Australian Category Seminar

A tensor product of symmetric monoidal categories. A tensor product of symmetric monoidal categories $A \square B$ is introduced with the universal property strict : $A \square B \to C$ is in bijection with $A \to$ SMon(B, C) where $F = (F, F', F'')$ is strict if F' and F'' are identities and Smon(B,C) is the (symmetric monoidal) category of symmetric monoidal functors from A to C. More formally, there is an adjunction $- \square B$ left adjoint to SMon(B,-): SMon_strict -> SMon. A related adjunction is $- \square' B$ left adjoint to SMon_strongmaps(B, -) : SMon_strict -> SMon_strong. Replacing strict morphisms by strong morphisms we get an equivalence between SMon_strong(A \square' B, C) and SMon_string(A, SMon_strong(B, C)). This $\square'$ gives a monoidal struciure for SMon_strong. A semi-ring is a symmetric monoidal cat V with a strong map $V \square' V \to V$ satisfying the appropriate conditions. Then V may be used to replace $R^+$ in the construction of Banach algebras as in last weeks talk.