure is investigated on the 2-category SMC with objects symmetric monoidal categories, with 1-cells symmetric monoidal functors and 2-cells monoidal natural transformations. It is shown to induce a symmetric monoidal closed structure on the category SMC/∼ quotient of SMC by the congruence ∼ generated by its 2-cells. This is Theorem 22.2. A similar result was found by M.Hyland and J.Power in their paper [Hypo02] which treats in particular the 2-category with objects symmetric monoidal categories, but with pseudo-functors as 1-cells, and monoidal natural transformations as 2-cells. This was done though in a general 2-categorical setting extending A.Kock’s work on commutative monads. Hyland and Power identified a 2-categorical structure namely that of pseudo-monoidal closed 2-category, on the 2category of T -algebras with their pseudo morphisms, for any pseudo-commutative doctrine T on Cat. Their pseudo monoidal closed structure generalises the Eilenberg-Kelly’s monoidal closed structure [EiKe66]. When considering general monoidal functors rather than pseudo ones, the problem of the existence of an adequate tensor seems more difficult. Precisely the tensor for T algebras in [Hypo02] occurs as a weak version of adjoint to the internal hom. This is not clear that this point generalises to the 2-categories of T -algebras with their lax morphisms. This is why the tensor in SMC is defined in this paper in an ad-hoc and more concrete way by means of a generating graph and relations. The categorical connection with the internal hom is established a posteriori. Note that the tensors here and in [Hypo02] are not isomorphic as symmetric monoidal categories. The unit in SMC was also problematic. Eventually the author’s view is that the case of the symmetric monoidal categories together with the monoidal functors should be elucidated